1 Problem Unipolar Induction via a Rotating Magnetized Cylinder


A conducting cylinder of radius R with permanent magnetization density M0 parallel to its axis when at rest is rotated about that axis with angular velocity ω = ω ˆz with respect to the lab frame. A voltmeter with very high internal resistance is connected to the rotating cylinder via wires with sliding contacts, one of which (C1) is on the axis of the cylinder and the other (C2) is on the circumference, as shown below. Deduce the voltage V observed on the voltmeter by a lab-frame analysis as well as by an analysis in the rotating frame. You may assume that the velocity ωR is small compared to the speed of light c. Comment on the electric polarization density P in the cylinder should it have (relative) permittivity ɛ that differs from unity. This configuration of unipolar induction was first considered by Faraday in 1851 [1], who also considered the case of the magnetized cylinder at rest while the voltmeter and contact wires rotated around the axis of the cylinder. 1

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