IV. On the induction of electric currents in infinite plates and spherical shells

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Abstract
1. In Vol. XX. (1872) of the Proceedings of the Royal Society (pp. 160-168) is a beautiful paper by the late Professor Clerk Maxwell giving an investigation of the induction of currents in an infinite plane sheet of uniform conductivity. For the purposes of the investigation the sheet is supposed infinitely thin; and when it is at rest and influenced by a varying external magnetic system, the effect of the currents induced in it is found to be equivalent to an infinite train of images, at the sheet, of the external system, which, after being formed, move off to infinity with uniform velocity. When the external system revolves uniformly round an axis normal to the sheet, the effect is shown to be the same as if the sheet itself revolved round the axis and the magnetic system remained fixed. The images will then lie in a spiral trail in the form of a helix whose axis is perpendicular to the sheet. This theory was afterwards reproduced in his ‘Treatise on Electricity and Magnetism,’ and the latter part proved directly from the equations. The analysis there given is somewhat difficult to follow, though it is doubtless possible to present it in a more logically exact form. The problem of the induction of currents has also been treated by Felici (Tertolini’s ‘Annali,’ 1853-54) and by Jochmann (Crelle, 1864, and Pogg. Ann., 1864). Jochmann has solved the case of a sphere which rotates uniformly in a magnetic field symmetrical about the axis of revolution and finds that no currents will be generated in it, but that there will be a certain distribution of free electricity throughout its interior and over its surface. He has also handled the case of an infinite plate of finite thickness, which revolves uniformly round a normal, by neglecting the inductive action of the currents on themselves, and shows that the conditions of the problem may then be satisfied by a system of currents parallel to the faces of the plate; he has also traced the forms of the current and equipotential lines in some simple cases. The solution, however, as Maxwell has shown in the case of a thin copper disc, can be true only for very small values of the angular velocity.

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