A dime is suspended from a thread and hung between the poles...

A dime is suspended from a thread and hung between the poles of a strong horseshoe magnet as shown in Figure P31.74. The dime rotates at constant angular speed $\omega$ about a vertical axis. Letting $\theta$ represent the angle between the direction of $\mathbf{B}$ and the normal to the face of the dime, sketch a graph of the torque due to induced currents as a function of $\theta$ for $0<\theta<2 \pi$.

Key Concepts

Electromagnetic Induction

This is the phenomenon by which a changing magnetic flux through a conductor induces an electromotive force (EMF) and consequently a current. In rotating systems, such as a spinning conductor in a magnetic field, variations in the orientation relative to the magnetic field cause the magnetic flux through the conductor to change continuously, leading to induced currents.

Eddy Currents

These are currents that are induced in bulk conductors when subjected to changing magnetic fields. They circulate in loops within the conductor and generate their own magnetic fields, which in turn affect the motion of the conductor. The presence of these currents is a key factor in producing the observed torque in rotating conductive objects.

Lenz's Law

This law states that the direction of the induced current is such that its magnetic field opposes the change in the magnetic flux that produced it. It is essential in determining the direction of the induced torque and explains why the induced currents in the rotating conductor generate forces that resist the motion.

Magnetic Damping and Induced Torque

When a conductor rotates in a magnetic field, the eddy currents generated by electromagnetic induction interact with the external magnetic field to produce forces (and thus torque) that oppose the motion. The magnitude of this induced torque depends on the angle between the magnetic field and the normal to the conductor's surface, usually resulting in a periodic, sinusoidal dependence on the angle, reflecting the rotational symmetry of the setup.

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