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XVI. Note on the Induction of Electric Currents, in an Infi- 

 nite Plane Current Sheet, which is rotating in a Field of 

 Magnetic Force. By A. B. Basset, M.A.* 



1. npHE determination of the currents induced in a thin 

 J- plane conducting sheet of infinite extent, which is 

 rotating about an axis perpendicular to itself in a field of 

 magnetic force, may be effected either by employing the 

 equations of electromotive force referred to moving axes, or 

 by dealing with the problem directly by means of the method 

 of images. The former method has been employed by Max- 

 wellf and Prof. C. NivenJ, and is probably the preferable one 

 when the currents cannot be conveniently represented by a 

 system of images ; but when the magnetic field is produced 

 by a system of magnets or currents, the problem can be easily 

 solved by finding an analytical expression for the magnetic 

 potential of the moving trail of images, to which the currents 

 are equivalent. 



In the present note I have employed the latter method of 

 dealing with the problem, and have thence deduced Niven's 

 results ; and I have also worked out the solution in the case 

 of a current sheet rotating about an axis, in the presence of 

 a short magnet which is fixed at right angles to the axis of 

 revolution. 



2. To solve the problem by means of images, let us choose 

 the instant at which we desire to observe the currents as the 

 origin of the time, and reckon the time backwards from this 

 epoch. 



Adopting Maxwell's notation, let F (z, t) be the value of 

 P' at time r ago (where — dY'jhz is the magnetic potential 

 of the inducing system), and let the system P' be suddenly 

 introduced and kept at rest ; a system of currents will be 

 instantaneously generated for which the value of P (where 

 '—d¥/dz is the magnetic potential of currents) isf(z, t), the 

 form of the function / being determined from the fact that it 

 is the image of F with respect to the sheet. 



This system of currents, as soon as it has been generated, 

 will begin to decay, the law of decay being such that, at the 

 end of an interval T, 



P=/(*+BT,t), 



where 27rK is the specific resistance of the sheet. 



If, therefore, at the end of an interval dr the system P / be 



* Communicated by the Author 



t Electricity and Magnetism, art. 668. 



% Phil. Trans. 1881. 



