WEBVTT 00:00:00.070 --> 00:00:03.263 Place one positive point charge in empty space. 00:00:03.460 --> 00:00:05.095 At every surrounding point, 00:00:05.178 --> 00:00:08.418 its electric field points directly away from the charge. 00:00:08.615 --> 00:00:10.888 These blue arrows represent field lines, 00:00:10.971 --> 00:00:14.641 a drawing that lets us follow the direction of the field through space. 00:00:15.437 --> 00:00:17.398 The picture is three dimensional. 00:00:17.481 --> 00:00:20.056 Every direction away from the charge is equivalent, 00:00:20.139 --> 00:00:23.263 and no radial direction is preferred over another. 00:00:24.060 --> 00:00:26.789 Now surround the charge with a small sphere. 00:00:26.986 --> 00:00:30.107 Every blue line leaving the charge crosses that sphere once, 00:00:30.190 --> 00:00:32.373 from its inside to its outside. 00:00:32.686 --> 00:00:34.948 If we use the lines as a counting picture, 00:00:35.031 --> 00:00:37.818 the outward flux is the number of crossings. 00:00:38.522 --> 00:00:42.912 The lines are not physical threads, and their drawn number is arbitrary. 00:00:43.224 --> 00:00:46.392 Still, the crossing picture captures something real: 00:00:46.475 --> 00:00:50.782 a surface facing the field receives positive flux, and a closed surface 00:00:50.782 --> 00:00:54.139 surrounding the source intercepts the whole outward pattern. 00:00:55.051 --> 00:00:57.896 Replace the small sphere with a larger one. 00:00:58.093 --> 00:01:02.494 The same radial lines cross it, so the crossing count has not changed. 00:01:02.946 --> 00:01:04.535 They are farther apart now, 00:01:04.618 --> 00:01:08.310 which is the picture's way of saying that the electric field is weaker. 00:01:09.107 --> 00:01:13.438 The radius grew, and the sphere's area grew as radius squared. 00:01:13.634 --> 00:01:17.815 But the field of a point charge weakened as one over radius squared. 00:01:18.011 --> 00:01:23.039 Twice the radius gives four times the area and one quarter of the field strength. 00:01:23.236 --> 00:01:25.373 Their product stays fixed. 00:01:26.169 --> 00:01:28.272 A sphere is not essential. 00:01:28.468 --> 00:01:33.009 Dent the surface here, bulge it there, and keep the charge enclosed. 00:01:33.205 --> 00:01:37.293 Every ray that leaves the charge still has to cross the closed boundary. 00:01:37.490 --> 00:01:40.669 The local crossing angles and local field strengths change, 00:01:40.752 --> 00:01:43.074 but the total outward count does not. 00:01:44.726 --> 00:01:47.420 For a sphere of radius r, the point-charge 00:01:47.420 --> 00:01:53.899 field has magnitude one over four pi epsilon zero, times q over r squared. 00:01:54.696 --> 00:01:57.645 The sphere's area is four pi r squared. 00:01:57.842 --> 00:01:59.897 This is exactly the geometrical growth 00:01:59.897 --> 00:02:03.068 that compensates for the inverse-square weakening. 00:02:03.864 --> 00:02:06.280 Multiply field strength by area. 00:02:06.476 --> 00:02:10.585 The r squared in the area cancels the r squared in the denominator, 00:02:10.668 --> 00:02:13.025 and four pi cancels as well. 00:02:13.822 --> 00:02:17.561 What remains is q divided by epsilon zero. 00:02:17.665 --> 00:02:19.616 It has no radius in it. 00:02:19.813 --> 00:02:23.157 The calculation therefore agrees with the crossing picture: 00:02:23.261 --> 00:02:27.708 every surrounding sphere receives the same total outward electric flux. 00:02:28.412 --> 00:02:33.091 The lumpy surface needs a more careful sum because its field strength and angle vary 00:02:33.091 --> 00:02:34.740 from patch to patch. 00:02:34.843 --> 00:02:38.069 But once those local contributions are counted correctly, 00:02:38.152 --> 00:02:41.520 the same enclosed charge gives the same total flux. 00:02:42.317 --> 00:02:46.392 That invariant total is the idea Gauss's law will make exact. 00:02:46.589 --> 00:02:51.396 The next step is to define what one small crossing contributes, including the angle 00:02:51.396 --> 00:02:53.556 at which the field meets the surface. 00:02:54.636 --> 00:02:57.551 Zoom in on one small piece of a surface. 00:02:57.747 --> 00:03:02.156 Its area is delta A, and the yellow arrow is the outward unit normal, 00:03:02.240 --> 00:03:04.842 the direction perpendicular to that patch. 00:03:06.056 --> 00:03:10.341 Let the electric field meet the patch at an angle theta from the normal. 00:03:10.538 --> 00:03:14.358 The field crosses most effectively when it points along the normal. 00:03:15.155 --> 00:03:19.776 The patch's flux is E dot n hat times delta A. 00:03:19.880 --> 00:03:24.803 Equivalently, it is field strength times area times cosine theta. 00:03:25.000 --> 00:03:28.426 The cosine selects the normal component of the field. 00:03:29.222 --> 00:03:31.588 If the field runs parallel to the patch, 00:03:31.672 --> 00:03:35.214 theta is ninety degrees and the cosine is zero. 00:03:35.410 --> 00:03:38.449 Lines may skim along the surface, but none cross it, 00:03:38.533 --> 00:03:41.100 so that part contributes no flux. 00:03:42.013 --> 00:03:45.810 If the field points outward, the dot product is positive. 00:03:46.007 --> 00:03:49.305 If it points inward, the dot product is negative. 00:03:49.502 --> 00:03:54.228 Flux therefore counts signed crossings, outward minus inward. 00:03:55.624 --> 00:03:58.864 A curved surface is built from many small patches. 00:03:59.061 --> 00:04:01.999 Add the local dot products over all of them. 00:04:02.795 --> 00:04:07.916 As the patches become arbitrarily small, the sum becomes a surface integral. 00:04:08.368 --> 00:04:11.593 This is the precise definition of electric flux. 00:04:11.677 --> 00:04:12.954 Unlike the number of lines 00:04:12.954 --> 00:04:17.506 an artist draws, the integral has a fixed numerical meaning. 00:04:18.558 --> 00:04:22.216 Return to a point charge and a spherical surface around it. 00:04:22.412 --> 00:04:24.395 Symmetry makes the field radial, 00:04:24.479 --> 00:04:27.974 so it points along the outward normal everywhere on the sphere. 00:04:28.771 --> 00:04:32.219 Symmetry also makes the field strength the same at every point 00:04:32.219 --> 00:04:34.426 of a sphere with radius r. 00:04:34.530 --> 00:04:37.174 We can therefore take E outside the integral, 00:04:37.258 --> 00:04:40.475 leaving the total area four pi r squared. 00:04:41.271 --> 00:04:44.256 Insert the inverse-square point-charge field. 00:04:44.360 --> 00:04:46.923 The area growth cancels the field's weakening, 00:04:47.007 --> 00:04:49.782 leaving q divided by epsilon zero. 00:04:50.695 --> 00:04:54.643 Now deform the closed surface without moving it across the charge. 00:04:54.840 --> 00:04:59.670 Some patches tilt, some move closer, and others move farther away. 00:05:00.122 --> 00:05:05.266 The local dot products change, but the net signed crossing count cannot change. 00:05:06.063 --> 00:05:08.874 Several charges simply add their fields. 00:05:09.070 --> 00:05:12.577 A charge inside contributes its full outward flux. 00:05:13.192 --> 00:05:18.007 A charge outside sends as much flux into the closed surface as it sends back out, 00:05:18.091 --> 00:05:20.855 so its net contribution is zero. 00:05:21.652 --> 00:05:23.789 The result is Gauss's law: 00:05:23.892 --> 00:05:27.666 the flux through any closed surface equals the net enclosed charge divided 00:05:27.666 --> 00:05:29.454 by epsilon zero. 00:05:29.767 --> 00:05:34.632 The surface may be imaginary, irregular, or placed wherever we choose. 00:05:36.029 --> 00:05:41.416 Written in full, the closed-surface integral of E dot n hat equals enclosed 00:05:41.416 --> 00:05:43.994 charge divided by epsilon zero. 00:05:44.698 --> 00:05:47.798 Notice what the law does and does not say. 00:05:47.995 --> 00:05:51.374 It always gives total flux from enclosed charge. 00:05:51.571 --> 00:05:55.562 It gives the electric field itself only when symmetry makes the field's direction 00:05:55.646 --> 00:05:59.338 and magnitude simple on a carefully chosen surface. 00:06:00.251 --> 00:06:02.516 That choice is the real technique. 00:06:02.712 --> 00:06:06.208 For a spherical charge distribution we will choose a sphere. 00:06:06.311 --> 00:06:08.657 For a line we will choose a cylinder. 00:06:08.854 --> 00:06:11.467 For a sheet we will choose a pillbox. 00:06:12.082 --> 00:06:16.552 In every case, the source geometry chooses the useful Gaussian surface. 00:06:17.614 --> 00:06:19.390 Take a solid insulating sphere 00:06:19.390 --> 00:06:24.209 of radius R, with total charge Q spread uniformly throughout its volume. 00:06:24.313 --> 00:06:28.385 Because the distribution looks the same after any rotation about its centre, 00:06:28.469 --> 00:06:31.105 the electric field must point radially. 00:06:31.902 --> 00:06:34.047 A short turn confirms the geometry. 00:06:34.131 --> 00:06:37.626 There is no preferred direction around the sphere, so at a fixed distance 00:06:37.626 --> 00:06:41.586 from the centre every point must have the same field magnitude. 00:06:42.382 --> 00:06:45.704 First ask for the field outside the charge distribution. 00:06:45.808 --> 00:06:49.903 Choose a spherical Gaussian surface of radius r greater than R, 00:06:49.987 --> 00:06:51.753 centred on the same point. 00:06:52.549 --> 00:06:56.241 On this Gaussian sphere, the field is everywhere normal to the surface 00:06:56.241 --> 00:06:59.644 and has one constant magnitude E of r. 00:06:59.841 --> 00:07:04.276 The flux is therefore E times four pi r squared. 00:07:05.073 --> 00:07:10.484 The Gaussian surface encloses the entire charged body, so the enclosed charge is Q. 00:07:11.280 --> 00:07:15.472 Gauss's law sets the flux equal to Q over epsilon zero. 00:07:15.576 --> 00:07:19.454 Solving for E gives the familiar inverse-square field. 00:07:20.251 --> 00:07:24.036 Outside a spherically symmetric distribution, the field is exactly 00:07:24.036 --> 00:07:27.392 the same as if all the charge were concentrated at the centre. 00:07:27.496 --> 00:07:31.966 Gauss's law makes that statement exact, not merely approximate. 00:07:33.270 --> 00:07:36.646 Now move the Gaussian sphere inside the charged body. 00:07:36.730 --> 00:07:39.084 Its radius r is less than R. 00:07:39.168 --> 00:07:41.301 The field still has spherical symmetry, 00:07:41.385 --> 00:07:44.776 but the surface now encloses only part of the total charge. 00:07:45.991 --> 00:07:50.415 Uniform volume charge density means total charge divided by total volume. 00:07:50.611 --> 00:07:54.896 The density rho is Q over four thirds pi R cubed. 00:07:55.321 --> 00:07:59.374 The smaller Gaussian sphere has volume four thirds pi r cubed. 00:07:59.477 --> 00:08:03.321 Multiply that volume by rho to find the charge it encloses. 00:08:04.118 --> 00:08:09.158 The common factors cancel, leaving Q times r cubed over R cubed. 00:08:09.354 --> 00:08:11.119 This fraction is simply the fraction 00:08:11.119 --> 00:08:14.614 of the charged volume lying inside the Gaussian surface. 00:08:15.318 --> 00:08:21.251 Gauss's law again says E times four pi r squared equals enclosed charge 00:08:21.251 --> 00:08:23.051 over epsilon zero. 00:08:23.248 --> 00:08:26.418 One power of r survives after division. 00:08:26.901 --> 00:08:29.766 The interior field is proportional to r. 00:08:29.850 --> 00:08:34.076 It is zero at the centre, where all directions balance, and it grows linearly 00:08:34.076 --> 00:08:36.991 as the Gaussian sphere encloses more charge. 00:08:38.295 --> 00:08:40.746 Put the two regions on one graph. 00:08:40.942 --> 00:08:43.285 Distance is measured in units of R, 00:08:43.369 --> 00:08:46.632 and field strength in units of its value at the surface. 00:08:47.428 --> 00:08:50.726 Inside, the field rises in a straight line from zero. 00:08:50.830 --> 00:08:55.494 In normalized form, E over E at the surface equals r over R. 00:08:55.578 --> 00:08:58.107 Enclosed charge grows as r cubed, 00:08:58.191 --> 00:09:02.836 while Gaussian area grows as r squared, leaving one power of r. 00:09:03.540 --> 00:09:07.917 At r equals R, the inside and outside formulas agree. 00:09:08.114 --> 00:09:12.236 There is no jump in the field because the charge fills a volume rather than sitting 00:09:12.236 --> 00:09:15.069 in an infinitesimally thin surface layer. 00:09:15.866 --> 00:09:20.940 Outside, E over the surface field equals R squared over r squared. 00:09:21.137 --> 00:09:23.200 The enclosed charge has stopped growing, 00:09:23.284 --> 00:09:26.896 while the Gaussian area continues to grow as r squared. 00:09:27.693 --> 00:09:30.248 The method was the same in both regions: 00:09:30.444 --> 00:09:33.437 use the source symmetry to choose a concentric sphere, 00:09:33.521 --> 00:09:38.305 make E constant on that surface, and then count only the charge actually enclosed. 00:09:39.446 --> 00:09:41.338 Now stretch the charge distribution 00:09:41.338 --> 00:09:45.937 into an ideal infinite line with uniform charge per length lambda. 00:09:46.133 --> 00:09:48.663 Translation along the line changes nothing, 00:09:48.746 --> 00:09:51.231 and rotation around it changes nothing. 00:09:52.028 --> 00:09:56.290 Those symmetries force the electric field to point directly away from the line. 00:09:56.486 --> 00:09:59.770 Its magnitude can depend on perpendicular distance r, 00:09:59.853 --> 00:10:03.232 but not on position along the line or angle around it. 00:10:04.029 --> 00:10:06.421 Choose a cylinder centred on the line. 00:10:06.525 --> 00:10:10.447 Its curved side is everywhere the same distance r from the charge, 00:10:10.530 --> 00:10:13.457 so E has one constant magnitude there. 00:10:14.253 --> 00:10:16.379 Let the solid settle into view. 00:10:16.831 --> 00:10:19.441 The charged line is the cylinder's axis, 00:10:19.524 --> 00:10:22.764 and the blue field arrows point through its curved wall. 00:10:23.677 --> 00:10:26.996 On the two end caps, the outward normals point along the line, 00:10:27.079 --> 00:10:30.122 while the electric field points radially away from it. 00:10:30.225 --> 00:10:34.498 Their dot product is zero, so the caps contribute no flux. 00:10:35.295 --> 00:10:37.618 Only the curved side contributes. 00:10:37.814 --> 00:10:41.365 Its area is circumference two pi r times length L, 00:10:41.448 --> 00:10:44.839 so the flux is E times two pi r L. 00:10:45.218 --> 00:10:48.385 The cylinder encloses a length L of line charge. 00:10:48.468 --> 00:10:53.334 Charge per length lambda times L gives enclosed charge lambda L. 00:10:53.875 --> 00:10:59.077 Apply Gauss's law. The length L appears on both sides and cancels. 00:10:59.274 --> 00:11:03.431 Solving leaves lambda over two pi epsilon zero r. 00:11:04.227 --> 00:11:08.988 The line field falls as one over r, not one over r squared. 00:11:09.185 --> 00:11:10.867 As the cylinder expands, 00:11:10.950 --> 00:11:15.165 its relevant area per unit length grows only in proportion to r. 00:11:16.561 --> 00:11:20.321 Next spread charge uniformly across an ideal infinite sheet, 00:11:20.404 --> 00:11:22.843 with surface charge density sigma. 00:11:23.040 --> 00:11:26.338 Sliding anywhere within the sheet cannot change the field. 00:11:27.134 --> 00:11:30.955 Rotating the sheet within its own plane also changes nothing. 00:11:31.151 --> 00:11:34.308 The only distinguished direction is perpendicular to the sheet, 00:11:34.391 --> 00:11:37.503 so the field must point normally away on both sides. 00:11:38.300 --> 00:11:41.842 Choose a short cylindrical pillbox that straddles the sheet. 00:11:42.038 --> 00:11:44.997 Its flat caps are parallel to the charge distribution, 00:11:45.080 --> 00:11:47.240 and its curved wall joins them. 00:11:48.037 --> 00:11:50.522 A small turn shows the construction. 00:11:50.718 --> 00:11:55.363 The field passes through the two caps, while it runs parallel to the curved wall. 00:11:56.160 --> 00:12:00.593 The curved wall contributes zero flux because its normal lies within the sheet 00:12:00.676 --> 00:12:03.045 while E is perpendicular to it. 00:12:03.242 --> 00:12:07.793 Each cap contributes E A, so the total flux is two E A. 00:12:08.555 --> 00:12:13.328 The pillbox encloses sheet area A, so it encloses charge sigma A. 00:12:14.089 --> 00:12:18.815 Gauss's law gives two E A equals sigma A over epsilon zero. 00:12:18.977 --> 00:12:23.552 The cap area cancels, leaving E equals sigma over two epsilon zero. 00:12:24.314 --> 00:12:26.518 There is no distance in the answer. 00:12:26.601 --> 00:12:28.422 For an ideal infinite sheet, 00:12:28.505 --> 00:12:33.020 moving the caps farther away does not spread a fixed bundle over a growing area. 00:12:33.103 --> 00:12:36.122 The same cap area intercepts the same flux. 00:12:37.519 --> 00:12:39.702 Put the three geometries together. 00:12:39.899 --> 00:12:44.065 A point spreads flux over a sphere whose area grows as r squared, 00:12:44.148 --> 00:12:47.550 so its field has the form a constant over r squared. 00:12:48.347 --> 00:12:51.680 A line spreads flux over the curved wall of a cylinder. 00:12:51.876 --> 00:12:57.857 Per unit length, that area grows as r, so the field has the form a constant over r. 00:12:58.282 --> 00:13:01.348 A sheet sends flux through two equal caps. 00:13:01.451 --> 00:13:04.480 Their area does not change when the pillbox grows taller, 00:13:04.563 --> 00:13:07.071 so the ideal sheet field is constant. 00:13:07.868 --> 00:13:11.200 Gauss's law was identical in all three cases. 00:13:11.397 --> 00:13:15.217 What changed was the symmetry, and symmetry determined the surface 00:13:15.217 --> 00:13:19.745 on which E became constant and the unwanted pieces contributed zero. 00:13:20.804 --> 00:13:23.336 A conductor contains mobile charge. 00:13:23.532 --> 00:13:26.166 If an electric field existed inside the metal, 00:13:26.249 --> 00:13:29.373 that charge would feel a force and begin to move. 00:13:29.569 --> 00:13:33.459 A state with moving charge is not electrostatic equilibrium. 00:13:33.885 --> 00:13:37.809 The mobile charge rearranges on an extremely short time scale. 00:13:37.913 --> 00:13:41.441 Its own electric field opposes the field that drove the motion, 00:13:41.524 --> 00:13:43.242 and equilibrium is reached only 00:13:43.242 --> 00:13:47.283 when the net electric field inside the conducting material is zero. 00:13:47.894 --> 00:13:53.130 That statement comes from the physics of mobile charge, not from Gauss's law alone. 00:13:53.443 --> 00:13:57.937 Gauss's law now tells us an important consequence of the zero field. 00:13:58.734 --> 00:14:03.657 Draw any closed Gaussian surface lying completely inside the conducting material. 00:14:03.761 --> 00:14:08.464 Because E is zero at every point on it, its electric flux is zero. 00:14:09.260 --> 00:14:10.607 Gauss's law then says 00:14:10.607 --> 00:14:15.484 that the net charge enclosed by every such interior surface is zero. 00:14:16.188 --> 00:14:20.530 Excess charge therefore cannot remain distributed through the bulk of the metal 00:14:20.530 --> 00:14:22.911 in electrostatic equilibrium. 00:14:23.107 --> 00:14:26.649 It moves outward and comes to rest on the conductor's surface. 00:14:27.562 --> 00:14:32.381 Outside the conductor, those surface charges can produce an electric field. 00:14:32.693 --> 00:14:34.491 Inside the conducting material, 00:14:34.574 --> 00:14:38.383 they have arranged themselves so that their combined field cancels. 00:14:39.180 --> 00:14:41.967 The surface distribution need not be uniform. 00:14:42.419 --> 00:14:45.123 Charge crowds more strongly near sharp points, 00:14:45.206 --> 00:14:48.759 where the surrounding field can become especially large. 00:14:49.072 --> 00:14:53.809 But the equilibrium condition inside the metal remains E equals zero. 00:14:55.206 --> 00:14:59.456 Now replace the simple conductor with the metal body of a hard-top car. 00:14:59.559 --> 00:15:03.473 The body forms a conducting shell around the passenger compartment. 00:15:04.385 --> 00:15:06.604 Suppose lightning strikes the roof. 00:15:06.800 --> 00:15:10.818 The lightning delivers charge and a large current to the exterior metal. 00:15:11.615 --> 00:15:13.657 The charge spreads over the outside, 00:15:13.740 --> 00:15:17.572 and the current finds conducting paths along the exterior body. 00:15:18.186 --> 00:15:20.230 The metal shell carries the dangerous 00:15:20.230 --> 00:15:24.166 electrical disturbance around the passenger space rather than through it. 00:15:24.870 --> 00:15:28.781 The passenger compartment is therefore close to one electric potential, 00:15:28.864 --> 00:15:32.197 with a strongly reduced electric field inside. 00:15:32.649 --> 00:15:36.562 This shielding behavior is often called the Faraday-cage effect. 00:15:37.266 --> 00:15:40.224 The protection comes from the continuous metal shell, 00:15:40.307 --> 00:15:42.584 not primarily from the rubber tires. 00:15:42.896 --> 00:15:45.718 During a storm, occupants should remain inside 00:15:45.718 --> 00:15:50.293 with windows closed and avoid touching metal parts connected to the exterior. 00:15:51.089 --> 00:15:54.234 The same principle is used deliberately in shielded rooms, 00:15:54.317 --> 00:15:58.648 cable coverings, and metal enclosures around sensitive electronics. 00:15:58.845 --> 00:16:00.562 Conductors rearrange charge 00:16:00.645 --> 00:16:04.860 so that their protected interiors experience very little electric field. 00:16:06.256 --> 00:16:08.751 Three ideas organize the whole lecture. 00:16:08.834 --> 00:16:13.256 First, electric flux is the signed amount of electric field crossing a surface, 00:16:13.339 --> 00:16:17.055 and the flux through a closed surface counts net enclosed charge. 00:16:17.851 --> 00:16:21.810 Second, Gauss's law becomes a field-solving tool only 00:16:21.810 --> 00:16:24.609 when symmetry chooses a useful surface: 00:16:24.713 --> 00:16:30.391 a sphere for spherical charge, a cylinder for a line, and a pillbox for a sheet. 00:16:31.443 --> 00:16:33.160 Third, mobile charge 00:16:33.243 --> 00:16:37.434 in a conductor rearranges until the field inside the metal is zero. 00:16:37.631 --> 00:16:42.322 Excess charge lives on the surface, and a closed metal body redirects an external 00:16:42.322 --> 00:16:45.109 electrical disturbance around its interior. 00:16:46.161 --> 00:16:48.936 Flux made the counting picture precise. 00:16:49.133 --> 00:16:52.152 Gauss's law connected that count to charge. 00:16:52.465 --> 00:16:55.552 Symmetry turned the law into three electric fields, 00:16:55.635 --> 00:16:59.943 and electrostatic equilibrium turned it into protection inside a conductor.