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Unipolar Induction

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Unipolar Induction via a Rotating Conducting Magnetized Cylinder Kirk T McDonald Joseph Henry Laboratories Princeton University Princeton NJ 08544 August 17 2008 updated November 13 2012 1 Problem 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 z 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 R is small compared to the speed of light c Comment on the electric polarization density P in the cylinder should it have relative permittivity that di ers from unity This con guration of unipolar induction was rst 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 2 2 1 Solution Analysis Using a Comoving Inertial Frame As discussed in 2 3 4 5 the best approach to an understanding of lab frame electrodynamics of a rotating system is via a comoving inertial frame corresponding to some point in the rotating system We follow Minkowski 2 in arguing that the local magnetization at a point P in the rotating cylinder equals the rest value M0 according to an observer in the inertial frame that is instantaneously comoving with point P That is M M0 where the superscript indicates quantities observed in the comoving inertial frame Similarly we expect that the electric polarization P near point P in the comoving inertial frame equals that of the magnetized cylinder in an inertial rest frame namely P 0 Writing v as the velocity of point P in the lab frame the eld transformations to the comoving inertial frame are

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