4 Prof. J. Larmor on Electromagnetic Induction 



We must add the proper surface distribution: for instance, if 

 the conductor be a sphere, the outside potential which corre- 

 sponds to the given value at the surface is 



t n a ^ 5 3 cos 2 6 — 1 , 1 a? 



^i=C . - — i coy^a? . 5-jp +i ft>7o-> 



p 6p p 



where p is the radial line, and cf> its inclination to the axis. 

 Thus the surface-density 



" 47r \dp dp J 



8 



IT 



The arbitrary constant C allows us to superpose any free dis- 

 tribution. If the charge of the body was originally zero, we 

 may give it such a value that the charge shall remain zero ; 

 but if the axis of rotation is uninsulated, the condition is that 

 C is zero. The above agrees with Jochmann's results. 



For the case of a flat disk rotating about an axis perpendi- 

 cular to its plane in any uniform field of force, we may divide 

 the force into two components — one parallel to the plane of 

 the disk, which produces no induction, on account of the thin- 

 ness of the sheet, and the other perpendicular to it, whose effect 

 has just been estimated. Thus Arago's rotations will not 

 occur in a uniform field. 



By connecting one terminal of a wire to the axis and 

 making the other terminal rub along the circumference, in 

 Faraday's manner, we utilize the difference of potential to pro- 

 duce a current in the external circuit. 



Of course these static charges are of an exceedingly minute 

 character. We have found ^ in electromagnetic units ; and 

 to reduce to electrostatic units we must divide by the reducing 

 factor v, which is approximately equal to the velocity of light: 



the electric volume-density is therefore — ~- electrostatic 



units, and the surface-density must also be divided by v to 

 reduce it to electrostatic units. 



III. We now proceed to the case of a spherical conducting 

 shell at rest in a magnetic field. 



Let <I> be the current-function in the sheet : the value of <5? 

 at any point will be the strength of the equivalent magnetic 

 shell "at that point. It is well known (Maxwell, art. 670) that 

 the magnetic potential in space not occupied by the sheet is 



