{"version":1,"lectureId":"01M14TZVE71NE21PCDF8EKB09N","attempt":1,"publication":{"slug":"faraday-lenz-and-the-magnet-that-will-not-fall","title":"Why a Magnet Falls Slowly Through a Copper Pipe","subject":"physics","summary":"A visual introduction to electromagnetic braking, beginning with one conducting loop and the magnetic flux through it. Faraday's law gives the induced current, Lenz's law fixes its direction, and the current's magnetic field produces a force opposing the magnet's motion. The copper pipe is then treated as many such loops, leading to a terminal speed where electromagnetic drag balances weight. The same chain of ideas closes the lecture in a practical setting: eddy-current brakes that deliberately convert a train's kinetic energy into heat.","metaDescription":"See how changing magnetic flux, Lenz's law, and electromagnetic drag make a magnet fall slowly through copper and power train brakes.","transcript":"Drop an ordinary metal object through a vertical pipe and gravity wins quickly. Drop a strong magnet through a copper pipe and something startling happens: the magnet can take several seconds to emerge. Copper is not attracted to the magnet, so what is holding it back? Here is the experiment in side view. The yellow walls are copper. The red and blue body is the magnet, and its weight points down the pipe. Release it. The magnet still falls, so copper has not suspended it. But the fall is slow and controlled rather than almost free. Whatever the copper does, it acts only while the magnetic field is moving relative to the metal. A pipe is complicated, so begin with one thin conducting loop. We see the loop edge on at the left. At the right we will record the magnetic flux through its enclosed area. Flux measures how much magnetic field passes through the loop, including its direction. Far from the loop, the magnet contributes little flux. As it approaches, the magnitude grows. Continue through the loop. The signed flux changes rapidly, crosses through zero as the magnetic geometry reverses, and then weakens again as the magnet moves away on the other side. Write that measurement as magnetic flux, phi B. It is the surface integral of the magnetic field dotted with an oriented area element. The crucial fact is not merely that the loop has flux. The flux changes with time because the magnet and loop move relative to one another. Motion has therefore created the condition needed for electromagnetic induction. A changing magnetic flux has an electrical consequence. Faraday's law says that the induced electromotive force equals minus the time rate of change of flux. Copper conducts, but it has resistance R. Ohm's law therefore turns the induced voltage into a current. Faster flux change gives more voltage; lower resistance gives more current. Now the minus sign matters. It does not mean that current is somehow negative. It fixes the direction through Lenz's law: the induced current opposes the change in flux. Take a north pole approaching the loop from above. The magnet's field points downward through the loop, and that downward flux is becoming stronger. The loop answers with an upward field, opposing that increase. Viewed from above, an upward field requires counterclockwise conventional current. That induced field makes the upper face of the loop behave like a north pole. It repels the approaching north pole of the magnet, so the force on the falling magnet points upward. Nothing here says that copper is a permanent magnet. Motion changed the flux, the changing flux drove a current, and the current temporarily created the magnetic field. Before the magnet reaches a loop, the flux magnitude is increasing. The induced field resists that approach, so the loop repels the magnet. After the magnet passes, the flux is decreasing. The current reverses to preserve the disappearing flux, and the loop attracts the receding magnet. The magnetic details reverse, but the force still points upward. Approaching loops push back. Receding loops pull back. In both cases the induced force opposes the motion that caused the changing flux. The direction can also be checked with energy. Drag force dotted with velocity is negative, so the electromagnetic force removes mechanical energy from the falling magnet. That energy has not vanished. Current flows through resistive copper, so electrical power I squared R becomes heat. The pipe warms by a tiny amount while the magnet slows. So Lenz's law is not an extra rule pasted onto Faraday's law. Its minus sign protects energy conservation: the induced effect fights its cause rather than helping the magnet accelerate itself. A real pipe is not one loop. Imagine slicing its wall into many narrow rings. Each yellow ring is a conducting path around the pipe, and each one can carry its own induced current. Together those rings form the cylindrical copper wall. A gentle turn makes the stack visible as a three-dimensional pipe rather than as a bundle of flat lines. Place the magnet inside. Gravity pulls downward, while the combined electromagnetic drag from the surrounding currents points upward. A ring below the magnet is being approached. A ring above has just been left behind. Their induced currents run in opposite senses, because one flux is strengthening while the other is weakening. Yet both rings oppose the fall. Add the effects of all the rings and the magnet experiences a smooth upward drag throughout the pipe. Watch the same magnet continue downward while both force arrows travel with it. Now ask how that drag depends on speed. Moving faster changes the flux faster. Faraday's law then gives a larger voltage, a larger current, and a larger opposing magnetic force. For a fixed magnet and pipe, and over the useful low-speed range, gather the geometry and electrical resistance into one constant k. Then the drag magnitude is approximately k times v. Take downward as positive. Newton's second law says mass times acceleration equals the downward weight, m g, minus the upward electromagnetic drag, k v. At first v is small, so weight wins and the magnet accelerates. As v increases, the drag grows. The rising curve shows the speed approaching a limiting value. Terminal speed is reached when acceleration becomes zero. Then the forces balance: m g equals k v sub t. Solve that one line. The terminal speed is m g divided by k. A heavier magnet tends to fall faster; stronger magnetic coupling or lower copper resistance increases k and lowers the terminal speed. The balance is stable. Below terminal speed, weight is larger than drag and the magnet speeds up. Above terminal speed, drag is larger than weight and the magnet slows down. At terminal speed the magnet still loses gravitational potential energy. Each second, weight supplies power m g v sub t, and the many loop currents dissipate the same total power as heat. That is why the magnet does not hover and why it does not keep accelerating. It descends steadily, converting gravitational energy into many tiny resistive losses distributed along the copper wall. The copper-pipe experiment looks like a curiosity, but engineers use the same effect deliberately. An eddy-current brake places strong magnets close to a conducting rail or metal braking surface. The magnets do not have to touch the rail. Their field reaches across the gap into the conductor. As the train moves right, each patch of rail experiences changing magnetic flux. Closed circulating currents form within the bulk metal. These are eddy currents, the extended-sheet version of the current in our single copper loop. Lenz's law fixes their direction. The currents create magnetic fields that oppose the passing magnet pattern, so the force on the train points left, opposite its velocity. Watch the assembly move along the rail. The field pattern, eddy currents, and braking force travel with the active region, while the conducting rail itself remains fixed. The causal chain is exactly the one we built for the pipe. Relative speed produces changing flux. Changing flux produces current. Current produces the opposing braking force. The train's kinetic energy becomes electrical energy in the eddy currents and then resistive heat in the rail or brake disc. There is no mystery energy sink and no ordinary magnetic attraction to copper. As speed falls, the flux changes more slowly, so induced current and braking force weaken. At rest the motion-driven eddy currents disappear. Real trains therefore combine this smooth, low-wear method with other braking systems that can hold the vehicle still. Three statements carry the whole lecture. First, changing flux induces current in a conductor. Copper need not be a permanent magnet. Second, the induced current makes its own magnetic field. Lenz's law gives the direction that opposes the relative motion. Third, the lost mechanical energy becomes heat. In the pipe that energy conversion makes a falling magnet descend at terminal speed. On a train, the same conversion is useful braking. So the magnet falls slowly not because copper is secretly magnetic, but because motion continually creates currents whose magnetic effects resist that motion. The pipe demonstrates the law. The train brake puts it to work.","watch":{"version":1,"scenes":[{"title":"The Copper Pipe Puzzle","start":0,"end":106.66145833333334,"objects":{"card":"a Title that says \"Introductory Electromagnetism — Why a Magnet Falls Slowly Through a Copper Pipe\"","comparison_heading":"a Heading that says \"Replace the Pipe by One Loop\"","fall_arrow":"a Vector [green] labelled \"arrow(v)\" drawn in pipe_view (start=(4.7, 2.8), end=(4.7, 1.0))","fall_height":"a VariableNumber (initial_value=2.7)","flux_arrow":"a Vector [magenta] labelled \"Phi_B\" drawn in loop_view (end=(0.0, ((-2.4 * magnet_z) / (((magnet_z * magnet_z) + 0.9) ** 1.…)","flux_axes":"an Axes (x_range=(-3.0, 3.0), y_range=(-1.4, 1.4), x_ticks_every=1.0)","flux_caption":"a Tex [text] that says \"Flux through that loop\"","flux_curve":"a FunctionPlot [magenta] drawn in flux_axes (function=, x_range=(-3.0, 3.0))","flux_heading":"a Heading that says \"Magnetic Flux Through the Loop\"","flux_point":"a PlotPoint [yellow] drawn in flux_axes (target='flux_curve', x=)","flux_result":"a Math [text] that says \"$upright(\"motion\") arrow.r upright(\"changing flux\")$\"","flux_work":"a Derivation [text] that says \"$Phi_B = integral_S bold(B) dot dif bold(A) \\ frac(dif Phi_B, dif t) eq.not 0$\"","loop_caption":"a Tex [text] that says \"One conducting loop\"","loop_edge":"a Line [yellow] labelled \"upright(\"copper loop, edge on\")\" drawn in loop_view (start=(-1.45, 0.0), end=(1.45, 0.0))","loop_view":"a Figure (x_range=(-2.4, 2.4), y_range=(-3.2, 3.4), aspect=(4.8, 6.6))","magnet_z":"a VariableNumber (initial_value=2.4)","motion_arrow":"a Vector [green] labelled \"arrow(v)\" drawn in loop_view (start=(1.75, 2.7), end=(1.75, 1.15))","north_half":"a Polygon [red] drawn in loop_view (vertices=((-0.48, ), (0.48, ), (3.45,