498 Displacement Currents [OH. xvn 



acting on an electromagnetic unit of charge), will be CX, CY, CZ, so that we 

 have the relations 



etc ............................... (532), 



and equation (526) becomes 



' ~ 7 , * f\ r 



\ r C dt/ dy dz 



Thus the present system of equations differs from that previously obtained, 

 in which the displacement-current was neglected, by the presence of the term 



K dX 



-^ -=. To form an estimate of the relative importance of this term, let us 



examine the case of an alternating current in which the time factor is e ipt . 

 We may as usual replace - by ip t and equation (533) becomes 



V r C r dy dz' 



Thus neglecting the displacement-current amounts to neglecting the 

 ratio Kipr/4t7rC 2 . Clearly the neglect of this ratio produces the greatest 

 error in problems in which" r is large (conductors of high resistance) and in 

 which p is large (rapidly changing fields). On substituting numerical values 

 it will be found that in problems of conduction through metals, the neglect 

 of the factor Kiprl^irC* produces a quite inappreciable error unless p is com- 

 parable with 10 15 i.e. unless we are dealing with oscillating fields of which 

 the frequency is comparable with that of light-waves. Thus the effect of 

 the displacement-current in metals has been inappreciable in the problems 

 so far discussed, so that the neglect of this effect may be regarded as 

 justifiable. The matter stands differently as regards the problems to be 

 discussed in the next chapter, in which the oscillations of the field are 

 identical with those of light-waves. 



EQUATIONS FOR AN ISOTROPIC DIELECTRIC. 



574. The equations assume special importance when the medium is 

 isotropic and non-conducting. There can be no conduction-current, so that 

 we put u = v = w = 0. We also put 



4>7rf= KX, etc., a = /-tot, etc. 

 The equations now become 



^<OT = 87 _8/3\ ^ da _dZ dY\ 



C dt ~ dy fc\ " G ~di = lfy~ ~fo 



KdY da dy\ p, d8 dX dZ 



KdZ_ = d@_fa\ _*y_^_?* 



G dt dx dyl ~C~dt~lte~l)y 



