(in Conducting Sheets and Solid Bodies. 11 



Remembering that the magnetic potential is — (Pr), we see 

 that the effect of the sphere is to introduce a new varying 

 field with phase increased by a greater period, and which in 

 other respects bears the same relation to the old as the electro- 

 static field produced by an uninsulated sphere of the same 

 radius bears to the inducing field, w r hen the strength of the 



former is diminished in the ratio 7 — rr\73 — , 1 w ~ — r~o\ • • 



(n + l)(2n + l)(2w + 3) cr 



By combining the proper harmonic solutions, any possible 

 case can of course be represented. 



The form of the solution obtained in equation (4) suggests 

 another way of looking at the problem. The first term gives 

 a field equal and opposite to the inducing field at each instant; 

 the following terms represent its decay. Since V 2 Po = 0, 

 that opposite field can only be due to a surface distribution of 

 currents. Thus, if at each instant we suppose a system of 

 currents to start in the superficial layer of the body which 

 neutralizes for internal space the effect of the outside changes, 

 the actual state of the body is that produced by these currents 

 soaking into it and decaying by their own mutual action. The 

 equation of decay (3) with P omitted is the same as the 

 equation for the diffusion of heat from the surface into the 

 body, though the boundary conditions are different; the cor- 

 responding thermal diffusivity (conductivity divided by den- 

 sity) will be cr/47r. The value of this for copper is about 

 130; and as the actual heat diffusivity for copper is about 1*2, 

 we see that to have penetration into the solid of the same order 

 in both cases the oscillations must be about 100 times quicker 

 for the electrical case. 



V. We now proceed to consider the practically important 

 case of a conductor in motion in a magnetic field. If you 

 consider any closed circuit in the conductor, it is clear 

 that the electromotive force round it depends only on the 

 change produced by its motion in the number of tubes of 

 force that it encloses, and is therefore quite independent of 

 whether the relative motion of the conductor and the field be 

 ascribed to the conductor or to the magnetic field, or to both 

 conjointly. Therefore the currents induced in the body are 

 derived from the same equations, whether the axes are fixed, 

 or are moving in any manner , uniform or not. But in the 

 case of an unclosed circuit there is a difference introduced in 

 the value of the electrostatic potential. In fact such an open 

 line which is at rest relatively to the moving axes is displaced 

 across the field, owing to the motion of the axes : if you sup- 



