534-537] Rapidly alternating currents 467 



.(479). 



If we eliminate the current-components from the system of equations 

 (477) and (478), we obtain 



and similar equations are satisfied by b and c. 



536. The equation which has been found to be satisfied by u, v, w, 

 a, ft and 7 is the well-known equation of conduction of heat. Thus 

 we see that the currents induced in a mass of metal, as well as the com- 

 ponents of the magnetic field associated with these currents, will diffuse 

 through the metal in the same way as heat diffuses through a uniform 

 conductor. 



Rapidly alternating currents. 



537. The equations assume a form of special interest when the currents 

 are alternating currents of high frequency. We may assume each component 

 of current to be proportional to e ipt (cf. 514), and may then replace the 



operator -- by the multiplier ip. The equations now assume the form 



(481), 



T 



and if p is so large that it may be treated as infinite, these equations assume 

 the simple form 



u v = w 0, 



a = b = c = 0. 



Thus for currents of infinite frequency, there is neither current nor 

 magnetic field in the interior. The currents are confined to the surface, 

 and the only part of the conductor which comes into play at all is a thin 

 skin on the surface. 



Equations (481) enable us to form an estimate of the thickness of this 

 skin when the frequency of the currents is very great without being actually 

 infinite. 



At a point on the surface of the conductor, let us take rectangular 

 axes so that the direction of the current is that of Ox while the normal to 

 the surface is Oz. If the thickness of the skin is very small, we need not 



302 



