Forced Vibrations of Electromagnetic Systems, 367 



39. Effect of uniformly magnetizing a Conducting Sphere 

 surrounded by a Nonconducting Dielectric. — Here, of course, 

 it is the lines of E that are circles centred upon the axis, both 

 inside and outside. Let h be the impressed magnetic force, and 

 hv the surface-density of its vorticity, at r=a, outside which 

 the medium is nonconducting, and inside a conducting dielec- 

 tric. The differential equation of E OJ the surface value of the 

 tensor of E at r = a, is [compare (124), § 19] 



in which r=a, and /j, and q are to have the proper values on 

 the two sides of the surface. 

 Now, by (111), 



W'fW=-q{l + (qr)- 1 {l + qry i } . . . (240) 



in the case of m=l, (first order), here considered. This re- 

 fers to the external dielectric, in which q =p/v. Let v = <x> , 

 making 



W / /W=-a~ 1 (241) 



This assumption is justifiable when the sphere has sensible 

 conductivity, on account of the slowness of action it creates 

 in comparison with the rapidity of propagation in the dielec- 

 tric outside. Then (239) becomes, 



_^ = _J gift sinh qS\ + 1 / 1 _ 1 \ / 212 x 



E fi^acosh q 1 a — (q 1 a)~ 1 sinhq x a pa\fJL Pi) 



if /jl is the external and fa the internal inductivity, and q^ 

 the internal q. When the inductivities are equal, there is a 

 material simplification, leading to 



r ^asinh^a v ' 



where q l = { (47T&! -f Cip)fap}?. First let c : = 0, in the conduc- 

 tor, making q 1 2 = 4:7Tfak 1 p——s 2 , say. Then 



-n 1 cos sa— (sa)' 1 sin sa 1 , aM M\ 



L a = — -. — -. — r-^-A hv. . . (244) 



47rA; 1 a (sa)" 1 sin sa v ' 



From this we see that sin sa = is the determinantal equation 

 of normal systems. The slowest is 



sa=7r, or — p~ ] = 4// 1 & 1 a 2 /7r. . . . (245) 



This time-constant is about (1250) -1 second if the sphere be 

 of copper of 1 centim. radius ; about 8 seconds if of 1 metre 

 radius, and about 10 million years if of the size of the earth. 



