# Electric Flux and Gauss’s Law: How Symmetry Reveals Electric Fields > Electric flux begins as a signed count of field lines crossing a surface and becomes the surface integral in Gauss’s law. The lecture develops that law from a point charge, then uses spherical, cylindrical, and planar symmetry to find the fields of a uniformly charged solid sphere, an infinite line, and an infinite sheet. It closes with electrostatic conductors, surface charge, shielding, and the protection provided by a metal car body during lightning. - Canonical watch page: [Electric Flux and Gauss’s Law: How Symmetry Reveals Electric Fields](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Physics - Published: 2026-08-29T00:40:47.000Z - Updated: 2026-08-29T00:40:47.000Z - Duration: PT1029S (17 minutes 9 seconds) - Chapters: 5 - Views: 0 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14TZWMWWK33R8DXJ7Z859ME/1/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14TZWMWWK33R8DXJ7Z859ME/1/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14TZWMWWK33R8DXJ7Z859ME/1/dark/poster.jpg) ## Description Build electric flux and Gauss’s law visually, apply symmetry to spheres, lines and sheets, then explain conductors and lightning shielding. ## Chapters - [00:00–02:57.108 · Flux That Refuses to Change](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0) - [02:57.108–06:14.171 · From Crossings to Gauss's Law](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=177.10831249999995) - [06:14.171–09:46.464 · A Uniformly Charged Sphere](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=374.1709166666666) - [09:46.464–13:36.753 · Lines, Sheets, and the Surfaces They Choose](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=586.4642083333332) - [13:36.753–17:09 · Conductors, Surface Charge, and the Car](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=816.7534166666665) ## Transcript ### [00:00 · Flux That Refuses to Change](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0) Place one positive point charge in empty space. At every surrounding point, its electric field points directly away from the charge. These blue arrows represent field lines, a drawing that lets us follow the direction of the field through space. The picture is three dimensional. Every direction away from the charge is equivalent, and no radial direction is preferred over another. Now surround the charge with a small sphere. Every blue line leaving the charge crosses that sphere once, from its inside to its outside. If we use the lines as a counting picture, the outward flux is the number of crossings. The lines are not physical threads, and their drawn number is arbitrary. Still, the crossing picture captures something real: a surface facing the field receives positive flux, and a closed surface surrounding the source intercepts the whole outward pattern. Replace the small sphere with a larger one. The same radial lines cross it, so the crossing count has not changed. They are farther apart now, which is the picture's way of saying that the electric field is weaker. The radius grew, and the sphere's area grew as radius squared. But the field of a point charge weakened as one over radius squared. Twice the radius gives four times the area and one quarter of the field strength. Their product stays fixed. A sphere is not essential. Dent the surface here, bulge it there, and keep the charge enclosed. Every ray that leaves the charge still has to cross the closed boundary. The local crossing angles and local field strengths change, but the total outward count does not. For a sphere of radius r, the point-charge field has magnitude one over four pi epsilon zero, times q over r squared. The sphere's area is four pi r squared. This is exactly the geometrical growth that compensates for the inverse-square weakening. Multiply field strength by area. The r squared in the area cancels the r squared in the denominator, and four pi cancels as well. What remains is q divided by epsilon zero. It has no radius in it. The calculation therefore agrees with the crossing picture: every surrounding sphere receives the same total outward electric flux. The lumpy surface needs a more careful sum because its field strength and angle vary from patch to patch. But once those local contributions are counted correctly, the same enclosed charge gives the same total flux. That invariant total is the idea Gauss's law will make exact. The next step is to define what one small crossing contributes, including the angle at which the field meets the surface. ### [02:57.108 · From Crossings to Gauss's Law](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=177.10831249999995) Zoom in on one small piece of a surface. Its area is delta A, and the yellow arrow is the outward unit normal, the direction perpendicular to that patch. Let the electric field meet the patch at an angle theta from the normal. The field crosses most effectively when it points along the normal. The patch's flux is E dot n hat times delta A. Equivalently, it is field strength times area times cosine theta. The cosine selects the normal component of the field. If the field runs parallel to the patch, theta is ninety degrees and the cosine is zero. Lines may skim along the surface, but none cross it, so that part contributes no flux. If the field points outward, the dot product is positive. If it points inward, the dot product is negative. Flux therefore counts signed crossings, outward minus inward. A curved surface is built from many small patches. Add the local dot products over all of them. As the patches become arbitrarily small, the sum becomes a surface integral. This is the precise definition of electric flux. Unlike the number of lines an artist draws, the integral has a fixed numerical meaning. Return to a point charge and a spherical surface around it. Symmetry makes the field radial, so it points along the outward normal everywhere on the sphere. Symmetry also makes the field strength the same at every point of a sphere with radius r. We can therefore take E outside the integral, leaving the total area four pi r squared. Insert the inverse-square point-charge field. The area growth cancels the field's weakening, leaving q divided by epsilon zero. Now deform the closed surface without moving it across the charge. Some patches tilt, some move closer, and others move farther away. The local dot products change, but the net signed crossing count cannot change. Several charges simply add their fields. A charge inside contributes its full outward flux. A charge outside sends as much flux into the closed surface as it sends back out, so its net contribution is zero. The result is Gauss's law: the flux through any closed surface equals the net enclosed charge divided by epsilon zero. The surface may be imaginary, irregular, or placed wherever we choose. Written in full, the closed-surface integral of E dot n hat equals enclosed charge divided by epsilon zero. Notice what the law does and does not say. It always gives total flux from enclosed charge. It gives the electric field itself only when symmetry makes the field's direction and magnitude simple on a carefully chosen surface. That choice is the real technique. For a spherical charge distribution we will choose a sphere. For a line we will choose a cylinder. For a sheet we will choose a pillbox. In every case, the source geometry chooses the useful Gaussian surface. ### [06:14.171 · A Uniformly Charged Sphere](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=374.1709166666666) Take a solid insulating sphere of radius R, with total charge Q spread uniformly throughout its volume. Because the distribution looks the same after any rotation about its centre, the electric field must point radially. A short turn confirms the geometry. There is no preferred direction around the sphere, so at a fixed distance from the centre every point must have the same field magnitude. First ask for the field outside the charge distribution. Choose a spherical Gaussian surface of radius r greater than R, centred on the same point. On this Gaussian sphere, the field is everywhere normal to the surface and has one constant magnitude E of r. The flux is therefore E times four pi r squared. The Gaussian surface encloses the entire charged body, so the enclosed charge is Q. Gauss's law sets the flux equal to Q over epsilon zero. Solving for E gives the familiar inverse-square field. Outside a spherically symmetric distribution, the field is exactly the same as if all the charge were concentrated at the centre. Gauss's law makes that statement exact, not merely approximate. Now move the Gaussian sphere inside the charged body. Its radius r is less than R. The field still has spherical symmetry, but the surface now encloses only part of the total charge. Uniform volume charge density means total charge divided by total volume. The density rho is Q over four thirds pi R cubed. The smaller Gaussian sphere has volume four thirds pi r cubed. Multiply that volume by rho to find the charge it encloses. The common factors cancel, leaving Q times r cubed over R cubed. This fraction is simply the fraction of the charged volume lying inside the Gaussian surface. Gauss's law again says E times four pi r squared equals enclosed charge over epsilon zero. One power of r survives after division. The interior field is proportional to r. It is zero at the centre, where all directions balance, and it grows linearly as the Gaussian sphere encloses more charge. Put the two regions on one graph. Distance is measured in units of R, and field strength in units of its value at the surface. Inside, the field rises in a straight line from zero. In normalized form, E over E at the surface equals r over R. Enclosed charge grows as r cubed, while Gaussian area grows as r squared, leaving one power of r. At r equals R, the inside and outside formulas agree. There is no jump in the field because the charge fills a volume rather than sitting in an infinitesimally thin surface layer. Outside, E over the surface field equals R squared over r squared. The enclosed charge has stopped growing, while the Gaussian area continues to grow as r squared. The method was the same in both regions: use the source symmetry to choose a concentric sphere, make E constant on that surface, and then count only the charge actually enclosed. ### [09:46.464 · Lines, Sheets, and the Surfaces They Choose](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=586.4642083333332) Now stretch the charge distribution into an ideal infinite line with uniform charge per length lambda. Translation along the line changes nothing, and rotation around it changes nothing. Those symmetries force the electric field to point directly away from the line. Its magnitude can depend on perpendicular distance r, but not on position along the line or angle around it. Choose a cylinder centred on the line. Its curved side is everywhere the same distance r from the charge, so E has one constant magnitude there. Let the solid settle into view. The charged line is the cylinder's axis, and the blue field arrows point through its curved wall. On the two end caps, the outward normals point along the line, while the electric field points radially away from it. Their dot product is zero, so the caps contribute no flux. Only the curved side contributes. Its area is circumference two pi r times length L, so the flux is E times two pi r L. The cylinder encloses a length L of line charge. Charge per length lambda times L gives enclosed charge lambda L. Apply Gauss's law. The length L appears on both sides and cancels. Solving leaves lambda over two pi epsilon zero r. The line field falls as one over r, not one over r squared. As the cylinder expands, its relevant area per unit length grows only in proportion to r. Next spread charge uniformly across an ideal infinite sheet, with surface charge density sigma. Sliding anywhere within the sheet cannot change the field. Rotating the sheet within its own plane also changes nothing. The only distinguished direction is perpendicular to the sheet, so the field must point normally away on both sides. Choose a short cylindrical pillbox that straddles the sheet. Its flat caps are parallel to the charge distribution, and its curved wall joins them. A small turn shows the construction. The field passes through the two caps, while it runs parallel to the curved wall. The curved wall contributes zero flux because its normal lies within the sheet while E is perpendicular to it. Each cap contributes E A, so the total flux is two E A. The pillbox encloses sheet area A, so it encloses charge sigma A. Gauss's law gives two E A equals sigma A over epsilon zero. The cap area cancels, leaving E equals sigma over two epsilon zero. There is no distance in the answer. For an ideal infinite sheet, moving the caps farther away does not spread a fixed bundle over a growing area. The same cap area intercepts the same flux. Put the three geometries together. A point spreads flux over a sphere whose area grows as r squared, so its field has the form a constant over r squared. A line spreads flux over the curved wall of a cylinder. Per unit length, that area grows as r, so the field has the form a constant over r. A sheet sends flux through two equal caps. Their area does not change when the pillbox grows taller, so the ideal sheet field is constant. Gauss's law was identical in all three cases. What changed was the symmetry, and symmetry determined the surface on which E became constant and the unwanted pieces contributed zero. ### [13:36.753 · Conductors, Surface Charge, and the Car](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=816.7534166666665) A conductor contains mobile charge. If an electric field existed inside the metal, that charge would feel a force and begin to move. A state with moving charge is not electrostatic equilibrium. The mobile charge rearranges on an extremely short time scale. Its own electric field opposes the field that drove the motion, and equilibrium is reached only when the net electric field inside the conducting material is zero. That statement comes from the physics of mobile charge, not from Gauss's law alone. Gauss's law now tells us an important consequence of the zero field. Draw any closed Gaussian surface lying completely inside the conducting material. Because E is zero at every point on it, its electric flux is zero. Gauss's law then says that the net charge enclosed by every such interior surface is zero. Excess charge therefore cannot remain distributed through the bulk of the metal in electrostatic equilibrium. It moves outward and comes to rest on the conductor's surface. Outside the conductor, those surface charges can produce an electric field. Inside the conducting material, they have arranged themselves so that their combined field cancels. The surface distribution need not be uniform. Charge crowds more strongly near sharp points, where the surrounding field can become especially large. But the equilibrium condition inside the metal remains E equals zero. Now replace the simple conductor with the metal body of a hard-top car. The body forms a conducting shell around the passenger compartment. Suppose lightning strikes the roof. The lightning delivers charge and a large current to the exterior metal. The charge spreads over the outside, and the current finds conducting paths along the exterior body. The metal shell carries the dangerous electrical disturbance around the passenger space rather than through it. The passenger compartment is therefore close to one electric potential, with a strongly reduced electric field inside. This shielding behavior is often called the Faraday-cage effect. The protection comes from the continuous metal shell, not primarily from the rubber tires. During a storm, occupants should remain inside with windows closed and avoid touching metal parts connected to the exterior. The same principle is used deliberately in shielded rooms, cable coverings, and metal enclosures around sensitive electronics. Conductors rearrange charge so that their protected interiors experience very little electric field. Three ideas organize the whole lecture. First, electric flux is the signed amount of electric field crossing a surface, and the flux through a closed surface counts net enclosed charge. Second, Gauss's law becomes a field-solving tool only when symmetry chooses a useful surface: a sphere for spherical charge, a cylinder for a line, and a pillbox for a sheet. Third, mobile charge in a conductor rearranges until the field inside the metal is zero. Excess charge lives on the surface, and a closed metal body redirects an external electrical disturbance around its interior. Flux made the counting picture precise. Gauss's law connected that count to charge. Symmetry turned the law into three electric fields, and electrostatic equilibrium turned it into protection inside a conductor. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M14TZWMWWK33R8DXJ7Z859ME/1/semantic.json) Record version: 1. Render attempt: 1. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [Flux That Refuses to Change](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0) Span: 00:00–02:57.108 (0s–177.10831249999995s). #### Objects - charge: a Point \[red\] labelled "q" drawn in space (location=(0.0, 0.0, 0.0)) - equation\_heading: a Heading that says "Weakening Field, Growing Area" - heading: a Heading that says "A Point Charge and Its Field" - large\_sphere: a Sphere \[green\] drawn in space (radius=1.65, opacity=0.14) - lumpy: a Surface \[magenta\] drawn in space (function=\, u\_range=(0.0, 3.141592653589793), v\_range=(0.0, 6.283185307179586)) - rays: a Vector \[blue\] drawn in space (start=(0.18, 0.0, 0.0), end=(2.35, 0.0, 0.0)) - rays\_10: a Vector \[blue\] drawn in space (start=(-0.12725999999999998, -0.12725999999999998, 0.0), end=(-1.6614499999999999, -1.6614499999999999, 0.0)) - rays\_11: a Vector \[blue\] drawn in space (start=(0.12725999999999998, 0.0, 0.12725999999999998), end=(1.6614499999999999, 0.0, 1.6614499999999999)) - rays\_12: a Vector \[blue\] drawn in space (start=(-0.12725999999999998, 0.0, 0.12725999999999998), end=(-1.6614499999999999, 0.0, 1.6614499999999999)) - rays\_13: a Vector \[blue\] drawn in space (start=(0.0, 0.12725999999999998, -0.12725999999999998), end=(0.0, 1.6614499999999999, -1.6614499999999999)) - rays\_14: a Vector \[blue\] drawn in space (start=(0.0, -0.12725999999999998, -0.12725999999999998), end=(0.0, -1.6614499999999999, -1.6614499999999999)) - rays\_2: a Vector \[blue\] drawn in space (start=(-0.18, 0.0, 0.0), end=(-2.35, 0.0, 0.0)) - rays\_3: a Vector \[blue\] drawn in space (start=(0.0, 0.18, 0.0), end=(0.0, 2.35, 0.0)) - rays\_4: a Vector \[blue\] drawn in space (start=(0.0, -0.18, 0.0), end=(0.0, -2.35, 0.0)) - rays\_5: a Vector \[blue\] drawn in space (start=(0.0, 0.0, 0.18), end=(0.0, 0.0, 2.35)) - rays\_6: a Vector \[blue\] drawn in space (start=(0.0, 0.0, -0.18), end=(0.0, 0.0, -2.35)) - rays\_7: a Vector \[blue\] drawn in space (start=(0.12725999999999998, 0.12725999999999998, 0.0), end=(1.6614499999999999, 1.6614499999999999, 0.0)) - rays\_8: a Vector \[blue\] drawn in space (start=(-0.12725999999999998, 0.12725999999999998, 0.0), end=(-1.6614499999999999, 1.6614499999999999, 0.0)) - rays\_9: a Vector \[blue\] drawn in space (start=(0.12725999999999998, -0.12725999999999998, 0.0), end=(1.6614499999999999, -1.6614499999999999, 0.0)) - result: a Math \[text\] that says "$Phi\_E = frac(q, epsilon\_0)$" - small\_sphere: a Sphere \[yellow\] drawn in space (radius=0.9, opacity=0.18) - space: an Axes3D (x\_range=(-2.6, 2.6), y\_range=(-2.6, 2.6), z\_range=(-2.6, 2.6)) - work: a Derivation \[text\] that says "$E(r) = frac(1, 4 pi epsilon\_0) frac(q, r^2) \\ A(r) = 4 pi r^2 \\ E(r) A(r) = frac(1, 4 pi epsilon\_0) frac(q, r^2) 4 pi r^2 \\ Phi\_E = frac(q, epsilon\_0)$" #### Beats ##### [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0) Narration: Place one positive point charge in empty space. At every surrounding point, its electric field points directly away from the charge. These blue arrows represent field lines, a drawing that lets us follow the direction of the field through space. Board: Empty. Actions: - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): heading is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): space is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): charge is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_2 is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_3 is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_4 is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_5 is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_6 is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_7 is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_8 is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_9 is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_10 is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_11 is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_12 is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_13 is shown on the screen, written out. - [00:00](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=0): rays\_14 is shown on the screen, written out. ##### [00:15.508](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=15.5075) Narration: The picture is three dimensional. Every direction away from the charge is equivalent, and no radial direction is preferred over another. Board: space — an Axes3D (x\_range=(-2.6, 2.6), y\_range=(-2.6, 2.6), z\_range=(-2.6, 2.6)); heading — a Heading that says "A Point Charge and Its Field"; charge — a Point \[red\] labelled "q" drawn in space (location=(0.0, 0.0, 0.0)); rays — a Vector \[blue\] drawn in space (start=(0.18, 0.0, 0.0), end=(2.35, 0.0, 0.0)); rays\_2 — a Vector \[blue\] drawn in space (start=(-0.18, 0.0, 0.0), end=(-2.35, 0.0, 0.0)); rays\_3 — a Vector \[blue\] drawn in space (start=(0.0, 0.18, 0.0), end=(0.0, 2.35, 0.0)); rays\_4 — a Vector \[blue\] drawn in space (start=(0.0, -0.18, 0.0), end=(0.0, -2.35, 0.0)); rays\_5 — a Vector \[blue\] drawn in space (start=(0.0, 0.0, 0.18), end=(0.0, 0.0, 2.35)); rays\_6 — a Vector \[blue\] drawn in space (start=(0.0, 0.0, -0.18), end=(0.0, 0.0, -2.35)); rays\_7 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, 0.12725999999999998, 0.0), end=(1.6614499999999999, 1.6614499999999999, 0.0)); rays\_8 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, 0.12725999999999998, 0.0), end=(-1.6614499999999999, 1.6614499999999999, 0.0)); rays\_9 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, -0.12725999999999998, 0.0), end=(1.6614499999999999, -1.6614499999999999, 0.0)); rays\_10 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, -0.12725999999999998, 0.0), end=(-1.6614499999999999, -1.6614499999999999, 0.0)); rays\_11 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, 0.0, 0.12725999999999998), end=(1.6614499999999999, 0.0, 1.6614499999999999)); rays\_12 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, 0.0, 0.12725999999999998), end=(-1.6614499999999999, 0.0, 1.6614499999999999)); rays\_13 — a Vector \[blue\] drawn in space (start=(0.0, 0.12725999999999998, -0.12725999999999998), end=(0.0, 1.6614499999999999, -1.6614499999999999)); rays\_14 — a Vector \[blue\] drawn in space (start=(0.0, -0.12725999999999998, -0.12725999999999998), end=(0.0, -1.6614499999999999, -1.6614499999999999)) Actions: - [00:15.508](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=15.5075): space turns in its own slot. ##### [00:24.803](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=24.8035) Narration: Now surround the charge with a small sphere. Every blue line leaving the charge crosses that sphere once, from its inside to its outside. If we use the lines as a counting picture, the outward flux is the number of crossings. Board: Unchanged from the preceding beat in this scene. Actions: - [00:27.079](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=27.078999999999997): small\_sphere is shown on the screen, faded in. ##### [00:40.195](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=40.195) Narration: The lines are not physical threads, and their drawn number is arbitrary. Still, the crossing picture captures something real: a surface facing the field receives positive flux, and a closed surface surrounding the source intercepts the whole outward pattern. Board: space — an Axes3D (x\_range=(-2.6, 2.6), y\_range=(-2.6, 2.6), z\_range=(-2.6, 2.6)); heading — a Heading that says "A Point Charge and Its Field"; charge — a Point \[red\] labelled "q" drawn in space (location=(0.0, 0.0, 0.0)); rays — a Vector \[blue\] drawn in space (start=(0.18, 0.0, 0.0), end=(2.35, 0.0, 0.0)); rays\_2 — a Vector \[blue\] drawn in space (start=(-0.18, 0.0, 0.0), end=(-2.35, 0.0, 0.0)); rays\_3 — a Vector \[blue\] drawn in space (start=(0.0, 0.18, 0.0), end=(0.0, 2.35, 0.0)); rays\_4 — a Vector \[blue\] drawn in space (start=(0.0, -0.18, 0.0), end=(0.0, -2.35, 0.0)); rays\_5 — a Vector \[blue\] drawn in space (start=(0.0, 0.0, 0.18), end=(0.0, 0.0, 2.35)); rays\_6 — a Vector \[blue\] drawn in space (start=(0.0, 0.0, -0.18), end=(0.0, 0.0, -2.35)); rays\_7 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, 0.12725999999999998, 0.0), end=(1.6614499999999999, 1.6614499999999999, 0.0)); rays\_8 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, 0.12725999999999998, 0.0), end=(-1.6614499999999999, 1.6614499999999999, 0.0)); rays\_9 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, -0.12725999999999998, 0.0), end=(1.6614499999999999, -1.6614499999999999, 0.0)); rays\_10 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, -0.12725999999999998, 0.0), end=(-1.6614499999999999, -1.6614499999999999, 0.0)); rays\_11 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, 0.0, 0.12725999999999998), end=(1.6614499999999999, 0.0, 1.6614499999999999)); rays\_12 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, 0.0, 0.12725999999999998), end=(-1.6614499999999999, 0.0, 1.6614499999999999)); rays\_13 — a Vector \[blue\] drawn in space (start=(0.0, 0.12725999999999998, -0.12725999999999998), end=(0.0, 1.6614499999999999, -1.6614499999999999)); rays\_14 — a Vector \[blue\] drawn in space (start=(0.0, -0.12725999999999998, -0.12725999999999998), end=(0.0, -1.6614499999999999, -1.6614499999999999)); small\_sphere — a Sphere \[yellow\] drawn in space (radius=0.9, opacity=0.18) Actions: - [00:52.629](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=52.629): small\_sphere is indicated — a transient flash. ##### [00:57.536](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=57.536) Narration: Replace the small sphere with a larger one. The same radial lines cross it, so the crossing count has not changed. They are farther apart now, which is the picture's way of saying that the electric field is weaker. Board: Unchanged from the preceding beat in this scene. Actions: - [00:57.536](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=57.536): small\_sphere is hidden from the screen. - [00:59.313](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=59.313): large\_sphere is shown on the screen, faded in. ##### [01:11.162](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=71.1625) Narration: The radius grew, and the sphere's area grew as radius squared. But the field of a point charge weakened as one over radius squared. Twice the radius gives four times the area and one quarter of the field strength. Their product stays fixed. Board: space — an Axes3D (x\_range=(-2.6, 2.6), y\_range=(-2.6, 2.6), z\_range=(-2.6, 2.6)); heading — a Heading that says "A Point Charge and Its Field"; charge — a Point \[red\] labelled "q" drawn in space (location=(0.0, 0.0, 0.0)); rays — a Vector \[blue\] drawn in space (start=(0.18, 0.0, 0.0), end=(2.35, 0.0, 0.0)); rays\_2 — a Vector \[blue\] drawn in space (start=(-0.18, 0.0, 0.0), end=(-2.35, 0.0, 0.0)); rays\_3 — a Vector \[blue\] drawn in space (start=(0.0, 0.18, 0.0), end=(0.0, 2.35, 0.0)); rays\_4 — a Vector \[blue\] drawn in space (start=(0.0, -0.18, 0.0), end=(0.0, -2.35, 0.0)); rays\_5 — a Vector \[blue\] drawn in space (start=(0.0, 0.0, 0.18), end=(0.0, 0.0, 2.35)); rays\_6 — a Vector \[blue\] drawn in space (start=(0.0, 0.0, -0.18), end=(0.0, 0.0, -2.35)); rays\_7 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, 0.12725999999999998, 0.0), end=(1.6614499999999999, 1.6614499999999999, 0.0)); rays\_8 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, 0.12725999999999998, 0.0), end=(-1.6614499999999999, 1.6614499999999999, 0.0)); rays\_9 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, -0.12725999999999998, 0.0), end=(1.6614499999999999, -1.6614499999999999, 0.0)); rays\_10 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, -0.12725999999999998, 0.0), end=(-1.6614499999999999, -1.6614499999999999, 0.0)); rays\_11 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, 0.0, 0.12725999999999998), end=(1.6614499999999999, 0.0, 1.6614499999999999)); rays\_12 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, 0.0, 0.12725999999999998), end=(-1.6614499999999999, 0.0, 1.6614499999999999)); rays\_13 — a Vector \[blue\] drawn in space (start=(0.0, 0.12725999999999998, -0.12725999999999998), end=(0.0, 1.6614499999999999, -1.6614499999999999)); rays\_14 — a Vector \[blue\] drawn in space (start=(0.0, -0.12725999999999998, -0.12725999999999998), end=(0.0, -1.6614499999999999, -1.6614499999999999)); large\_sphere — a Sphere \[green\] drawn in space (radius=1.65, opacity=0.14) Actions: - [01:25.768](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=85.768): large\_sphere is indicated — a transient flash. ##### [01:28.04](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=88.03999999999999) Narration: A sphere is not essential. Dent the surface here, bulge it there, and keep the charge enclosed. Every ray that leaves the charge still has to cross the closed boundary. The local crossing angles and local field strengths change, but the total outward count does not. Board: Unchanged from the preceding beat in this scene. Actions: - [01:28.04](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=88.03999999999999): large\_sphere is hidden from the screen. - [01:30.664](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=90.66399999999999): lumpy is shown on the screen, faded in. - [01:44.956](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=104.95599999999999): space moves to a new place on the board. - [01:44.956](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=104.95599999999999): heading is hidden from the screen — left the board. ##### [01:46.156](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=106.15599999999999) Narration: For a sphere of radius r, the point-charge field has magnitude one over four pi epsilon zero, times q over r squared. Board: space — an Axes3D (x\_range=(-2.6, 2.6), y\_range=(-2.6, 2.6), z\_range=(-2.6, 2.6)); charge — a Point \[red\] labelled "q" drawn in space (location=(0.0, 0.0, 0.0)); rays — a Vector \[blue\] drawn in space (start=(0.18, 0.0, 0.0), end=(2.35, 0.0, 0.0)); rays\_2 — a Vector \[blue\] drawn in space (start=(-0.18, 0.0, 0.0), end=(-2.35, 0.0, 0.0)); rays\_3 — a Vector \[blue\] drawn in space (start=(0.0, 0.18, 0.0), end=(0.0, 2.35, 0.0)); rays\_4 — a Vector \[blue\] drawn in space (start=(0.0, -0.18, 0.0), end=(0.0, -2.35, 0.0)); rays\_5 — a Vector \[blue\] drawn in space (start=(0.0, 0.0, 0.18), end=(0.0, 0.0, 2.35)); rays\_6 — a Vector \[blue\] drawn in space (start=(0.0, 0.0, -0.18), end=(0.0, 0.0, -2.35)); rays\_7 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, 0.12725999999999998, 0.0), end=(1.6614499999999999, 1.6614499999999999, 0.0)); rays\_8 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, 0.12725999999999998, 0.0), end=(-1.6614499999999999, 1.6614499999999999, 0.0)); rays\_9 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, -0.12725999999999998, 0.0), end=(1.6614499999999999, -1.6614499999999999, 0.0)); rays\_10 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, -0.12725999999999998, 0.0), end=(-1.6614499999999999, -1.6614499999999999, 0.0)); rays\_11 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, 0.0, 0.12725999999999998), end=(1.6614499999999999, 0.0, 1.6614499999999999)); rays\_12 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, 0.0, 0.12725999999999998), end=(-1.6614499999999999, 0.0, 1.6614499999999999)); rays\_13 — a Vector \[blue\] drawn in space (start=(0.0, 0.12725999999999998, -0.12725999999999998), end=(0.0, 1.6614499999999999, -1.6614499999999999)); rays\_14 — a Vector \[blue\] drawn in space (start=(0.0, -0.12725999999999998, -0.12725999999999998), end=(0.0, -1.6614499999999999, -1.6614499999999999)); lumpy — a Surface \[magenta\] drawn in space (function=\, u\_range=(0.0, 3.141592653589793), v\_range=(0.0, 6.283185307179586)) Actions: - [01:46.156](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=106.15599999999999): equation\_heading is shown on the screen, written out. - [01:49.848](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=109.84799999999998): work is shown on the screen, written out. ##### [01:56.462](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=116.46199999999999) Narration: The sphere's area is four pi r squared. This is exactly the geometrical growth that compensates for the inverse-square weakening. Board: space — an Axes3D (x\_range=(-2.6, 2.6), y\_range=(-2.6, 2.6), z\_range=(-2.6, 2.6)); charge — a Point \[red\] labelled "q" drawn in space (location=(0.0, 0.0, 0.0)); rays — a Vector \[blue\] drawn in space (start=(0.18, 0.0, 0.0), end=(2.35, 0.0, 0.0)); rays\_2 — a Vector \[blue\] drawn in space (start=(-0.18, 0.0, 0.0), end=(-2.35, 0.0, 0.0)); rays\_3 — a Vector \[blue\] drawn in space (start=(0.0, 0.18, 0.0), end=(0.0, 2.35, 0.0)); rays\_4 — a Vector \[blue\] drawn in space (start=(0.0, -0.18, 0.0), end=(0.0, -2.35, 0.0)); rays\_5 — a Vector \[blue\] drawn in space (start=(0.0, 0.0, 0.18), end=(0.0, 0.0, 2.35)); rays\_6 — a Vector \[blue\] drawn in space (start=(0.0, 0.0, -0.18), end=(0.0, 0.0, -2.35)); rays\_7 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, 0.12725999999999998, 0.0), end=(1.6614499999999999, 1.6614499999999999, 0.0)); rays\_8 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, 0.12725999999999998, 0.0), end=(-1.6614499999999999, 1.6614499999999999, 0.0)); rays\_9 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, -0.12725999999999998, 0.0), end=(1.6614499999999999, -1.6614499999999999, 0.0)); rays\_10 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, -0.12725999999999998, 0.0), end=(-1.6614499999999999, -1.6614499999999999, 0.0)); rays\_11 — a Vector \[blue\] drawn in space (start=(0.12725999999999998, 0.0, 0.12725999999999998), end=(1.6614499999999999, 0.0, 1.6614499999999999)); rays\_12 — a Vector \[blue\] drawn in space (start=(-0.12725999999999998, 0.0, 0.12725999999999998), end=(-1.6614499999999999, 0.0, 1.6614499999999999)); rays\_13 — a Vector \[blue\] drawn in space (start=(0.0, 0.12725999999999998, -0.12725999999999998), end=(0.0, 1.6614499999999999, -1.6614499999999999)); rays\_14 — a Vector \[blue\] drawn in space (start=(0.0, -0.12725999999999998, -0.12725999999999998), end=(0.0, -1.6614499999999999, -1.6614499999999999)); lumpy — a Surface \[magenta\] drawn in space (function=\, u\_range=(0.0, 3.141592653589793), v\_range=(0.0, 6.283185307179586)); equation\_heading — a Heading that says "Weakening Field, Growing Area" Actions: - [01:57.31](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=117.30999999999999): work is shown on the screen, written out. ##### [02:5.386](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=125.38649999999998) Narration: Multiply field strength by area. The r squared in the area cancels the r squared in the denominator, and four pi cancels as well. Board: Unchanged from the preceding beat in this scene. Actions: - [02:5.735](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=125.73499999999997): work is shown on the screen, written out. - [02:10.228](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=130.22799999999998): work (the "r^2" part) is slashed through — it cancels. - [02:10.228](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=130.22799999999998): work (the "r^2#2" part) is slashed through — it cancels. ##### [02:15.774](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=135.77399999999997) Narration: What remains is q divided by epsilon zero. It has no radius in it. The calculation therefore agrees with the crossing picture: every surrounding sphere receives the same total outward electric flux. Board: Unchanged from the preceding beat in this scene. Actions: - [02:16.354](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=136.35399999999996): work is shown on the screen, written out. - [02:27.268](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=147.26799999999997): A box is drawn around work. ##### [02:30.422](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=150.42199999999997) Narration: The lumpy surface needs a more careful sum because its field strength and angle vary from patch to patch. But once those local contributions are counted correctly, the same enclosed charge gives the same total flux. Board: Unchanged from the preceding beat in this scene. Actions: - [02:30.944](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=150.94399999999996): lumpy is indicated — a transient flash. ##### [02:44.477](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=164.47749999999996) Narration: That invariant total is the idea Gauss's law will make exact. The next step is to define what one small crossing contributes, including the angle at which the field meets the surface. Board: Unchanged from the preceding beat in this scene. Actions: - [02:45.244](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=165.24399999999994): result is shown on the screen, written out. - [02:55.817](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=175.8166458333333): A box is drawn around result. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): equation\_heading is hidden from the screen — left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): result is hidden from the screen — left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): space is hidden from the screen — left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): charge is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_2 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_3 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_4 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_5 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_6 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_7 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_8 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_9 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_10 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_11 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_12 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_13 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): rays\_14 is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): lumpy is hidden from the screen — space left the board. - [02:56.067](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=176.0666458333333): work is hidden from the screen — left the board. ### Scene 2: [From Crossings to Gauss's Law](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=177.10831249999995) Span: 02:57.108–06:14.171 (177.10831249999995s–374.1709166666666s). #### Objects - angle: an Angle \[green\] labelled "theta" drawn in patch\_picture (sides=((1.55, 0.95), (0.0, 1.45))) - field\_arrow: a Vector \[blue\] labelled "arrow(E)" drawn in patch\_picture (end=(1.55, 0.95)) - gauss\_definition: a Panel that says "The net electric flux through any closed surface equals the net charge enclosed by that surface divided by $epsilon\_0$." - gauss\_heading: a Heading that says "Gauss's Law" - gauss\_law: a Math \[text\] that says "$integral.double\_S arrow(E) dot hat(n) thin dif A = frac(q\_(upright("enc")), epsilon\_0)$" - heading: a Heading that says "The Point-Charge Count" - local\_formula: a Math \[text\] that says "$Delta Phi\_E = arrow(E) dot hat(n) thin Delta A = E Delta A cos theta$" - local\_heading: a Heading that says "One Small Patch" - local\_note: a Text \[text\] that says "Only the component of the electric field normal to the surface contributes." - normal\_arrow: a Vector \[yellow\] labelled "hat(n)" drawn in patch\_picture (end=(0.0, 1.45)) - parallel\_field: a Vector \[magenta\] labelled "arrow(E)" drawn in patch\_picture (start=(-1.55, -0.55), end=(0.75, -0.55)) - patch: a Polygon \[gray\] labelled "Delta A" drawn in patch\_picture (vertices=((-1.25, -0.25), (1.05, -0.25), (1.35, 0.25), (-0.95, 0.25)), fill\_opacity=0.18) - patch\_picture: a Figure (x\_range=(-2.4, 2.4), y\_range=(-1.7, 1.9), aspect=(5.0, 3.6)) - point\_result: a Math \[text\] that says "$Phi\_E = frac(q, epsilon\_0)$" - point\_work: a Derivation \[text\] that says "$Phi\_E &= integral.double\_S arrow(E) dot hat(n) thin dif A \\ &= E(r) 4 pi r^2 \\ &= frac(1, 4 pi epsilon\_0) frac(q, r^2) 4 pi r^2 \\ &= frac(q, epsilon\_0)$" - shell: a Sphere \[yellow\] drawn in sphere\_axes (radius=1.35, opacity=0.16) - source: a Point \[red\] labelled "q" drawn in sphere\_axes (location=(0.0, 0.0, 0.0)) - sphere\_axes: an Axes3D (x\_range=(-2.0, 2.0), y\_range=(-2.0, 2.0), z\_range=(-2.0, 2.0)) - sum\_heading: a Heading that says "Add Every Patch" - sum\_work: a Derivation \[text\] that says "$Phi\_E &= sum\_i arrow(E)\_i dot hat(n)\_i thin Delta A\_i \\ &arrow.r integral.double\_S arrow(E) dot hat(n) thin dif A$" #### Beats ##### [02:57.108](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=177.10831249999995) Narration: Zoom in on one small piece of a surface. Its area is delta A, and the yellow arrow is the outward unit normal, the direction perpendicular to that patch. Board: Empty. Actions: - [02:57.108](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=177.10831249999995): local\_heading is shown on the screen, written out. - [02:58.466](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=178.46631249999996): patch\_picture is shown on the screen, written out. - [03:0.603](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=180.60331249999996): patch is shown on the screen, written out. - [03:4.004](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=184.00431249999994): normal\_arrow is shown on the screen, written out. ##### [03:7.844](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=187.84381249999996) Narration: Let the electric field meet the patch at an angle theta from the normal. The field crosses most effectively when it points along the normal. Board: patch\_picture — a Figure (x\_range=(-2.4, 2.4), y\_range=(-1.7, 1.9), aspect=(5.0, 3.6)); local\_heading — a Heading that says "One Small Patch"; patch — a Polygon \[gray\] labelled "Delta A" drawn in patch\_picture (vertices=((-1.25, -0.25), (1.05, -0.25), (1.35, 0.25), (-0.95, 0.25)), fill\_opacity=0.18); normal\_arrow — a Vector \[yellow\] labelled "hat(n)" drawn in patch\_picture (end=(0.0, 1.45)) Actions: - [03:9.016](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=189.01631249999994): field\_arrow is shown on the screen, written out. - [03:10.189](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=190.18931249999994): angle is shown on the screen, written out. ##### [03:16.733](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=196.73281249999997) Narration: The patch's flux is E dot n hat times delta A. Equivalently, it is field strength times area times cosine theta. The cosine selects the normal component of the field. Board: patch\_picture — a Figure (x\_range=(-2.4, 2.4), y\_range=(-1.7, 1.9), aspect=(5.0, 3.6)); local\_heading — a Heading that says "One Small Patch"; patch — a Polygon \[gray\] labelled "Delta A" drawn in patch\_picture (vertices=((-1.25, -0.25), (1.05, -0.25), (1.35, 0.25), (-0.95, 0.25)), fill\_opacity=0.18); normal\_arrow — a Vector \[yellow\] labelled "hat(n)" drawn in patch\_picture (end=(0.0, 1.45)); field\_arrow — a Vector \[blue\] labelled "arrow(E)" drawn in patch\_picture (end=(1.55, 0.95)); angle — an Angle \[green\] labelled "theta" drawn in patch\_picture (sides=((1.55, 0.95), (0.0, 1.45))) Actions: - [03:17.545](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=197.54531249999997): patch\_picture moves to a new place on the board. - [03:17.545](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=197.54531249999997): local\_formula is shown on the screen, written out. - [03:24.651](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=204.65131249999996): local\_formula (the "cos theta" part) is emphasized. - [03:29.33](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=209.32981249999995): local\_formula (the "cos theta" part) is no longer emphasized. ##### [03:29.93](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=209.92981249999997) Narration: If the field runs parallel to the patch, theta is ninety degrees and the cosine is zero. Lines may skim along the surface, but none cross it, so that part contributes no flux. Board: local\_formula — a Math \[text\] that says "$Delta Phi\_E = arrow(E) dot hat(n) thin Delta A = E Delta A cos theta$"; patch\_picture — a Figure (x\_range=(-2.4, 2.4), y\_range=(-1.7, 1.9), aspect=(5.0, 3.6)); local\_heading — a Heading that says "One Small Patch"; patch — a Polygon \[gray\] labelled "Delta A" drawn in patch\_picture (vertices=((-1.25, -0.25), (1.05, -0.25), (1.35, 0.25), (-0.95, 0.25)), fill\_opacity=0.18); normal\_arrow — a Vector \[yellow\] labelled "hat(n)" drawn in patch\_picture (end=(0.0, 1.45)); field\_arrow — a Vector \[blue\] labelled "arrow(E)" drawn in patch\_picture (end=(1.55, 0.95)); angle — an Angle \[green\] labelled "theta" drawn in patch\_picture (sides=((1.55, 0.95), (0.0, 1.45))) Actions: - [03:31.23](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=211.23031249999997): parallel\_field is shown on the screen, written out. - [03:36.756](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=216.75631249999995): parallel\_field is indicated — a transient flash. - [03:41.493](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=221.49331249999994): parallel\_field is hidden from the screen. ##### [03:42.093](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=222.09331249999997) Narration: If the field points outward, the dot product is positive. If it points inward, the dot product is negative. Flux therefore counts signed crossings, outward minus inward. Board: Unchanged from the preceding beat in this scene. Actions: - [03:43.382](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=223.38231249999995): normal\_arrow is indicated — a transient flash. - [03:54.377](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=234.37681249999997): patch\_picture moves to a new place on the board. - [03:54.377](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=234.37681249999997): local\_formula is hidden from the screen — left the board. - [03:54.377](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=234.37681249999997): local\_heading is hidden from the screen — left the board. ##### [03:55.577](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=235.57681249999996) Narration: A curved surface is built from many small patches. Add the local dot products over all of them. Board: patch\_picture — a Figure (x\_range=(-2.4, 2.4), y\_range=(-1.7, 1.9), aspect=(5.0, 3.6)); patch — a Polygon \[gray\] labelled "Delta A" drawn in patch\_picture (vertices=((-1.25, -0.25), (1.05, -0.25), (1.35, 0.25), (-0.95, 0.25)), fill\_opacity=0.18); normal\_arrow — a Vector \[yellow\] labelled "hat(n)" drawn in patch\_picture (end=(0.0, 1.45)); field\_arrow — a Vector \[blue\] labelled "arrow(E)" drawn in patch\_picture (end=(1.55, 0.95)); angle — an Angle \[green\] labelled "theta" drawn in patch\_picture (sides=((1.55, 0.95), (0.0, 1.45))) Actions: - [03:55.577](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=235.57681249999996): sum\_heading is shown on the screen, written out. - [03:59.315](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=239.31531249999995): sum\_work is shown on the screen, written out. ##### [04:2.806](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=242.80631249999993) Narration: As the patches become arbitrarily small, the sum becomes a surface integral. This is the precise definition of electric flux. Unlike the number of lines an artist draws, the integral has a fixed numerical meaning. Board: patch\_picture — a Figure (x\_range=(-2.4, 2.4), y\_range=(-1.7, 1.9), aspect=(5.0, 3.6)); patch — a Polygon \[gray\] labelled "Delta A" drawn in patch\_picture (vertices=((-1.25, -0.25), (1.05, -0.25), (1.35, 0.25), (-0.95, 0.25)), fill\_opacity=0.18); normal\_arrow — a Vector \[yellow\] labelled "hat(n)" drawn in patch\_picture (end=(0.0, 1.45)); field\_arrow — a Vector \[blue\] labelled "arrow(E)" drawn in patch\_picture (end=(1.55, 0.95)); angle — an Angle \[green\] labelled "theta" drawn in patch\_picture (sides=((1.55, 0.95), (0.0, 1.45))); sum\_heading — a Heading that says "Add Every Patch" Actions: - [04:7.334](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=247.33431249999995): sum\_work is shown on the screen, written out. - [04:17.504](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=257.50431249999997): patch\_picture is hidden from the screen — left the board. - [04:17.504](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=257.50431249999997): patch is hidden from the screen — patch\_picture left the board. - [04:17.504](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=257.50431249999997): normal\_arrow is hidden from the screen — patch\_picture left the board. - [04:17.504](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=257.50431249999997): field\_arrow is hidden from the screen — patch\_picture left the board. - [04:17.504](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=257.50431249999997): angle is hidden from the screen — patch\_picture left the board. - [04:17.504](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=257.50431249999997): sum\_heading is hidden from the screen — left the board. - [04:17.504](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=257.50431249999997): sum\_work is hidden from the screen — left the board. ##### [04:18.104](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=258.10431249999993) Narration: Return to a point charge and a spherical surface around it. Symmetry makes the field radial, so it points along the outward normal everywhere on the sphere. Board: Empty. Actions: - [04:18.104](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=258.10431249999993): sphere\_axes is shown on the screen, written out. - [04:18.104](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=258.10431249999993): source is shown on the screen, written out. - [04:20.612](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=260.6123124999999): sphere\_axes moves to a new place on the board. - [04:20.612](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=260.6123124999999): point\_work is shown on the screen, written out. - [04:26.928](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=266.92831249999995): shell is shown on the screen, faded in. ##### [04:28.445](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=268.44481249999995) Narration: Symmetry also makes the field strength the same at every point of a sphere with radius r. We can therefore take E outside the integral, leaving the total area four pi r squared. Board: sphere\_axes — an Axes3D (x\_range=(-2.0, 2.0), y\_range=(-2.0, 2.0), z\_range=(-2.0, 2.0)); source — a Point \[red\] labelled "q" drawn in sphere\_axes (location=(0.0, 0.0, 0.0)); shell — a Sphere \[yellow\] drawn in sphere\_axes (radius=1.35, opacity=0.16) Actions: - [04:38.162](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=278.1623125): point\_work is shown on the screen, written out. ##### [04:40.759](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=280.75931249999996) Narration: Insert the inverse-square point-charge field. The area growth cancels the field's weakening, leaving q divided by epsilon zero. Board: Unchanged from the preceding beat in this scene. Actions: - [04:41.107](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=281.1073124999999): point\_work is shown on the screen, written out. - [04:47.005](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=287.00531249999995): point\_work is shown on the screen, written out. ##### [04:50.241](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=290.24081249999995) Narration: Now deform the closed surface without moving it across the charge. Some patches tilt, some move closer, and others move farther away. The local dot products change, but the net signed crossing count cannot change. Board: Unchanged from the preceding beat in this scene. Actions: - [04:51.529](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=291.52931249999995): shell is indicated — a transient flash. ##### [05:4.772](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=304.77231249999994) Narration: Several charges simply add their fields. A charge inside contributes its full outward flux. A charge outside sends as much flux into the closed surface as it sends back out, so its net contribution is zero. Board: Unchanged from the preceding beat in this scene. Actions: - [05:8.58](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=308.58031249999993): source is indicated — a transient flash. ##### [05:19.281](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=319.28131249999996) Narration: The result is Gauss's law: the flux through any closed surface equals the net enclosed charge divided by epsilon zero. The surface may be imaginary, irregular, or placed wherever we choose. Board: Unchanged from the preceding beat in this scene. Actions: - [05:19.942](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=319.94231249999996): point\_result is shown on the screen, written out. - [05:31.645](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=331.64531249999993): A box is drawn around point\_result. - [05:32.551](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=332.55081249999995): point\_result is hidden from the screen — left the board. - [05:32.551](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=332.55081249999995): point\_work is hidden from the screen — left the board. - [05:32.551](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=332.55081249999995): sphere\_axes is hidden from the screen — left the board. - [05:32.551](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=332.55081249999995): source is hidden from the screen — sphere\_axes left the board. - [05:32.551](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=332.55081249999995): shell is hidden from the screen — sphere\_axes left the board. ##### [05:33.751](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=333.75081249999994) Narration: Written in full, the closed-surface integral of E dot n hat equals enclosed charge divided by epsilon zero. Board: Empty. Actions: - [05:33.751](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=333.75081249999994): gauss\_heading is shown on the screen, written out. - [05:36.293](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=336.29331249999996): gauss\_law is shown on the screen, written out. - [05:38.65](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=338.6503124999999): gauss\_definition is shown on the screen, written out. ##### [05:42.06](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=342.05981249999996) Narration: Notice what the law does and does not say. It always gives total flux from enclosed charge. It gives the electric field itself only when symmetry makes the field's direction and magnitude simple on a carefully chosen surface. Board: gauss\_law — a Math \[text\] that says "$integral.double\_S arrow(E) dot hat(n) thin dif A = frac(q\_(upright("enc")), epsilon\_0)$"; gauss\_definition — a Panel that says "The net electric flux through any closed surface equals the net charge enclosed by that surface divided by $epsilon\_0$."; gauss\_heading — a Heading that says "Gauss's Law" Actions: - [05:50.964](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=350.9643125): gauss\_law is indicated — a transient flash. ##### [05:56.846](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=356.84631249999995) Narration: That choice is the real technique. For a spherical charge distribution we will choose a sphere. For a line we will choose a cylinder. For a sheet we will choose a pillbox. In every case, the source geometry chooses the useful Gaussian surface. Board: Unchanged from the preceding beat in this scene. Actions: - [06:12.055](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=372.0553124999999): A box is drawn around gauss\_law. - [06:13.129](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=373.12924999999996): gauss\_definition is hidden from the screen — left the board. - [06:13.129](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=373.12924999999996): gauss\_heading is hidden from the screen — left the board. - [06:13.129](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=373.12924999999996): gauss\_law is hidden from the screen — left the board. ### Scene 3: [A Uniformly Charged Sphere](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=374.1709166666666) Span: 06:14.171–09:46.464 (374.1709166666666s–586.4642083333332s). #### Objects - axes: an Axes3D (x\_range=(-2.8, 2.8), y\_range=(-2.8, 2.8), z\_range=(-2.8, 2.8)) - centre: a Point \[red\] labelled "O" drawn in axes (location=(0.0, 0.0, 0.0)) - charged\_body: a Sphere \[red\] drawn in axes (radius=1.2, opacity=0.24) - field\_curve: a FunctionPlot \[blue\] drawn in graph (function=\, x\_range=(0.0, 3.0)) - gaussian: a Sphere \[yellow\] drawn in axes (radius=\, opacity=0.11) - gaussian\_radius: a VariableNumber (initial\_value=2.1) - graph: an Axes (x\_range=(0.0, 3.0), y\_range=(0.0, 1.2), x\_ticks\_every=1.0) - graph\_heading: a Heading that says "The Complete Field" - inside\_answer: a Math \[text\] that says "$E\_(upright("in"))(r) = frac(Q, 4 pi epsilon\_0 R^3) r$" - inside\_heading: a Heading that says "Inside: $0 \<= r \<= R$" - inside\_label: a Math \[text\] that says "$frac(E, E\_R) = frac(r, R)$" - inside\_work: a Derivation \[text\] that says "$rho = frac(Q, frac(4, 3) pi R^3) \\ q\_(upright("enc")) = rho frac(4, 3) pi r^3 \\ &= Q frac(r^3, R^3) \\ E(r) 4 pi r^2 = frac(Q r^3, epsilon\_0 R^3)$" - math: a Math \[text\] that says "$upright("total charge") = Q, quad upright("radius") = R$" - outside\_answer: a Math \[text\] that says "$E\_(upright("out"))(r) = frac(1, 4 pi epsilon\_0) frac(Q, r^2)$" - outside\_heading: a Heading that says "Outside: $r \>= R$" - outside\_label: a Math \[text\] that says "$frac(E, E\_R) = frac(R^2, r^2)$" - outside\_work: a Derivation \[text\] that says "$Phi\_E &= E(r) 4 pi r^2 \\ q\_(upright("enc")) &= Q \\ E(r) 4 pi r^2 &= frac(Q, epsilon\_0)$" - radial\_arrows: a Vector \[blue\] drawn in axes (start=(1.25, 0.0, 0.0), end=(2.15, 0.0, 0.0)) - radial\_arrows\_2: a Vector \[blue\] drawn in axes (start=(-1.25, 0.0, 0.0), end=(-2.15, 0.0, 0.0)) - radial\_arrows\_3: a Vector \[blue\] drawn in axes (start=(0.0, 1.25, 0.0), end=(0.0, 2.15, 0.0)) - radial\_arrows\_4: a Vector \[blue\] drawn in axes (start=(0.0, -1.25, 0.0), end=(0.0, -2.15, 0.0)) - radial\_arrows\_5: a Vector \[blue\] drawn in axes (start=(0.0, 0.0, 1.25), end=(0.0, 0.0, 2.15)) - radial\_arrows\_6: a Vector \[blue\] drawn in axes (start=(0.0, 0.0, -1.25), end=(0.0, 0.0, -2.15)) - sphere\_heading: a Heading that says "A Uniformly Charged Solid Sphere" - surface\_point: a PlotPoint \[yellow\] labelled "r=R" drawn in graph (target='field\_curve', x=1.0) #### Beats ##### [06:14.171](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=374.1709166666666) Narration: Take a solid insulating sphere of radius R, with total charge Q spread uniformly throughout its volume. Because the distribution looks the same after any rotation about its centre, the electric field must point radially. Board: Empty. Actions: - [06:14.171](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=374.1709166666666): sphere\_heading is shown on the screen, written out. - [06:14.171](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=374.1709166666666): axes is shown on the screen, written out. - [06:15.657](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=375.6569166666666): charged\_body is shown on the screen, faded in. - [06:24.515](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=384.5149166666666): centre is shown on the screen, written out. - [06:26.826](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=386.82591666666656): radial\_arrows is shown on the screen, written out. - [06:26.826](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=386.82591666666656): radial\_arrows\_2 is shown on the screen, written out. - [06:26.826](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=386.82591666666656): radial\_arrows\_3 is shown on the screen, written out. - [06:26.826](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=386.82591666666656): radial\_arrows\_4 is shown on the screen, written out. - [06:26.826](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=386.82591666666656): radial\_arrows\_5 is shown on the screen, written out. - [06:26.826](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=386.82591666666656): radial\_arrows\_6 is shown on the screen, written out. ##### [06:28.32](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=388.3199166666666) Narration: A short turn confirms the geometry. There is no preferred direction around the sphere, so at a fixed distance from the centre every point must have the same field magnitude. Board: axes — an Axes3D (x\_range=(-2.8, 2.8), y\_range=(-2.8, 2.8), z\_range=(-2.8, 2.8)); sphere\_heading — a Heading that says "A Uniformly Charged Solid Sphere"; charged\_body — a Sphere \[red\] drawn in axes (radius=1.2, opacity=0.24); centre — a Point \[red\] labelled "O" drawn in axes (location=(0.0, 0.0, 0.0)); radial\_arrows — a Vector \[blue\] drawn in axes (start=(1.25, 0.0, 0.0), end=(2.15, 0.0, 0.0)); radial\_arrows\_2 — a Vector \[blue\] drawn in axes (start=(-1.25, 0.0, 0.0), end=(-2.15, 0.0, 0.0)); radial\_arrows\_3 — a Vector \[blue\] drawn in axes (start=(0.0, 1.25, 0.0), end=(0.0, 2.15, 0.0)); radial\_arrows\_4 — a Vector \[blue\] drawn in axes (start=(0.0, -1.25, 0.0), end=(0.0, -2.15, 0.0)); radial\_arrows\_5 — a Vector \[blue\] drawn in axes (start=(0.0, 0.0, 1.25), end=(0.0, 0.0, 2.15)); radial\_arrows\_6 — a Vector \[blue\] drawn in axes (start=(0.0, 0.0, -1.25), end=(0.0, 0.0, -2.15)) Actions: - [06:28.32](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=388.3199166666666): axes turns in its own slot. ##### [06:39.868](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=399.8679166666666) Narration: First ask for the field outside the charge distribution. Choose a spherical Gaussian surface of radius r greater than R, centred on the same point. Board: Unchanged from the preceding beat in this scene. Actions: - [06:44.721](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=404.7209166666666): gaussian is shown on the screen, faded in. - [06:50.015](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=410.01541666666657): axes moves to a new place on the board. - [06:50.015](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=410.01541666666657): sphere\_heading is hidden from the screen — left the board. ##### [06:50.615](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=410.6154166666666) Narration: On this Gaussian sphere, the field is everywhere normal to the surface and has one constant magnitude E of r. The flux is therefore E times four pi r squared. Board: axes — an Axes3D (x\_range=(-2.8, 2.8), y\_range=(-2.8, 2.8), z\_range=(-2.8, 2.8)); charged\_body — a Sphere \[red\] drawn in axes (radius=1.2, opacity=0.24); centre — a Point \[red\] labelled "O" drawn in axes (location=(0.0, 0.0, 0.0)); radial\_arrows — a Vector \[blue\] drawn in axes (start=(1.25, 0.0, 0.0), end=(2.15, 0.0, 0.0)); radial\_arrows\_2 — a Vector \[blue\] drawn in axes (start=(-1.25, 0.0, 0.0), end=(-2.15, 0.0, 0.0)); radial\_arrows\_3 — a Vector \[blue\] drawn in axes (start=(0.0, 1.25, 0.0), end=(0.0, 2.15, 0.0)); radial\_arrows\_4 — a Vector \[blue\] drawn in axes (start=(0.0, -1.25, 0.0), end=(0.0, -2.15, 0.0)); radial\_arrows\_5 — a Vector \[blue\] drawn in axes (start=(0.0, 0.0, 1.25), end=(0.0, 0.0, 2.15)); radial\_arrows\_6 — a Vector \[blue\] drawn in axes (start=(0.0, 0.0, -1.25), end=(0.0, 0.0, -2.15)); gaussian — a Sphere \[yellow\] drawn in axes (radius=\, opacity=0.11) Actions: - [06:50.615](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=410.6154166666666): outside\_heading is shown on the screen, written out. - [06:58.22](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=418.21991666666656): outside\_work is shown on the screen, written out. ##### [07:2.57](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=422.5699166666666) Narration: The Gaussian surface encloses the entire charged body, so the enclosed charge is Q. Board: axes — an Axes3D (x\_range=(-2.8, 2.8), y\_range=(-2.8, 2.8), z\_range=(-2.8, 2.8)); charged\_body — a Sphere \[red\] drawn in axes (radius=1.2, opacity=0.24); centre — a Point \[red\] labelled "O" drawn in axes (location=(0.0, 0.0, 0.0)); radial\_arrows — a Vector \[blue\] drawn in axes (start=(1.25, 0.0, 0.0), end=(2.15, 0.0, 0.0)); radial\_arrows\_2 — a Vector \[blue\] drawn in axes (start=(-1.25, 0.0, 0.0), end=(-2.15, 0.0, 0.0)); radial\_arrows\_3 — a Vector \[blue\] drawn in axes (start=(0.0, 1.25, 0.0), end=(0.0, 2.15, 0.0)); radial\_arrows\_4 — a Vector \[blue\] drawn in axes (start=(0.0, -1.25, 0.0), end=(0.0, -2.15, 0.0)); radial\_arrows\_5 — a Vector \[blue\] drawn in axes (start=(0.0, 0.0, 1.25), end=(0.0, 0.0, 2.15)); radial\_arrows\_6 — a Vector \[blue\] drawn in axes (start=(0.0, 0.0, -1.25), end=(0.0, 0.0, -2.15)); gaussian — a Sphere \[yellow\] drawn in axes (radius=\, opacity=0.11); outside\_heading — a Heading that says "Outside: $r \>= R$" Actions: - [07:6.97](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=426.96991666666656): outside\_work is shown on the screen, written out. ##### [07:9.37](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=429.3699166666666) Narration: Gauss's law sets the flux equal to Q over epsilon zero. Solving for E gives the familiar inverse-square field. Board: Unchanged from the preceding beat in this scene. Actions: - [07:9.718](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=429.7179166666666): outside\_work is shown on the screen, written out. - [07:14.269](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=434.2689166666666): outside\_answer is shown on the screen, written out. ##### [07:18.433](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=438.4334166666666) Narration: Outside a spherically symmetric distribution, the field is exactly the same as if all the charge were concentrated at the centre. Gauss's law makes that statement exact, not merely approximate. Board: axes — an Axes3D (x\_range=(-2.8, 2.8), y\_range=(-2.8, 2.8), z\_range=(-2.8, 2.8)); charged\_body — a Sphere \[red\] drawn in axes (radius=1.2, opacity=0.24); centre — a Point \[red\] labelled "O" drawn in axes (location=(0.0, 0.0, 0.0)); radial\_arrows — a Vector \[blue\] drawn in axes (start=(1.25, 0.0, 0.0), end=(2.15, 0.0, 0.0)); radial\_arrows\_2 — a Vector \[blue\] drawn in axes (start=(-1.25, 0.0, 0.0), end=(-2.15, 0.0, 0.0)); radial\_arrows\_3 — a Vector \[blue\] drawn in axes (start=(0.0, 1.25, 0.0), end=(0.0, 2.15, 0.0)); radial\_arrows\_4 — a Vector \[blue\] drawn in axes (start=(0.0, -1.25, 0.0), end=(0.0, -2.15, 0.0)); radial\_arrows\_5 — a Vector \[blue\] drawn in axes (start=(0.0, 0.0, 1.25), end=(0.0, 0.0, 2.15)); radial\_arrows\_6 — a Vector \[blue\] drawn in axes (start=(0.0, 0.0, -1.25), end=(0.0, 0.0, -2.15)); gaussian — a Sphere \[yellow\] drawn in axes (radius=\, opacity=0.11); outside\_answer — a Math \[text\] that says "$E\_(upright("out"))(r) = frac(1, 4 pi epsilon\_0) frac(Q, r^2)$"; outside\_heading — a Heading that says "Outside: $r \>= R$" Actions: - [07:22.718](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=442.7179166666666): A box is drawn around outside\_answer. - [07:26.793](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=446.79291666666654): The box around outside\_answer is lifted. - [07:31.437](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=451.4369166666666): outside\_answer is hidden from the screen — left the board. - [07:31.437](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=451.4369166666666): outside\_heading is hidden from the screen — left the board. - [07:31.437](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=451.4369166666666): outside\_work is hidden from the screen — left the board. ##### [07:32.637](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=452.6369166666666) Narration: Now move the Gaussian sphere inside the charged body. Its radius r is less than R. The field still has spherical symmetry, but the surface now encloses only part of the total charge. Board: axes — an Axes3D (x\_range=(-2.8, 2.8), y\_range=(-2.8, 2.8), z\_range=(-2.8, 2.8)); charged\_body — a Sphere \[red\] drawn in axes (radius=1.2, opacity=0.24); centre — a Point \[red\] labelled "O" drawn in axes (location=(0.0, 0.0, 0.0)); radial\_arrows — a Vector \[blue\] drawn in axes (start=(1.25, 0.0, 0.0), end=(2.15, 0.0, 0.0)); radial\_arrows\_2 — a Vector \[blue\] drawn in axes (start=(-1.25, 0.0, 0.0), end=(-2.15, 0.0, 0.0)); radial\_arrows\_3 — a Vector \[blue\] drawn in axes (start=(0.0, 1.25, 0.0), end=(0.0, 2.15, 0.0)); radial\_arrows\_4 — a Vector \[blue\] drawn in axes (start=(0.0, -1.25, 0.0), end=(0.0, -2.15, 0.0)); radial\_arrows\_5 — a Vector \[blue\] drawn in axes (start=(0.0, 0.0, 1.25), end=(0.0, 0.0, 2.15)); radial\_arrows\_6 — a Vector \[blue\] drawn in axes (start=(0.0, 0.0, -1.25), end=(0.0, 0.0, -2.15)); gaussian — a Sphere \[yellow\] drawn in axes (radius=\, opacity=0.11) Actions: - [07:32.637](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=452.6369166666666): inside\_heading is shown on the screen, written out. - [07:34.703](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=454.7029166666666): gaussian is redrawn as the numbers it depends on change. - [07:34.703](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=454.7029166666666): gaussian\_radius ticks to 0.75. ##### [07:45.474](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=465.4739166666666) Narration: Uniform volume charge density means total charge divided by total volume. The density rho is Q over four thirds pi R cubed. Board: axes — an Axes3D (x\_range=(-2.8, 2.8), y\_range=(-2.8, 2.8), z\_range=(-2.8, 2.8)); charged\_body — a Sphere \[red\] drawn in axes (radius=1.2, opacity=0.24); centre — a Point \[red\] labelled "O" drawn in axes (location=(0.0, 0.0, 0.0)); radial\_arrows — a Vector \[blue\] drawn in axes (start=(1.25, 0.0, 0.0), end=(2.15, 0.0, 0.0)); radial\_arrows\_2 — a Vector \[blue\] drawn in axes (start=(-1.25, 0.0, 0.0), end=(-2.15, 0.0, 0.0)); radial\_arrows\_3 — a Vector \[blue\] drawn in axes (start=(0.0, 1.25, 0.0), end=(0.0, 2.15, 0.0)); radial\_arrows\_4 — a Vector \[blue\] drawn in axes (start=(0.0, -1.25, 0.0), end=(0.0, -2.15, 0.0)); radial\_arrows\_5 — a Vector \[blue\] drawn in axes (start=(0.0, 0.0, 1.25), end=(0.0, 0.0, 2.15)); radial\_arrows\_6 — a Vector \[blue\] drawn in axes (start=(0.0, 0.0, -1.25), end=(0.0, 0.0, -2.15)); gaussian — a Sphere \[yellow\] drawn in axes (radius=\, opacity=0.11); inside\_heading — a Heading that says "Inside: $0 \<= r \<= R$" Actions: - [07:47.041](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=467.0409166666666): inside\_work is shown on the screen, written out. ##### [07:55.072](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=475.0719166666666) Narration: The smaller Gaussian sphere has volume four thirds pi r cubed. Multiply that volume by rho to find the charge it encloses. Board: Unchanged from the preceding beat in this scene. Actions: - [07:57.382](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=477.3819166666666): inside\_work is shown on the screen, written out. ##### [08:4.948](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=484.9484166666666) Narration: The common factors cancel, leaving Q times r cubed over R cubed. This fraction is simply the fraction of the charged volume lying inside the Gaussian surface. Board: Unchanged from the preceding beat in this scene. Actions: - [08:7.143](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=487.14291666666657): inside\_work is shown on the screen, written out. ##### [08:16.705](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=496.70541666666657) Narration: Gauss's law again says E times four pi r squared equals enclosed charge over epsilon zero. One power of r survives after division. Board: Unchanged from the preceding beat in this scene. Actions: - [08:17.054](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=497.05391666666657): inside\_work is shown on the screen, written out. ##### [08:28.753](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=508.7529166666666) Narration: The interior field is proportional to r. It is zero at the centre, where all directions balance, and it grows linearly as the Gaussian sphere encloses more charge. Board: Unchanged from the preceding beat in this scene. Actions: - [08:30.274](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=510.2739166666666): inside\_answer is shown on the screen, written out. - [08:36.195](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=516.1949166666666): A box is drawn around inside\_answer. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): axes is hidden from the screen — left the board. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): charged\_body is hidden from the screen — axes left the board. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): centre is hidden from the screen — axes left the board. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): radial\_arrows is hidden from the screen — axes left the board. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): radial\_arrows\_2 is hidden from the screen — axes left the board. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): radial\_arrows\_3 is hidden from the screen — axes left the board. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): radial\_arrows\_4 is hidden from the screen — axes left the board. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): radial\_arrows\_5 is hidden from the screen — axes left the board. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): radial\_arrows\_6 is hidden from the screen — axes left the board. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): gaussian is hidden from the screen — axes left the board. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): inside\_answer is hidden from the screen — left the board. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): inside\_heading is hidden from the screen — left the board. - [08:39.992](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=519.9919166666666): inside\_work is hidden from the screen — left the board. ##### [08:41.192](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=521.1919166666665) Narration: Put the two regions on one graph. Distance is measured in units of R, and field strength in units of its value at the surface. Board: Empty. Actions: - [08:41.192](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=521.1919166666665): graph\_heading is shown on the screen, written out. - [08:42.945](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=522.9449166666666): graph is shown on the screen, written out. - [08:42.945](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=522.9449166666666): field\_curve is shown on the screen, written out. ##### [08:50.301](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=530.3014166666666) Narration: Inside, the field rises in a straight line from zero. In normalized form, E over E at the surface equals r over R. Enclosed charge grows as r cubed, while Gaussian area grows as r squared, leaving one power of r. Board: graph — an Axes (x\_range=(0.0, 3.0), y\_range=(0.0, 1.2), x\_ticks\_every=1.0); graph\_heading — a Heading that says "The Complete Field"; field\_curve — a FunctionPlot \[blue\] drawn in graph (function=\, x\_range=(0.0, 3.0)) Actions: - [08:52.635](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=532.6349166666665): field\_curve is indicated — a transient flash. - [08:54.887](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=534.8869166666666): graph moves to a new place on the board. - [08:54.887](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=534.8869166666666): inside\_label is shown on the screen, written out. ##### [09:7.654](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=547.6544166666665) Narration: At r equals R, the inside and outside formulas agree. There is no jump in the field because the charge fills a volume rather than sitting in an infinitesimally thin surface layer. Board: inside\_label — a Math \[text\] that says "$frac(E, E\_R) = frac(r, R)$"; graph — an Axes (x\_range=(0.0, 3.0), y\_range=(0.0, 1.2), x\_ticks\_every=1.0); graph\_heading — a Heading that says "The Complete Field"; field\_curve — a FunctionPlot \[blue\] drawn in graph (function=\, x\_range=(0.0, 3.0)) Actions: - [09:8.305](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=548.3049166666665): surface\_point is shown on the screen, written out. ##### [09:20.642](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=560.6424166666666) Narration: Outside, E over the surface field equals R squared over r squared. The enclosed charge has stopped growing, while the Gaussian area continues to grow as r squared. Board: inside\_label — a Math \[text\] that says "$frac(E, E\_R) = frac(r, R)$"; graph — an Axes (x\_range=(0.0, 3.0), y\_range=(0.0, 1.2), x\_ticks\_every=1.0); graph\_heading — a Heading that says "The Complete Field"; field\_curve — a FunctionPlot \[blue\] drawn in graph (function=\, x\_range=(0.0, 3.0)); surface\_point — a PlotPoint \[yellow\] labelled "r=R" drawn in graph (target='field\_curve', x=1.0) Actions: - [09:20.991](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=560.9909166666665): outside\_label is shown on the screen, written out. ##### [09:33.351](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=573.3514166666666) Narration: The method was the same in both regions: use the source symmetry to choose a concentric sphere, make E constant on that surface, and then count only the charge actually enclosed. Board: inside\_label — a Math \[text\] that says "$frac(E, E\_R) = frac(r, R)$"; outside\_label — a Math \[text\] that says "$frac(E, E\_R) = frac(R^2, r^2)$"; graph — an Axes (x\_range=(0.0, 3.0), y\_range=(0.0, 1.2), x\_ticks\_every=1.0); graph\_heading — a Heading that says "The Complete Field"; field\_curve — a FunctionPlot \[blue\] drawn in graph (function=\, x\_range=(0.0, 3.0)); surface\_point — a PlotPoint \[yellow\] labelled "r=R" drawn in graph (target='field\_curve', x=1.0) Actions: - [09:34.733](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=574.7329166666666): surface\_point is indicated — a transient flash. - [09:45.423](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=585.4225416666666): graph is hidden from the screen — left the board. - [09:45.423](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=585.4225416666666): field\_curve is hidden from the screen — graph left the board. - [09:45.423](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=585.4225416666666): surface\_point is hidden from the screen — graph left the board. - [09:45.423](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=585.4225416666666): graph\_heading is hidden from the screen — left the board. - [09:45.423](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=585.4225416666666): inside\_label is hidden from the screen — left the board. - [09:45.423](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=585.4225416666666): outside\_label is hidden from the screen — left the board. ### Scene 4: [Lines, Sheets, and the Surfaces They Choose](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=586.4642083333332) Span: 09:46.464–13:36.753 (586.4642083333332s–816.7534166666665s). #### Objects - charged\_line: a Line \[red\] labelled "lambda" drawn in line\_axes (start=(0.0, 0.0, -2.4), end=(0.0, 0.0, 2.4)) - charged\_sheet: a Plane \[red\] labelled "sigma" drawn in sheet\_axes (size=3.7, opacity=0.22) - comparison\_heading: a Heading that says "Geometry Controls the Distance Law" - line\_answer: a Math \[text\] that says "$E\_(upright("line"))(r) = frac(lambda, 2 pi epsilon\_0 r)$" - line\_arrows: a Vector \[blue\] drawn in line\_axes (start=(0.15, 0.0, -0.8), end=(1.7, 0.0, -0.8)) - line\_arrows\_2: a Vector \[blue\] drawn in line\_axes (start=(-0.15, 0.0, -0.8), end=(-1.7, 0.0, -0.8)) - line\_arrows\_3: a Vector \[blue\] drawn in line\_axes (start=(0.0, 0.15, 0.0), end=(0.0, 1.7, 0.0)) - line\_arrows\_4: a Vector \[blue\] drawn in line\_axes (start=(0.0, -0.15, 0.0), end=(0.0, -1.7, 0.0)) - line\_arrows\_5: a Vector \[blue\] drawn in line\_axes (start=(0.15, 0.0, 0.8), end=(1.7, 0.0, 0.8)) - line\_arrows\_6: a Vector \[blue\] drawn in line\_axes (start=(-0.15, 0.0, 0.8), end=(-1.7, 0.0, 0.8)) - line\_axes: an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-2.6, 2.6)) - line\_caption: a Text \[text\] that says "Line: area per length grows like $r$." - line\_case: a Math \[text\] that says "$E(r) = frac(C, r)$" - line\_cylinder: a Cylinder \[yellow\] drawn in line\_axes (start=(0.0, 0.0, -1.35), end=(0.0, 0.0, 1.35), fill\_end\_circles=False) - line\_heading: a Heading that says "An Infinite Uniform Line Charge" - line\_work: a Derivation \[text\] that says "$Phi\_E &= E (2 pi r L) \\ q\_(upright("enc")) &= lambda L \\ E (2 pi r L) &= frac(lambda L, epsilon\_0)$" - pillbox: a Cylinder \[yellow\] drawn in sheet\_axes (start=(0.0, 0.0, -0.85), end=(0.0, 0.0, 0.85), fill\_end\_circles=False) - point\_caption: a Text \[text\] that says "Point: area grows like $r^2$." - point\_case: a Math \[text\] that says "$E(r) = frac(C, r^2)$" - sheet\_answer: a Math \[text\] that says "$E\_(upright("sheet")) = frac(sigma, 2 epsilon\_0)$" - sheet\_arrows: a Vector \[blue\] drawn in sheet\_axes (start=(-1.0, -0.5, 0.1), end=(-1.0, -0.5, 1.65)) - sheet\_arrows\_2: a Vector \[blue\] drawn in sheet\_axes (start=(0.0, 0.0, 0.1), end=(0.0, 0.0, 1.65)) - sheet\_arrows\_3: a Vector \[blue\] drawn in sheet\_axes (start=(1.0, 0.5, 0.1), end=(1.0, 0.5, 1.65)) - sheet\_arrows\_4: a Vector \[blue\] drawn in sheet\_axes (start=(-1.0, -0.5, -0.1), end=(-1.0, -0.5, -1.65)) - sheet\_arrows\_5: a Vector \[blue\] drawn in sheet\_axes (start=(0.0, 0.0, -0.1), end=(0.0, 0.0, -1.65)) - sheet\_arrows\_6: a Vector \[blue\] drawn in sheet\_axes (start=(1.0, 0.5, -0.1), end=(1.0, 0.5, -1.65)) - sheet\_axes: an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-2.2, 2.2)) - sheet\_caption: a Text \[text\] that says "Sheet: the two cap areas stay fixed." - sheet\_case: a Math \[text\] that says "$E(r) = C$" - sheet\_heading: a Heading that says "An Infinite Uniform Sheet" - sheet\_work: a Derivation \[text\] that says "$Phi\_E &= E A + E A = 2 E A \\ q\_(upright("enc")) &= sigma A \\ 2 E A &= frac(sigma A, epsilon\_0)$" #### Beats ##### [09:46.464](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=586.4642083333332) Narration: Now stretch the charge distribution into an ideal infinite line with uniform charge per length lambda. Translation along the line changes nothing, and rotation around it changes nothing. Board: Empty. Actions: - [09:46.464](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=586.4642083333332): line\_heading is shown on the screen, written out. - [09:46.464](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=586.4642083333332): line\_axes is shown on the screen, written out. - [09:49.843](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=589.8432083333332): charged\_line is shown on the screen, written out. ##### [09:59.011](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=599.0107083333332) Narration: Those symmetries force the electric field to point directly away from the line. Its magnitude can depend on perpendicular distance r, but not on position along the line or angle around it. Board: line\_axes — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-2.6, 2.6)); line\_heading — a Heading that says "An Infinite Uniform Line Charge"; charged\_line — a Line \[red\] labelled "lambda" drawn in line\_axes (start=(0.0, 0.0, -2.4), end=(0.0, 0.0, 2.4)) Actions: - [10:2.61](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=602.6102083333332): line\_arrows is shown on the screen, written out. - [10:2.61](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=602.6102083333332): line\_arrows\_2 is shown on the screen, written out. - [10:2.61](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=602.6102083333332): line\_arrows\_3 is shown on the screen, written out. - [10:2.61](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=602.6102083333332): line\_arrows\_4 is shown on the screen, written out. - [10:2.61](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=602.6102083333332): line\_arrows\_5 is shown on the screen, written out. - [10:2.61](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=602.6102083333332): line\_arrows\_6 is shown on the screen, written out. ##### [10:12.103](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=612.1032083333332) Narration: Choose a cylinder centred on the line. Its curved side is everywhere the same distance r from the charge, so E has one constant magnitude there. Board: line\_axes — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-2.6, 2.6)); line\_heading — a Heading that says "An Infinite Uniform Line Charge"; charged\_line — a Line \[red\] labelled "lambda" drawn in line\_axes (start=(0.0, 0.0, -2.4), end=(0.0, 0.0, 2.4)); line\_arrows — a Vector \[blue\] drawn in line\_axes (start=(0.15, 0.0, -0.8), end=(1.7, 0.0, -0.8)); line\_arrows\_2 — a Vector \[blue\] drawn in line\_axes (start=(-0.15, 0.0, -0.8), end=(-1.7, 0.0, -0.8)); line\_arrows\_3 — a Vector \[blue\] drawn in line\_axes (start=(0.0, 0.15, 0.0), end=(0.0, 1.7, 0.0)); line\_arrows\_4 — a Vector \[blue\] drawn in line\_axes (start=(0.0, -0.15, 0.0), end=(0.0, -1.7, 0.0)); line\_arrows\_5 — a Vector \[blue\] drawn in line\_axes (start=(0.15, 0.0, 0.8), end=(1.7, 0.0, 0.8)); line\_arrows\_6 — a Vector \[blue\] drawn in line\_axes (start=(-0.15, 0.0, 0.8), end=(-1.7, 0.0, 0.8)) Actions: - [10:12.858](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=612.8582083333332): line\_cylinder is shown on the screen, faded in. ##### [10:22.154](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=622.1537083333332) Narration: Let the solid settle into view. The charged line is the cylinder's axis, and the blue field arrows point through its curved wall. Board: line\_axes — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-2.6, 2.6)); line\_heading — a Heading that says "An Infinite Uniform Line Charge"; charged\_line — a Line \[red\] labelled "lambda" drawn in line\_axes (start=(0.0, 0.0, -2.4), end=(0.0, 0.0, 2.4)); line\_arrows — a Vector \[blue\] drawn in line\_axes (start=(0.15, 0.0, -0.8), end=(1.7, 0.0, -0.8)); line\_arrows\_2 — a Vector \[blue\] drawn in line\_axes (start=(-0.15, 0.0, -0.8), end=(-1.7, 0.0, -0.8)); line\_arrows\_3 — a Vector \[blue\] drawn in line\_axes (start=(0.0, 0.15, 0.0), end=(0.0, 1.7, 0.0)); line\_arrows\_4 — a Vector \[blue\] drawn in line\_axes (start=(0.0, -0.15, 0.0), end=(0.0, -1.7, 0.0)); line\_arrows\_5 — a Vector \[blue\] drawn in line\_axes (start=(0.15, 0.0, 0.8), end=(1.7, 0.0, 0.8)); line\_arrows\_6 — a Vector \[blue\] drawn in line\_axes (start=(-0.15, 0.0, 0.8), end=(-1.7, 0.0, 0.8)); line\_cylinder — a Cylinder \[yellow\] drawn in line\_axes (start=(0.0, 0.0, -1.35), end=(0.0, 0.0, 1.35), fill\_end\_circles=False) Actions: - [10:22.154](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=622.1537083333332): line\_axes turns in its own slot. ##### [10:31.102](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=631.1017083333332) Narration: On the two end caps, the outward normals point along the line, while the electric field points radially away from it. Their dot product is zero, so the caps contribute no flux. Board: Unchanged from the preceding beat in this scene. Actions: - [10:31.891](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=631.8912083333332): line\_cylinder is indicated — a transient flash. ##### [10:43.219](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=643.2192083333332) Narration: Only the curved side contributes. Its area is circumference two pi r times length L, so the flux is E times two pi r L. Board: Unchanged from the preceding beat in this scene. Actions: - [10:50.533](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=650.5332083333332): line\_axes moves to a new place on the board. - [10:50.533](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=650.5332083333332): line\_work is shown on the screen, written out. ##### [10:54.157](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=654.1572083333332) Narration: The cylinder encloses a length L of line charge. Charge per length lambda times L gives enclosed charge lambda L. Board: Unchanged from the preceding beat in this scene. Actions: - [11:0.595](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=660.5952083333332): line\_work is shown on the screen, written out. ##### [11:3.308](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=663.3077083333332) Narration: Apply Gauss's law. The length L appears on both sides and cancels. Solving leaves lambda over two pi epsilon zero r. Board: Unchanged from the preceding beat in this scene. Actions: - [11:4.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=664.0622083333332): line\_work is shown on the screen, written out. - [11:9.507](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=669.5072083333332): line\_answer is shown on the screen, written out. ##### [11:13.323](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=673.3232083333332) Narration: The line field falls as one over r, not one over r squared. As the cylinder expands, its relevant area per unit length grows only in proportion to r. Board: line\_answer — a Math \[text\] that says "$E\_(upright("line"))(r) = frac(lambda, 2 pi epsilon\_0 r)$"; line\_axes — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-2.6, 2.6)); line\_heading — a Heading that says "An Infinite Uniform Line Charge"; charged\_line — a Line \[red\] labelled "lambda" drawn in line\_axes (start=(0.0, 0.0, -2.4), end=(0.0, 0.0, 2.4)); line\_arrows — a Vector \[blue\] drawn in line\_axes (start=(0.15, 0.0, -0.8), end=(1.7, 0.0, -0.8)); line\_arrows\_2 — a Vector \[blue\] drawn in line\_axes (start=(-0.15, 0.0, -0.8), end=(-1.7, 0.0, -0.8)); line\_arrows\_3 — a Vector \[blue\] drawn in line\_axes (start=(0.0, 0.15, 0.0), end=(0.0, 1.7, 0.0)); line\_arrows\_4 — a Vector \[blue\] drawn in line\_axes (start=(0.0, -0.15, 0.0), end=(0.0, -1.7, 0.0)); line\_arrows\_5 — a Vector \[blue\] drawn in line\_axes (start=(0.15, 0.0, 0.8), end=(1.7, 0.0, 0.8)); line\_arrows\_6 — a Vector \[blue\] drawn in line\_axes (start=(-0.15, 0.0, 0.8), end=(-1.7, 0.0, 0.8)); line\_cylinder — a Cylinder \[yellow\] drawn in line\_axes (start=(0.0, 0.0, -1.35), end=(0.0, 0.0, 1.35), fill\_end\_circles=False) Actions: - [11:14.89](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=674.8902083333332): A box is drawn around line\_answer. - [11:24.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=684.0622083333332): line\_answer is hidden from the screen — left the board. - [11:24.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=684.0622083333332): line\_axes is hidden from the screen — left the board. - [11:24.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=684.0622083333332): charged\_line is hidden from the screen — line\_axes left the board. - [11:24.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=684.0622083333332): line\_arrows is hidden from the screen — line\_axes left the board. - [11:24.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=684.0622083333332): line\_arrows\_2 is hidden from the screen — line\_axes left the board. - [11:24.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=684.0622083333332): line\_arrows\_3 is hidden from the screen — line\_axes left the board. - [11:24.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=684.0622083333332): line\_arrows\_4 is hidden from the screen — line\_axes left the board. - [11:24.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=684.0622083333332): line\_arrows\_5 is hidden from the screen — line\_axes left the board. - [11:24.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=684.0622083333332): line\_arrows\_6 is hidden from the screen — line\_axes left the board. - [11:24.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=684.0622083333332): line\_cylinder is hidden from the screen — line\_axes left the board. - [11:24.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=684.0622083333332): line\_heading is hidden from the screen — left the board. - [11:24.062](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=684.0622083333332): line\_work is hidden from the screen — left the board. ##### [11:25.262](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=685.2622083333332) Narration: Next spread charge uniformly across an ideal infinite sheet, with surface charge density sigma. Sliding anywhere within the sheet cannot change the field. Board: Empty. Actions: - [11:25.262](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=685.2622083333332): sheet\_heading is shown on the screen, written out. - [11:25.262](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=685.2622083333332): sheet\_axes is shown on the screen, written out. - [11:29.151](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=689.1512083333332): charged\_sheet is shown on the screen, faded in. ##### [11:36.439](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=696.4387083333332) Narration: Rotating the sheet within its own plane also changes nothing. The only distinguished direction is perpendicular to the sheet, so the field must point normally away on both sides. Board: sheet\_axes — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-2.2, 2.2)); sheet\_heading — a Heading that says "An Infinite Uniform Sheet"; charged\_sheet — a Plane \[red\] labelled "sigma" drawn in sheet\_axes (size=3.7, opacity=0.22) Actions: - [11:42.824](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=702.8242083333332): sheet\_arrows is shown on the screen, written out. - [11:42.824](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=702.8242083333332): sheet\_arrows\_2 is shown on the screen, written out. - [11:42.824](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=702.8242083333332): sheet\_arrows\_3 is shown on the screen, written out. - [11:42.824](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=702.8242083333332): sheet\_arrows\_4 is shown on the screen, written out. - [11:42.824](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=702.8242083333332): sheet\_arrows\_5 is shown on the screen, written out. - [11:42.824](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=702.8242083333332): sheet\_arrows\_6 is shown on the screen, written out. ##### [11:48.15](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=708.1497083333333) Narration: Choose a short cylindrical pillbox that straddles the sheet. Its flat caps are parallel to the charge distribution, and its curved wall joins them. Board: sheet\_axes — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-2.2, 2.2)); sheet\_heading — a Heading that says "An Infinite Uniform Sheet"; charged\_sheet — a Plane \[red\] labelled "sigma" drawn in sheet\_axes (size=3.7, opacity=0.22); sheet\_arrows — a Vector \[blue\] drawn in sheet\_axes (start=(-1.0, -0.5, 0.1), end=(-1.0, -0.5, 1.65)); sheet\_arrows\_2 — a Vector \[blue\] drawn in sheet\_axes (start=(0.0, 0.0, 0.1), end=(0.0, 0.0, 1.65)); sheet\_arrows\_3 — a Vector \[blue\] drawn in sheet\_axes (start=(1.0, 0.5, 0.1), end=(1.0, 0.5, 1.65)); sheet\_arrows\_4 — a Vector \[blue\] drawn in sheet\_axes (start=(-1.0, -0.5, -0.1), end=(-1.0, -0.5, -1.65)); sheet\_arrows\_5 — a Vector \[blue\] drawn in sheet\_axes (start=(0.0, 0.0, -0.1), end=(0.0, 0.0, -1.65)); sheet\_arrows\_6 — a Vector \[blue\] drawn in sheet\_axes (start=(1.0, 0.5, -0.1), end=(1.0, 0.5, -1.65)) Actions: - [11:49.81](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=709.8102083333332): pillbox is shown on the screen, faded in. ##### [11:58.293](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=718.2927083333332) Narration: A small turn shows the construction. The field passes through the two caps, while it runs parallel to the curved wall. Board: sheet\_axes — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-2.2, 2.2)); sheet\_heading — a Heading that says "An Infinite Uniform Sheet"; charged\_sheet — a Plane \[red\] labelled "sigma" drawn in sheet\_axes (size=3.7, opacity=0.22); sheet\_arrows — a Vector \[blue\] drawn in sheet\_axes (start=(-1.0, -0.5, 0.1), end=(-1.0, -0.5, 1.65)); sheet\_arrows\_2 — a Vector \[blue\] drawn in sheet\_axes (start=(0.0, 0.0, 0.1), end=(0.0, 0.0, 1.65)); sheet\_arrows\_3 — a Vector \[blue\] drawn in sheet\_axes (start=(1.0, 0.5, 0.1), end=(1.0, 0.5, 1.65)); sheet\_arrows\_4 — a Vector \[blue\] drawn in sheet\_axes (start=(-1.0, -0.5, -0.1), end=(-1.0, -0.5, -1.65)); sheet\_arrows\_5 — a Vector \[blue\] drawn in sheet\_axes (start=(0.0, 0.0, -0.1), end=(0.0, 0.0, -1.65)); sheet\_arrows\_6 — a Vector \[blue\] drawn in sheet\_axes (start=(1.0, 0.5, -0.1), end=(1.0, 0.5, -1.65)); pillbox — a Cylinder \[yellow\] drawn in sheet\_axes (start=(0.0, 0.0, -0.85), end=(0.0, 0.0, 0.85), fill\_end\_circles=False) Actions: - [11:58.293](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=718.2927083333332): sheet\_axes turns in its own slot. ##### [12:6.787](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=726.7872083333332) Narration: The curved wall contributes zero flux because its normal lies within the sheet while E is perpendicular to it. Each cap contributes E A, so the total flux is two E A. Board: Unchanged from the preceding beat in this scene. Actions: - [12:17.341](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=737.3412083333332): sheet\_axes moves to a new place on the board. - [12:17.341](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=737.3412083333332): sheet\_work is shown on the screen, written out. ##### [12:19.113](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=739.1132083333332) Narration: The pillbox encloses sheet area A, so it encloses charge sigma A. Board: Unchanged from the preceding beat in this scene. Actions: - [12:20.147](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=740.1472083333332): sheet\_work is shown on the screen, written out. ##### [12:24.642](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=744.6422083333332) Narration: Gauss's law gives two E A equals sigma A over epsilon zero. The cap area cancels, leaving E equals sigma over two epsilon zero. Board: Unchanged from the preceding beat in this scene. Actions: - [12:24.973](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=744.9732083333331): sheet\_work is shown on the screen, written out. - [12:31.393](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=751.3932083333332): sheet\_answer is shown on the screen, written out. ##### [12:35.133](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=755.1332083333332) Narration: There is no distance in the answer. For an ideal infinite sheet, moving the caps farther away does not spread a fixed bundle over a growing area. The same cap area intercepts the same flux. Board: sheet\_answer — a Math \[text\] that says "$E\_(upright("sheet")) = frac(sigma, 2 epsilon\_0)$"; sheet\_axes — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-2.2, 2.2)); sheet\_heading — a Heading that says "An Infinite Uniform Sheet"; charged\_sheet — a Plane \[red\] labelled "sigma" drawn in sheet\_axes (size=3.7, opacity=0.22); sheet\_arrows — a Vector \[blue\] drawn in sheet\_axes (start=(-1.0, -0.5, 0.1), end=(-1.0, -0.5, 1.65)); sheet\_arrows\_2 — a Vector \[blue\] drawn in sheet\_axes (start=(0.0, 0.0, 0.1), end=(0.0, 0.0, 1.65)); sheet\_arrows\_3 — a Vector \[blue\] drawn in sheet\_axes (start=(1.0, 0.5, 0.1), end=(1.0, 0.5, 1.65)); sheet\_arrows\_4 — a Vector \[blue\] drawn in sheet\_axes (start=(-1.0, -0.5, -0.1), end=(-1.0, -0.5, -1.65)); sheet\_arrows\_5 — a Vector \[blue\] drawn in sheet\_axes (start=(0.0, 0.0, -0.1), end=(0.0, 0.0, -1.65)); sheet\_arrows\_6 — a Vector \[blue\] drawn in sheet\_axes (start=(1.0, 0.5, -0.1), end=(1.0, 0.5, -1.65)); pillbox — a Cylinder \[yellow\] drawn in sheet\_axes (start=(0.0, 0.0, -0.85), end=(0.0, 0.0, 0.85), fill\_end\_circles=False) Actions: - [12:35.859](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=755.8592083333332): A box is drawn around sheet\_answer. - [12:48.026](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=768.0262083333332): sheet\_answer is hidden from the screen — left the board. - [12:48.026](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=768.0262083333332): sheet\_axes is hidden from the screen — left the board. - [12:48.026](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=768.0262083333332): charged\_sheet is hidden from the screen — sheet\_axes left the board. - [12:48.026](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=768.0262083333332): sheet\_arrows is hidden from the screen — sheet\_axes left the board. - [12:48.026](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=768.0262083333332): sheet\_arrows\_2 is hidden from the screen — sheet\_axes left the board. - [12:48.026](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=768.0262083333332): sheet\_arrows\_3 is hidden from the screen — sheet\_axes left the board. - [12:48.026](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=768.0262083333332): sheet\_arrows\_4 is hidden from the screen — sheet\_axes left the board. - [12:48.026](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=768.0262083333332): sheet\_arrows\_5 is hidden from the screen — sheet\_axes left the board. - [12:48.026](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=768.0262083333332): sheet\_arrows\_6 is hidden from the screen — sheet\_axes left the board. - [12:48.026](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=768.0262083333332): pillbox is hidden from the screen — sheet\_axes left the board. - [12:48.026](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=768.0262083333332): sheet\_heading is hidden from the screen — left the board. - [12:48.026](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=768.0262083333332): sheet\_work is hidden from the screen — left the board. ##### [12:49.226](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=769.2262083333331) Narration: Put the three geometries together. A point spreads flux over a sphere whose area grows as r squared, so its field has the form a constant over r squared. Board: Empty. Actions: - [12:49.226](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=769.2262083333331): comparison\_heading is shown on the screen, written out. - [12:52.419](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=772.4192083333331): point\_case is shown on the screen, written out. - [12:54.009](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=774.0092083333332): point\_caption is shown on the screen, written out. ##### [13:1.088](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=781.0882083333331) Narration: A line spreads flux over the curved wall of a cylinder. Per unit length, that area grows as r, so the field has the form a constant over r. Board: point\_case — a Math \[text\] that says "$E(r) = frac(C, r^2)$"; point\_caption — a Text \[text\] that says "Point: area grows like $r^2$."; comparison\_heading — a Heading that says "Geometry Controls the Distance Law" Actions: - [13:1.738](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=781.7382083333332): line\_case is shown on the screen, written out. - [13:4.292](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=784.2922083333332): line\_caption is shown on the screen, written out. ##### [13:12.415](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=792.4152083333331) Narration: A sheet sends flux through two equal caps. Their area does not change when the pillbox grows taller, so the ideal sheet field is constant. Board: point\_case — a Math \[text\] that says "$E(r) = frac(C, r^2)$"; point\_caption — a Text \[text\] that says "Point: area grows like $r^2$."; line\_case — a Math \[text\] that says "$E(r) = frac(C, r)$"; line\_caption — a Text \[text\] that says "Line: area per length grows like $r$."; comparison\_heading — a Heading that says "Geometry Controls the Distance Law" Actions: - [13:12.926](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=792.9262083333331): sheet\_case is shown on the screen, written out. - [13:14.865](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=794.8652083333332): sheet\_caption is shown on the screen, written out. ##### [13:22.872](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=802.8722083333332) Narration: Gauss's law was identical in all three cases. What changed was the symmetry, and symmetry determined the surface on which E became constant and the unwanted pieces contributed zero. Board: point\_case — a Math \[text\] that says "$E(r) = frac(C, r^2)$"; point\_caption — a Text \[text\] that says "Point: area grows like $r^2$."; line\_case — a Math \[text\] that says "$E(r) = frac(C, r)$"; line\_caption — a Text \[text\] that says "Line: area per length grows like $r$."; sheet\_case — a Math \[text\] that says "$E(r) = C$"; sheet\_caption — a Text \[text\] that says "Sheet: the two cap areas stay fixed."; comparison\_heading — a Heading that says "Geometry Controls the Distance Law" Actions: - [13:28.084](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=808.0842083333331): point\_case is indicated — a transient flash. - [13:28.284](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=808.2842083333331): line\_case is indicated — a transient flash. - [13:28.484](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=808.4842083333331): sheet\_case is indicated — a transient flash. - [13:35.712](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=815.7117499999998): comparison\_heading is hidden from the screen — left the board. - [13:35.712](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=815.7117499999998): line\_caption is hidden from the screen — left the board. - [13:35.712](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=815.7117499999998): line\_case is hidden from the screen — left the board. - [13:35.712](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=815.7117499999998): point\_caption is hidden from the screen — left the board. - [13:35.712](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=815.7117499999998): point\_case is hidden from the screen — left the board. - [13:35.712](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=815.7117499999998): sheet\_caption is hidden from the screen — left the board. - [13:35.712](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=815.7117499999998): sheet\_case is hidden from the screen — left the board. ### Scene 5: [Conductors, Surface Charge, and the Car](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=816.7534166666665) Span: 13:36.753–17:8.812 (816.7534166666665s–1028.8116041666665s). #### Objects - body: a Polygon \[blue\] labelled "upright("metal body")" drawn in car (vertices=((0.7, 1.0), (1.0, 2.3), (2.1, 2.6), (3.0, 3.6), (5.2, 3.6), (6…, fill\_opacity=0.22) - bolt\_one: a Line \[yellow\] drawn in car (start=(4.7, 5.1), end=(4.25, 4.45)) - bolt\_three: a Line \[yellow\] drawn in car (start=(4.65, 4.0), end=(4.25, 3.6)) - bolt\_two: a Line \[yellow\] drawn in car (start=(4.25, 4.45), end=(4.65, 4.0)) - cabin: a Polygon \[gray\] labelled "upright("passenger space")" drawn in car (vertices=((2.2, 2.5), (3.1, 3.4), (5.0, 3.4), (5.9, 2.5)), filled=False, dashed=True) - cabin\_field: a Math \[text\] that says "$arrow(E) approx arrow(0) quad upright("in the protected interior")$" - car: a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.2), aspect=(8.0, 5.2)) - car\_heading: a Heading that says "Why a Metal Car Protects Its Interior" - conductor\_heading: a Heading that says "Electrostatic Equilibrium in a Conductor" - conductor\_work: a Derivation \[text\] that says "$Phi\_E = integral.double\_S arrow(E) dot hat(n) thin dif A \\ &= 0 \\ q\_(upright("enc")) = epsilon\_0 Phi\_E = 0$" - equilibrium\_note: a Panel that says "In electrostatic equilibrium, mobile charge has finished rearranging. The electric field inside the conducting material is zero." - inner\_gaussian: a Circle \[gray\] labelled "S" drawn in metal\_picture (radius=0.72) - left\_current: a Vector \[red\] labelled "upright("surface current")" drawn in car (start=(4.2, 3.55), end=(1.15, 2.15)) - metal: a Circle \[blue\] drawn in metal\_picture (radius=1.35, filled=True) - metal\_picture: a Figure (x\_range=(-2.7, 2.7), y\_range=(-2.3, 2.3), aspect=(5.4, 4.6)) - outer\_fields: a Vector \[yellow\] drawn in metal\_picture (start=(1.42, 0.0), end=(2.25, 0.0)) - outer\_fields\_2: a Vector \[yellow\] drawn in metal\_picture (start=(-1.42, 0.0), end=(-2.25, 0.0)) - outer\_fields\_3: a Vector \[yellow\] drawn in metal\_picture (start=(0.0, 1.42), end=(0.0, 2.1)) - outer\_fields\_4: a Vector \[yellow\] drawn in metal\_picture (start=(0.0, -1.42), end=(0.0, -2.1)) - outer\_fields\_5: a Vector \[yellow\] drawn in metal\_picture (start=(1.0, 1.0), end=(1.62, 1.62)) - outer\_fields\_6: a Vector \[yellow\] drawn in metal\_picture (start=(-1.0, 1.0), end=(-1.62, 1.62)) - outer\_fields\_7: a Vector \[yellow\] drawn in metal\_picture (start=(1.0, -1.0), end=(1.62, -1.62)) - outer\_fields\_8: a Vector \[yellow\] drawn in metal\_picture (start=(-1.0, -1.0), end=(-1.62, -1.62)) - passenger\_left: a Point \[green\] drawn in car (location=(3.5, 2.65)) - passenger\_right: a Point \[green\] drawn in car (location=(4.55, 2.65)) - recap: a Block \[text\] that says "Flux through a closed surface counts enclosed charge. Symmetry chooses a surface on which the flux is easy to calculate. Mobile charge in a conductor rearranges until the field in the metal is zero." - recap\_heading: a Heading that says "What to Carry Away" - right\_current: a Vector \[red\] drawn in car (start=(4.35, 3.55), end=(6.85, 2.1)) - safety\_note: a Text \[text\] that says "The shell redirects charge and current around the passenger compartment. The protection comes from the metal body, not from the tires." - surface\_charges: a Point \[red\] drawn in metal\_picture (location=(1.35, 0.0)) - surface\_charges\_2: a Point \[red\] drawn in metal\_picture (location=(1.0341599982106204, 0.8677632730768281)) - surface\_charges\_3: a Point \[red\] drawn in metal\_picture (location=(0.23442503985035607, 1.3294904665664808)) - surface\_charges\_4: a Point \[red\] drawn in metal\_picture (location=(-0.6749999999999997, 1.1691342951089922)) - surface\_charges\_5: a Point \[red\] drawn in metal\_picture (location=(-1.2685850380609762, 0.461727193489653)) - surface\_charges\_6: a Point \[red\] drawn in metal\_picture (location=(-1.2685850380609764, -0.46172719348965274)) - surface\_charges\_7: a Point \[red\] drawn in metal\_picture (location=(-0.6750000000000006, -1.1691342951089918)) - surface\_charges\_8: a Point \[red\] drawn in metal\_picture (location=(0.23442503985035548, -1.329490466566481)) - surface\_charges\_9: a Point \[red\] drawn in metal\_picture (location=(1.03415999821062, -0.8677632730768285)) - surface\_heading: a Heading that says "Where the Excess Charge Goes" - surface\_result: a Math \[text\] that says "$q\_(upright("bulk")) = 0$" - wheel\_left: a Circle \[gray\] drawn in car (center=(2.0, 0.85), radius=0.55) - wheel\_right: a Circle \[gray\] drawn in car (center=(6.0, 0.85), radius=0.55) - zero\_field: a Math \[text\] that says "$arrow(E) = arrow(0) quad upright("inside conducting material")$" #### Beats ##### [13:36.753](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=816.7534166666665) Narration: A conductor contains mobile charge. If an electric field existed inside the metal, that charge would feel a force and begin to move. A state with moving charge is not electrostatic equilibrium. Board: Empty. Actions: - [13:36.753](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=816.7534166666665): conductor\_heading is shown on the screen, written out. - [13:36.916](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=816.9164166666665): metal\_picture is shown on the screen, written out. - [13:41.293](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=821.2934166666664): metal is shown on the screen, faded in. ##### [13:49.428](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=829.4279166666664) Narration: The mobile charge rearranges on an extremely short time scale. Its own electric field opposes the field that drove the motion, and equilibrium is reached only when the net electric field inside the conducting material is zero. Board: metal\_picture — a Figure (x\_range=(-2.7, 2.7), y\_range=(-2.3, 2.3), aspect=(5.4, 4.6)); conductor\_heading — a Heading that says "Electrostatic Equilibrium in a Conductor"; metal — a Circle \[blue\] drawn in metal\_picture (radius=1.35, filled=True) Actions: - [13:57.937](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=837.9374166666664): metal\_picture moves to a new place on the board. - [13:57.937](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=837.9374166666664): equilibrium\_note is shown on the screen, written out. - [14:2.21](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=842.2104166666664): zero\_field is shown on the screen, written out. ##### [14:3.669](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=843.6694166666665) Narration: That statement comes from the physics of mobile charge, not from Gauss's law alone. Gauss's law now tells us an important consequence of the zero field. Board: equilibrium\_note — a Panel that says "In electrostatic equilibrium, mobile charge has finished rearranging. The electric field inside the conducting material is zero."; zero\_field — a Math \[text\] that says "$arrow(E) = arrow(0) quad upright("inside conducting material")$"; metal\_picture — a Figure (x\_range=(-2.7, 2.7), y\_range=(-2.3, 2.3), aspect=(5.4, 4.6)); conductor\_heading — a Heading that says "Electrostatic Equilibrium in a Conductor"; metal — a Circle \[blue\] drawn in metal\_picture (radius=1.35, filled=True) Actions: - [14:12.167](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=852.1674166666664): zero\_field is indicated — a transient flash. - [14:13.351](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=853.3514166666664): metal\_picture moves to a new place on the board. - [14:13.351](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=853.3514166666664): conductor\_heading is hidden from the screen — left the board. - [14:13.351](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=853.3514166666664): equilibrium\_note is hidden from the screen — left the board. - [14:13.351](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=853.3514166666664): zero\_field is hidden from the screen — left the board. ##### [14:13.951](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=853.9514166666664) Narration: Draw any closed Gaussian surface lying completely inside the conducting material. Because E is zero at every point on it, its electric flux is zero. Board: metal\_picture — a Figure (x\_range=(-2.7, 2.7), y\_range=(-2.3, 2.3), aspect=(5.4, 4.6)); metal — a Circle \[blue\] drawn in metal\_picture (radius=1.35, filled=True) Actions: - [14:13.951](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=853.9514166666664): surface\_heading is shown on the screen, written out. - [14:15.298](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=855.2984166666664): inner\_gaussian is shown on the screen, written out. - [14:20.163](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=860.1634166666664): conductor\_work is shown on the screen, written out. - [14:22.578](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=862.5784166666665): conductor\_work is shown on the screen, written out. ##### [14:24.548](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=864.5479166666664) Narration: Gauss's law then says that the net charge enclosed by every such interior surface is zero. Board: metal\_picture — a Figure (x\_range=(-2.7, 2.7), y\_range=(-2.3, 2.3), aspect=(5.4, 4.6)); metal — a Circle \[blue\] drawn in metal\_picture (radius=1.35, filled=True); surface\_heading — a Heading that says "Where the Excess Charge Goes"; inner\_gaussian — a Circle \[gray\] labelled "S" drawn in metal\_picture (radius=0.72) Actions: - [14:27.032](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=867.0324166666664): conductor\_work is shown on the screen, written out. ##### [14:31.011](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=871.0109166666665) Narration: Excess charge therefore cannot remain distributed through the bulk of the metal in electrostatic equilibrium. It moves outward and comes to rest on the conductor's surface. Board: Unchanged from the preceding beat in this scene. Actions: - [14:34.087](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=874.0874166666664): surface\_result is shown on the screen, written out. - [14:40.159](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=880.1594166666664): surface\_charges is shown on the screen, written out. - [14:40.159](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=880.1594166666664): surface\_charges\_2 is shown on the screen, written out. - [14:40.159](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=880.1594166666664): surface\_charges\_3 is shown on the screen, written out. - [14:40.159](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=880.1594166666664): surface\_charges\_4 is shown on the screen, written out. - [14:40.159](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=880.1594166666664): surface\_charges\_5 is shown on the screen, written out. - [14:40.159](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=880.1594166666664): surface\_charges\_6 is shown on the screen, written out. - [14:40.159](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=880.1594166666664): surface\_charges\_7 is shown on the screen, written out. - [14:40.159](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=880.1594166666664): surface\_charges\_8 is shown on the screen, written out. - [14:40.159](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=880.1594166666664): surface\_charges\_9 is shown on the screen, written out. ##### [14:41.723](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=881.7234166666665) Narration: Outside the conductor, those surface charges can produce an electric field. Inside the conducting material, they have arranged themselves so that their combined field cancels. Board: metal\_picture — a Figure (x\_range=(-2.7, 2.7), y\_range=(-2.3, 2.3), aspect=(5.4, 4.6)); metal — a Circle \[blue\] drawn in metal\_picture (radius=1.35, filled=True); surface\_result — a Math \[text\] that says "$q\_(upright("bulk")) = 0$"; surface\_heading — a Heading that says "Where the Excess Charge Goes"; inner\_gaussian — a Circle \[gray\] labelled "S" drawn in metal\_picture (radius=0.72); surface\_charges — a Point \[red\] drawn in metal\_picture (location=(1.35, 0.0)); surface\_charges\_2 — a Point \[red\] drawn in metal\_picture (location=(1.0341599982106204, 0.8677632730768281)); surface\_charges\_3 — a Point \[red\] drawn in metal\_picture (location=(0.23442503985035607, 1.3294904665664808)); surface\_charges\_4 — a Point \[red\] drawn in metal\_picture (location=(-0.6749999999999997, 1.1691342951089922)); surface\_charges\_5 — a Point \[red\] drawn in metal\_picture (location=(-1.2685850380609762, 0.461727193489653)); surface\_charges\_6 — a Point \[red\] drawn in metal\_picture (location=(-1.2685850380609764, -0.46172719348965274)); surface\_charges\_7 — a Point \[red\] drawn in metal\_picture (location=(-0.6750000000000006, -1.1691342951089918)); surface\_charges\_8 — a Point \[red\] drawn in metal\_picture (location=(0.23442503985035548, -1.329490466566481)); surface\_charges\_9 — a Point \[red\] drawn in metal\_picture (location=(1.03415999821062, -0.8677632730768285)) Actions: - [14:42.071](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=882.0714166666664): outer\_fields is shown on the screen, written out. - [14:42.071](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=882.0714166666664): outer\_fields\_2 is shown on the screen, written out. - [14:42.071](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=882.0714166666664): outer\_fields\_3 is shown on the screen, written out. - [14:42.071](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=882.0714166666664): outer\_fields\_4 is shown on the screen, written out. - [14:42.071](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=882.0714166666664): outer\_fields\_5 is shown on the screen, written out. - [14:42.071](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=882.0714166666664): outer\_fields\_6 is shown on the screen, written out. - [14:42.071](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=882.0714166666664): outer\_fields\_7 is shown on the screen, written out. - [14:42.071](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=882.0714166666664): outer\_fields\_8 is shown on the screen, written out. ##### [14:52.946](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=892.9464166666664) Narration: The surface distribution need not be uniform. Charge crowds more strongly near sharp points, where the surrounding field can become especially large. But the equilibrium condition inside the metal remains E equals zero. Board: metal\_picture — a Figure (x\_range=(-2.7, 2.7), y\_range=(-2.3, 2.3), aspect=(5.4, 4.6)); metal — a Circle \[blue\] drawn in metal\_picture (radius=1.35, filled=True); surface\_result — a Math \[text\] that says "$q\_(upright("bulk")) = 0$"; surface\_heading — a Heading that says "Where the Excess Charge Goes"; inner\_gaussian — a Circle \[gray\] labelled "S" drawn in metal\_picture (radius=0.72); surface\_charges — a Point \[red\] drawn in metal\_picture (location=(1.35, 0.0)); surface\_charges\_2 — a Point \[red\] drawn in metal\_picture (location=(1.0341599982106204, 0.8677632730768281)); surface\_charges\_3 — a Point \[red\] drawn in metal\_picture (location=(0.23442503985035607, 1.3294904665664808)); surface\_charges\_4 — a Point \[red\] drawn in metal\_picture (location=(-0.6749999999999997, 1.1691342951089922)); surface\_charges\_5 — a Point \[red\] drawn in metal\_picture (location=(-1.2685850380609762, 0.461727193489653)); surface\_charges\_6 — a Point \[red\] drawn in metal\_picture (location=(-1.2685850380609764, -0.46172719348965274)); surface\_charges\_7 — a Point \[red\] drawn in metal\_picture (location=(-0.6750000000000006, -1.1691342951089918)); surface\_charges\_8 — a Point \[red\] drawn in metal\_picture (location=(0.23442503985035548, -1.329490466566481)); surface\_charges\_9 — a Point \[red\] drawn in metal\_picture (location=(1.03415999821062, -0.8677632730768285)); outer\_fields — a Vector \[yellow\] drawn in metal\_picture (start=(1.42, 0.0), end=(2.25, 0.0)); outer\_fields\_2 — a Vector \[yellow\] drawn in metal\_picture (start=(-1.42, 0.0), end=(-2.25, 0.0)); outer\_fields\_3 — a Vector \[yellow\] drawn in metal\_picture (start=(0.0, 1.42), end=(0.0, 2.1)); outer\_fields\_4 — a Vector \[yellow\] drawn in metal\_picture (start=(0.0, -1.42), end=(0.0, -2.1)); outer\_fields\_5 — a Vector \[yellow\] drawn in metal\_picture (start=(1.0, 1.0), end=(1.62, 1.62)); outer\_fields\_6 — a Vector \[yellow\] drawn in metal\_picture (start=(-1.0, 1.0), end=(-1.62, 1.62)); outer\_fields\_7 — a Vector \[yellow\] drawn in metal\_picture (start=(1.0, -1.0), end=(1.62, -1.62)); outer\_fields\_8 — a Vector \[yellow\] drawn in metal\_picture (start=(-1.0, -1.0), end=(-1.62, -1.62)) Actions: - [15:5.241](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=905.2414166666665): A box is drawn around surface\_result. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): conductor\_work is hidden from the screen — left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): metal\_picture is hidden from the screen — left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): metal is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): inner\_gaussian is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): surface\_charges is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): surface\_charges\_2 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): surface\_charges\_3 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): surface\_charges\_4 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): surface\_charges\_5 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): surface\_charges\_6 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): surface\_charges\_7 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): surface\_charges\_8 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): surface\_charges\_9 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): outer\_fields is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): outer\_fields\_2 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): outer\_fields\_3 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): outer\_fields\_4 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): outer\_fields\_5 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): outer\_fields\_6 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): outer\_fields\_7 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): outer\_fields\_8 is hidden from the screen — metal\_picture left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): surface\_heading is hidden from the screen — left the board. - [15:6.065](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=906.0654166666665): surface\_result is hidden from the screen — left the board. ##### [15:7.265](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=907.2654166666664) Narration: Now replace the simple conductor with the metal body of a hard-top car. The body forms a conducting shell around the passenger compartment. Board: Empty. Actions: - [15:7.265](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=907.2654166666664): car\_heading is shown on the screen, written out. - [15:7.265](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=907.2654166666664): car is shown on the screen, written out. - [15:9.912](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=909.9124166666664): body is shown on the screen, faded in. - [15:14.161](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=914.1614166666665): cabin is shown on the screen, written out. - [15:14.161](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=914.1614166666665): wheel\_left is shown on the screen, written out. - [15:14.161](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=914.1614166666665): wheel\_right is shown on the screen, written out. - [15:14.161](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=914.1614166666665): passenger\_left is shown on the screen, written out. - [15:14.161](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=914.1614166666665): passenger\_right is shown on the screen, written out. ##### [15:16.224](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=916.2244166666665) Narration: Suppose lightning strikes the roof. The lightning delivers charge and a large current to the exterior metal. Board: car — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.2), aspect=(8.0, 5.2)); car\_heading — a Heading that says "Why a Metal Car Protects Its Interior"; body — a Polygon \[blue\] labelled "upright("metal body")" drawn in car (vertices=((0.7, 1.0), (1.0, 2.3), (2.1, 2.6), (3.0, 3.6), (5.2, 3.6), (6…, fill\_opacity=0.22); cabin — a Polygon \[gray\] labelled "upright("passenger space")" drawn in car (vertices=((2.2, 2.5), (3.1, 3.4), (5.0, 3.4), (5.9, 2.5)), filled=False, dashed=True); wheel\_left — a Circle \[gray\] drawn in car (center=(2.0, 0.85), radius=0.55); wheel\_right — a Circle \[gray\] drawn in car (center=(6.0, 0.85), radius=0.55); passenger\_left — a Point \[green\] drawn in car (location=(3.5, 2.65)); passenger\_right — a Point \[green\] drawn in car (location=(4.55, 2.65)) Actions: - [15:17.06](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=917.0604166666665): bolt\_one is shown on the screen, written out. - [15:17.06](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=917.0604166666665): bolt\_two is shown on the screen, written out. - [15:17.06](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=917.0604166666665): bolt\_three is shown on the screen, written out. ##### [15:23.768](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=923.7679166666665) Narration: The charge spreads over the outside, and the current finds conducting paths along the exterior body. The metal shell carries the dangerous electrical disturbance around the passenger space rather than through it. Board: car — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.2), aspect=(8.0, 5.2)); car\_heading — a Heading that says "Why a Metal Car Protects Its Interior"; body — a Polygon \[blue\] labelled "upright("metal body")" drawn in car (vertices=((0.7, 1.0), (1.0, 2.3), (2.1, 2.6), (3.0, 3.6), (5.2, 3.6), (6…, fill\_opacity=0.22); cabin — a Polygon \[gray\] labelled "upright("passenger space")" drawn in car (vertices=((2.2, 2.5), (3.1, 3.4), (5.0, 3.4), (5.9, 2.5)), filled=False, dashed=True); wheel\_left — a Circle \[gray\] drawn in car (center=(2.0, 0.85), radius=0.55); wheel\_right — a Circle \[gray\] drawn in car (center=(6.0, 0.85), radius=0.55); passenger\_left — a Point \[green\] drawn in car (location=(3.5, 2.65)); passenger\_right — a Point \[green\] drawn in car (location=(4.55, 2.65)); bolt\_one — a Line \[yellow\] drawn in car (start=(4.7, 5.1), end=(4.25, 4.45)); bolt\_two — a Line \[yellow\] drawn in car (start=(4.25, 4.45), end=(4.65, 4.0)); bolt\_three — a Line \[yellow\] drawn in car (start=(4.65, 4.0), end=(4.25, 3.6)) Actions: - [15:25.52](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=925.5204166666665): left\_current is shown on the screen, written out. - [15:25.52](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=925.5204166666665): right\_current is shown on the screen, written out. ##### [15:36.617](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=936.6169166666665) Narration: The passenger compartment is therefore close to one electric potential, with a strongly reduced electric field inside. This shielding behavior is often called the Faraday-cage effect. Board: car — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.2), aspect=(8.0, 5.2)); car\_heading — a Heading that says "Why a Metal Car Protects Its Interior"; body — a Polygon \[blue\] labelled "upright("metal body")" drawn in car (vertices=((0.7, 1.0), (1.0, 2.3), (2.1, 2.6), (3.0, 3.6), (5.2, 3.6), (6…, fill\_opacity=0.22); cabin — a Polygon \[gray\] labelled "upright("passenger space")" drawn in car (vertices=((2.2, 2.5), (3.1, 3.4), (5.0, 3.4), (5.9, 2.5)), filled=False, dashed=True); wheel\_left — a Circle \[gray\] drawn in car (center=(2.0, 0.85), radius=0.55); wheel\_right — a Circle \[gray\] drawn in car (center=(6.0, 0.85), radius=0.55); passenger\_left — a Point \[green\] drawn in car (location=(3.5, 2.65)); passenger\_right — a Point \[green\] drawn in car (location=(4.55, 2.65)); bolt\_one — a Line \[yellow\] drawn in car (start=(4.7, 5.1), end=(4.25, 4.45)); bolt\_two — a Line \[yellow\] drawn in car (start=(4.25, 4.45), end=(4.65, 4.0)); bolt\_three — a Line \[yellow\] drawn in car (start=(4.65, 4.0), end=(4.25, 3.6)); left\_current — a Vector \[red\] labelled "upright("surface current")" drawn in car (start=(4.2, 3.55), end=(1.15, 2.15)); right\_current — a Vector \[red\] drawn in car (start=(4.35, 3.55), end=(6.85, 2.1)) Actions: - [15:41.4](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=941.4004166666665): cabin\_field is shown on the screen, written out. - [15:44.058](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=944.0584166666665): cabin is indicated — a transient flash. ##### [15:48.049](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=948.0489166666664) Narration: The protection comes from the continuous metal shell, not primarily from the rubber tires. During a storm, occupants should remain inside with windows closed and avoid touching metal parts connected to the exterior. Board: car — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.2), aspect=(8.0, 5.2)); cabin\_field — a Math \[text\] that says "$arrow(E) approx arrow(0) quad upright("in the protected interior")$"; car\_heading — a Heading that says "Why a Metal Car Protects Its Interior"; body — a Polygon \[blue\] labelled "upright("metal body")" drawn in car (vertices=((0.7, 1.0), (1.0, 2.3), (2.1, 2.6), (3.0, 3.6), (5.2, 3.6), (6…, fill\_opacity=0.22); cabin — a Polygon \[gray\] labelled "upright("passenger space")" drawn in car (vertices=((2.2, 2.5), (3.1, 3.4), (5.0, 3.4), (5.9, 2.5)), filled=False, dashed=True); wheel\_left — a Circle \[gray\] drawn in car (center=(2.0, 0.85), radius=0.55); wheel\_right — a Circle \[gray\] drawn in car (center=(6.0, 0.85), radius=0.55); passenger\_left — a Point \[green\] drawn in car (location=(3.5, 2.65)); passenger\_right — a Point \[green\] drawn in car (location=(4.55, 2.65)); bolt\_one — a Line \[yellow\] drawn in car (start=(4.7, 5.1), end=(4.25, 4.45)); bolt\_two — a Line \[yellow\] drawn in car (start=(4.25, 4.45), end=(4.65, 4.0)); bolt\_three — a Line \[yellow\] drawn in car (start=(4.65, 4.0), end=(4.25, 3.6)); left\_current — a Vector \[red\] labelled "upright("surface current")" drawn in car (start=(4.2, 3.55), end=(1.15, 2.15)); right\_current — a Vector \[red\] drawn in car (start=(4.35, 3.55), end=(6.85, 2.1)) Actions: - [15:48.56](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=948.5604166666665): safety\_note is shown on the screen, written out. ##### [16:1.664](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=961.6639166666664) Narration: The same principle is used deliberately in shielded rooms, cable coverings, and metal enclosures around sensitive electronics. Conductors rearrange charge so that their protected interiors experience very little electric field. Board: car — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.2), aspect=(8.0, 5.2)); cabin\_field — a Math \[text\] that says "$arrow(E) approx arrow(0) quad upright("in the protected interior")$"; safety\_note — a Text \[text\] that says "The shell redirects charge and current around the passenger compartment. The protection comes from the metal body, not from the tires."; car\_heading — a Heading that says "Why a Metal Car Protects Its Interior"; body — a Polygon \[blue\] labelled "upright("metal body")" drawn in car (vertices=((0.7, 1.0), (1.0, 2.3), (2.1, 2.6), (3.0, 3.6), (5.2, 3.6), (6…, fill\_opacity=0.22); cabin — a Polygon \[gray\] labelled "upright("passenger space")" drawn in car (vertices=((2.2, 2.5), (3.1, 3.4), (5.0, 3.4), (5.9, 2.5)), filled=False, dashed=True); wheel\_left — a Circle \[gray\] drawn in car (center=(2.0, 0.85), radius=0.55); wheel\_right — a Circle \[gray\] drawn in car (center=(6.0, 0.85), radius=0.55); passenger\_left — a Point \[green\] drawn in car (location=(3.5, 2.65)); passenger\_right — a Point \[green\] drawn in car (location=(4.55, 2.65)); bolt\_one — a Line \[yellow\] drawn in car (start=(4.7, 5.1), end=(4.25, 4.45)); bolt\_two — a Line \[yellow\] drawn in car (start=(4.25, 4.45), end=(4.65, 4.0)); bolt\_three — a Line \[yellow\] drawn in car (start=(4.65, 4.0), end=(4.25, 3.6)); left\_current — a Vector \[red\] labelled "upright("surface current")" drawn in car (start=(4.2, 3.55), end=(1.15, 2.15)); right\_current — a Vector \[red\] drawn in car (start=(4.35, 3.55), end=(6.85, 2.1)) Actions: - [16:12.229](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=972.2294166666665): cabin\_field is indicated — a transient flash. - [16:15.155](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=975.1549166666664): cabin\_field moves to a new place on the board. - [16:15.155](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=975.1549166666664): car moves to a new place on the board. - [16:15.155](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=975.1549166666664): car\_heading is hidden from the screen — left the board. - [16:15.155](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=975.1549166666664): safety\_note is hidden from the screen — left the board. - [16:15.155](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=975.1549166666664): recap is shown on the screen, written out. ##### [16:16.355](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=976.3549166666664) Narration: Three ideas organize the whole lecture. First, electric flux is the signed amount of electric field crossing a surface, and the flux through a closed surface counts net enclosed charge. Board: car — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.2), aspect=(8.0, 5.2)); cabin\_field — a Math \[text\] that says "$arrow(E) approx arrow(0) quad upright("in the protected interior")$"; body — a Polygon \[blue\] labelled "upright("metal body")" drawn in car (vertices=((0.7, 1.0), (1.0, 2.3), (2.1, 2.6), (3.0, 3.6), (5.2, 3.6), (6…, fill\_opacity=0.22); cabin — a Polygon \[gray\] labelled "upright("passenger space")" drawn in car (vertices=((2.2, 2.5), (3.1, 3.4), (5.0, 3.4), (5.9, 2.5)), filled=False, dashed=True); wheel\_left — a Circle \[gray\] drawn in car (center=(2.0, 0.85), radius=0.55); wheel\_right — a Circle \[gray\] drawn in car (center=(6.0, 0.85), radius=0.55); passenger\_left — a Point \[green\] drawn in car (location=(3.5, 2.65)); passenger\_right — a Point \[green\] drawn in car (location=(4.55, 2.65)); bolt\_one — a Line \[yellow\] drawn in car (start=(4.7, 5.1), end=(4.25, 4.45)); bolt\_two — a Line \[yellow\] drawn in car (start=(4.25, 4.45), end=(4.65, 4.0)); bolt\_three — a Line \[yellow\] drawn in car (start=(4.65, 4.0), end=(4.25, 3.6)); left\_current — a Vector \[red\] labelled "upright("surface current")" drawn in car (start=(4.2, 3.55), end=(1.15, 2.15)); right\_current — a Vector \[red\] drawn in car (start=(4.35, 3.55), end=(6.85, 2.1)); recap — a Block \[text\] that says "Flux through a closed surface counts enclosed charge. Symmetry chooses a surface on which the flux is easy to calculate. Mobile charge in a conductor rearranges until the field in the metal is zero." Actions: - [16:16.355](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=976.3549166666664): recap\_heading is shown on the screen, written out. - [16:19.489](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=979.4894166666664): recap (the "Flux through a closed surface counts enclosed charge." part) is indicated — a transient flash. ##### [16:27.775](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=987.7749166666665) Narration: Second, Gauss's law becomes a field-solving tool only when symmetry chooses a useful surface: a sphere for spherical charge, a cylinder for a line, and a pillbox for a sheet. Board: car — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.2), aspect=(8.0, 5.2)); cabin\_field — a Math \[text\] that says "$arrow(E) approx arrow(0) quad upright("in the protected interior")$"; body — a Polygon \[blue\] labelled "upright("metal body")" drawn in car (vertices=((0.7, 1.0), (1.0, 2.3), (2.1, 2.6), (3.0, 3.6), (5.2, 3.6), (6…, fill\_opacity=0.22); cabin — a Polygon \[gray\] labelled "upright("passenger space")" drawn in car (vertices=((2.2, 2.5), (3.1, 3.4), (5.0, 3.4), (5.9, 2.5)), filled=False, dashed=True); wheel\_left — a Circle \[gray\] drawn in car (center=(2.0, 0.85), radius=0.55); wheel\_right — a Circle \[gray\] drawn in car (center=(6.0, 0.85), radius=0.55); passenger\_left — a Point \[green\] drawn in car (location=(3.5, 2.65)); passenger\_right — a Point \[green\] drawn in car (location=(4.55, 2.65)); bolt\_one — a Line \[yellow\] drawn in car (start=(4.7, 5.1), end=(4.25, 4.45)); bolt\_two — a Line \[yellow\] drawn in car (start=(4.25, 4.45), end=(4.65, 4.0)); bolt\_three — a Line \[yellow\] drawn in car (start=(4.65, 4.0), end=(4.25, 3.6)); left\_current — a Vector \[red\] labelled "upright("surface current")" drawn in car (start=(4.2, 3.55), end=(1.15, 2.15)); right\_current — a Vector \[red\] drawn in car (start=(4.35, 3.55), end=(6.85, 2.1)); recap — a Block \[text\] that says "Flux through a closed surface counts enclosed charge. Symmetry chooses a surface on which the flux is easy to calculate. Mobile charge in a conductor rearranges until the field in the metal is zero."; recap\_heading — a Heading that says "What to Carry Away" Actions: - [16:28.076](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=988.0764166666664): recap (the "Symmetry chooses a surface on which the flux is easy to calculate." part) is indicated — a transient flash. ##### [16:40.344](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1000.3439166666665) Narration: Third, mobile charge in a conductor rearranges until the field inside the metal is zero. Excess charge lives on the surface, and a closed metal body redirects an external electrical disturbance around its interior. Board: Unchanged from the preceding beat in this scene. Actions: - [16:40.901](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1000.9014166666664): recap (the "Mobile charge in a conductor rearranges until the field in the metal is zero." part) is indicated — a transient flash. ##### [16:54.087](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1014.0869166666664) Narration: Flux made the counting picture precise. Gauss's law connected that count to charge. Symmetry turned the law into three electric fields, and electrostatic equilibrium turned it into protection inside a conductor. Board: Unchanged from the preceding beat in this scene. Actions: - [17:5.755](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1025.7554166666664): A box is drawn around cabin\_field. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): cabin\_field is hidden from the screen — left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): car is hidden from the screen — left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): body is hidden from the screen — car left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): cabin is hidden from the screen — car left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): wheel\_left is hidden from the screen — car left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): wheel\_right is hidden from the screen — car left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): passenger\_left is hidden from the screen — car left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): passenger\_right is hidden from the screen — car left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): bolt\_one is hidden from the screen — car left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): bolt\_two is hidden from the screen — car left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): bolt\_three is hidden from the screen — car left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): left\_current is hidden from the screen — car left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): right\_current is hidden from the screen — car left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): recap is hidden from the screen — left the board. - [17:7.77](https://academa.ai/lectures/gauss-law-and-what-flux-is-counting?t=1027.7699374999997): recap\_heading is hidden from the screen — left the board.