WEBVTT

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Drop an ordinary metal object through
a vertical pipe and gravity wins quickly.

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Drop a strong magnet through a copper
pipe and something startling happens:

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the magnet can take
several seconds to emerge.

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Copper is not attracted
to the magnet, so what is holding it back?

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Here is the experiment in side view.

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The yellow walls are copper.

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The red and blue body is the magnet,
and its weight points down the pipe.

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Release it.

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The magnet still falls,
so copper has not suspended it.

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But the fall is slow
and controlled rather than almost free.

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Whatever the copper does,

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it acts only while the magnetic
field is moving relative to the metal.

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A pipe is complicated, so begin
with one thin conducting loop.

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We see the loop edge on at the left.

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At the right we will record the magnetic
flux through its enclosed area.

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Flux measures how much magnetic
field passes through the loop,

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including its direction.

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Far from the loop,
the magnet contributes little flux.

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As it approaches, the magnitude grows.

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Continue through the loop.

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The signed flux changes rapidly, crosses
through zero as the magnetic geometry

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reverses, and then weakens again as
the magnet moves away on the other side.

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Write that measurement as magnetic
flux, phi B. It is the surface integral

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of the magnetic field dotted
with an oriented area element.

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The crucial fact is not merely that the
loop has flux. The flux changes with time

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because the magnet and loop
move relative to one another.

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Motion has therefore created the condition
needed for electromagnetic induction.

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A changing magnetic flux
has an electrical consequence.

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Faraday's law says that the induced

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electromotive force equals minus
the time rate of change of flux.

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Copper conducts, but it has resistance R.

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Ohm's law therefore turns
the induced voltage into a current.

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Faster flux change gives more voltage;

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lower resistance gives more current.

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Now the minus sign matters.

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It does not mean
that current is somehow negative.

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It fixes the direction through Lenz's law:

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the induced current
opposes the change in flux.

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Take a north pole
approaching the loop from above.

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The magnet's field points
downward through the loop,

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and that downward flux
is becoming stronger.

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The loop answers with an upward
field, opposing that increase.

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Viewed from above,

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an upward field requires
counterclockwise conventional current.

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That induced field makes the upper face
of the loop behave like a north pole.

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It repels the approaching north
pole of the magnet, so the force

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on the falling magnet points upward.

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Nothing here says
that copper is a permanent magnet.

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Motion changed the flux,
the changing flux drove a current,

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and the current temporarily
created the magnetic field.

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Before the magnet reaches a loop,
the flux magnitude is increasing.

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The induced field resists that approach,
so the loop repels the magnet.

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After the magnet passes,
the flux is decreasing.

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The current reverses
to preserve the disappearing flux,

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and the loop attracts the receding magnet.

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The magnetic details reverse,
but the force still points upward.

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Approaching loops push back.

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Receding loops pull back.

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In both cases the induced force opposes
the motion that caused the changing flux.

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The direction can also
be checked with energy.

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Drag force dotted
with velocity is negative,

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so the electromagnetic force removes
mechanical energy from the falling magnet.

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That energy has not vanished.

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Current flows through resistive copper, so
electrical power I squared R becomes heat.

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The pipe warms by a tiny
amount while the magnet slows.

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So Lenz's law is not an extra
rule pasted onto Faraday's law.

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Its minus sign protects
energy conservation:

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the induced effect fights its cause rather
than helping the magnet accelerate itself.

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A real pipe is not one loop. Imagine
slicing its wall into many narrow rings.

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Each yellow ring
is a conducting path around the pipe,

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and each one can carry
its own induced current.

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Together those rings form
the cylindrical copper wall.

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A gentle turn makes the stack visible
as a three-dimensional pipe rather than

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as a bundle of flat lines.

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Place the magnet inside.

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Gravity pulls downward, while the combined
electromagnetic drag from the surrounding

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currents points upward.

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A ring below the magnet
is being approached.

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A ring above has just been left behind.

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Their induced currents run in opposite
senses, because one flux is strengthening

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while the other is weakening.

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Yet both rings oppose the fall.

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Add the effects of all the rings

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and the magnet experiences a smooth
upward drag throughout the pipe.

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Watch the same magnet continue downward
while both force arrows travel with it.

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Now ask how that drag depends on speed.

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Moving faster changes the flux faster.

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Faraday's law then gives
a larger voltage, a larger current,

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and a larger opposing magnetic force.

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For a fixed magnet and pipe,
and over the useful low-speed range,

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gather the geometry and electrical
resistance into one constant k.

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Then the drag magnitude
is approximately k times v.

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Take downward as positive.

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Newton's second law says mass times
acceleration equals the downward weight,

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m g, minus the upward
electromagnetic drag, k v.

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At first v is small, so weight
wins and the magnet accelerates.

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As v increases, the drag grows.

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The rising curve shows the speed
approaching a limiting value.

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Terminal speed is reached
when acceleration becomes zero.

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Then the forces balance:
m g equals k v sub t.

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Solve that one line.

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The terminal speed is m g divided by k.

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A heavier magnet tends to fall faster;

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stronger magnetic coupling
or lower copper resistance increases k

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and lowers the terminal speed.

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The balance is stable.

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Below terminal speed, weight is larger
than drag and the magnet speeds up.

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Above terminal speed, drag is larger
than weight and the magnet slows down.

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At terminal speed the magnet still
loses gravitational potential energy.

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Each second, weight
supplies power m g v sub t,

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and the many loop currents
dissipate the same total power as heat.

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That is why the magnet does not hover
and why it does not keep accelerating.

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It descends steadily,
converting gravitational energy

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into many tiny resistive losses
distributed along the copper wall.

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The copper-pipe experiment
looks like a curiosity,

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but engineers use
the same effect deliberately.

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An eddy-current brake places strong
magnets close to a conducting rail

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or metal braking surface.

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The magnets do not have to touch the rail.

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Their field reaches across
the gap into the conductor.

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As the train moves right, each patch of
rail experiences changing magnetic flux.

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Closed circulating currents
form within the bulk metal.

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These are eddy currents,

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the extended-sheet version
of the current in our single copper loop.

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Lenz's law fixes their direction.

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The currents create magnetic fields
that oppose the passing magnet pattern,

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so the force on the train
points left, opposite its velocity.

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Watch the assembly move along the rail.

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The field pattern,
eddy currents, and braking force travel

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with the active region, while the
conducting rail itself remains fixed.

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The causal chain is exactly
the one we built for the pipe.

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Relative speed produces changing flux.

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Changing flux produces current.

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Current produces
the opposing braking force.

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The train's kinetic energy becomes
electrical energy in the eddy currents

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and then resistive heat
in the rail or brake disc.

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There is no mystery energy sink and no
ordinary magnetic attraction to copper.

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As speed falls,

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the flux changes more slowly, so induced
current and braking force weaken.

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At rest the motion-driven
eddy currents disappear.

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Real trains therefore combine this smooth,

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low-wear method with other braking
systems that can hold the vehicle still.

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Three statements carry the whole lecture.

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First, changing flux
induces current in a conductor.

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Copper need not be a permanent magnet.

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Second, the induced current
makes its own magnetic field.

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Lenz's law gives the direction
that opposes the relative motion.

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Third, the lost mechanical
energy becomes heat.

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In the pipe that energy conversion makes a
falling magnet descend at terminal speed.

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On a train, the same
conversion is useful braking.

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So the magnet falls slowly not because
copper is secretly magnetic, but because

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motion continually creates currents
whose magnetic effects resist that motion.

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The pipe demonstrates the law.

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The train brake puts it to work.