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# Magnetic Braking

When an electrical conductor, such as copper or aluminum, moves through the field of a permanent magnet or an electromagnet, electromagnetic induction creates eddy currents, which dissipate some of the kinetic energy into Joule heat and results in slowing the motion of the conductor. This principle is utilized in the construction of magnetic brakes. This Demonstration shows magnetic braking applied to a rotating metallic disk. This might, for example, serve to control resistance to motion in exercise machines. Magnetic braking can also find applications in roller coasters and railroad trains, in which the metallic conductor has the shape of a linear rail. In contrast to conventional friction brakes, there is no direct contact between interacting surfaces, which makes magnetic braking more reliable and reduces wear and tear.
Referring to the graphic, the sequence of events in magnetic braking can be described as follows. Assume the pole of the upper face of the magnet is the north pole. Then an electron of charge in the metal moving counterclockwise with velocity will experience a Lorentz force , where is the magnetic induction inside the metallic disk. This force is directed radially outward, away from the center of the disk. This can also be viewed as induced current, in terms of Faraday's law. The result is the creation of eddy currents, which, in this case, move radially outward directly between the magnetic poles and complete closed loops through regions in which the magnetic field is weaker. A second application of the Lorentz force law on the radially outward component of the electron's motion produces a clockwise force, which opposes the original motion of the electrons—hence the braking action. This is in conformity with Lenz's law, which states that an induced current is always in a direction that opposes the change causing it.
This Demonstration represents the effect only qualitatively. A more accurate analysis would involve solutions of time-dependent Maxwell's equations, taking into account geometric and electrical parameters of all the components. A laboratory simulation is described in the reference cited below. An approximate result for the torque on the disk is given by
,
where is the thickness of the rotating disk, is the area of a magnet pole face, is the average radial distance of the pole face from the center of the disk, is the conductivity of the metal, is the instantaneous angular velocity, and is the magnitude of the magnetic induction. Interestingly, a magnetic brake cannot itself bring the conductor to a perfect stop. The last bit of action has to be the effect of old-fashioned friction.

### DETAILS

Snapshot 1: metallic disk set into rotation
Snapshot 2: magnetic brake is applied; eddy currents are shown
Snapshot 3: disk brought to rest as eddy currents are damped out
Reference: H. D. Wiederick, N. Gauthier, D. A. Campbell, and R. Rochon, "Magnetic Braking: Simple Theory and Experiment," American Journal of Physics, 55(6), 1987 pp. 500–503.

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