537, 538] Plane Current-sheets 469 



Thus, if we denote the amplitude of the aggregate current by U, the 

 value of u will be U */ - 



The heat generated per unit time in a strip of unit width and unit 

 length is 



r I I u"dtdz 



Thus the resistance of the conductor is the same as would be the 



resistance for steady currents of a skin of depth 1 ' ' 



The results we have obtained will suffice to explain why it is that the conductors used 

 to convey rapidly alternating currents are made hollow, as also why it is that lightning 

 conductors are made of strips, rather than cylinders, of metal. 



PLANE CURRENT-SHEETS. 



538. We next examine the phenomenon of the induction of currents 

 in a plane sheet of metal. 



Let the plane of the current-sheet be taken to be z = 0. Let us intro- 

 duce a current-function <I>, which is to be defined for every point in the 

 sheet by the statement that the total strength of all the currents which 

 flow between the point and the boundary is <l>. Then the currents in the 

 sheet are known when the value of 4> is known at every point of the sheet. 

 If we assume that no electricity is introduced into, or removed from the 

 current-sheet, or allowed to accumulate at any point of it, then clearly <3> will 

 be a single-valued function of position on the sheet. 



The equation of the current-lines will be <3> = constant, and the line 

 4> = will be the boundary of the current-sheet. Between the lines <S> and 

 <E> 4- d<& we have a current of strength d& flowing in a closed circuit. The 

 magnetic field produced by this current is the same as that produced by 

 a magnetic shell of strength d<& coinciding with that part of the current- 

 sheet which is enclosed by this circuit, so that the magnetic effect of the 

 whole system of currents in the sheet is that of a shell coinciding with 

 the sheet and of variable strength <J>. This again may be replaced by a 

 distribution of magnetic poles of surface density 3>/e on the positive side of 

 the sheet, together with a distribution of surface density O/e on the 

 negative side of the sheet, where e is the thickness of the sheet. 



