468 Induction of Currents in Continuous Media [CH. xv 



consider any region except that in the immediate neighbourhood of the 

 origin, so that the problem is practically identical with that of current 

 flowing parallel to Ox in an infinite slab of metal having the plane Oxy 

 for a boundary. 



Equation (481) reduces in this case to 



and if we put = K 2 , the solution is 



u = Ae- 

 The value of K is found to be 



J **!!* 



^ r 



so that u = Ae 



and the condition that the current is to be confined to a thin skin may now 

 be expressed by the condition that u when ^ = oo , and is accordingly 

 .5 = 0. The multiplier A is independent of z t but will of course involve 

 the time through the factor e ipt \ let us put A = u e ipt , and we then have 

 the solution 



Rejecting the imaginary part, we are left with the real solution 



/2*w 



* 

 cos ( pt 



/Z-TT 



A/ - 



from which we see that, as we pass inwards from the surface of the con- 

 ductor, the phase of the current changes at a uniform rate, while its amplitude 

 decreases exponentially. 



We can best form an idea of the rate of decrease of the amplitude by considering a 

 concrete case. For copper we may take (in c.G. s. electromagnetic units) /Lt = l, r = 1600. 

 Thus for a current which alternates 1000 times per second, we have 



p = 27T x 1000, y = 5 approximately. 



It follows that at a depth of 1 cm. the current will be only e~ 6 or -0067 times its value 

 at the surface. Thus the current is practically confined to a skin of thickness 1 cm. 



rz<x> 

 The total current per unit width of the surface at time t is I udz, of 



J 2 = 



which the value is found to be 



u cos [pt 



