348 Steady Currents in continuous Media [CH. x 



currents have become steady, the value of V will be determined by 

 equation (325) and the boundary conditions, and the value of p is then given 

 by equation (326). 



From equations (325) and (326), we obtain 



,_ 1 f/i(^) + f S |F3 I 



4<7rT\dx dx^ ty ty dzdz^ '} 



The condition that p shall vanish, whatever the value of F, is that Kr shall 

 be constant throughout the dielectric : if this condition is satisfied the value 

 of p necessarily vanishes at every point for all systems of steady currents. 

 The most important case of this condition being satisfied occurs when the 

 dielectric is homogeneous throughout. If Kr is not constant throughout 

 the dielectric, equation (327) shews that we can have p at every point 

 provided the surfaces F = cons, and Kr = cons, cut one another at right 

 angles at every point, i.e. provided Kr is constant along every line of flow. 



We have already had an illustration ( 381) of the accumulation of 

 charge which occurs when the value of Kr varies in passing along a line 

 of flow. 



Time of Relaxation in a Homogeneous Dielectric. 



396. Let a homogeneous dielectric be charged so that the volume 

 density at any point is p. 



If any closed surface is taken inside the dielectric, the total charge 

 inside this surface must be 



1 1 1 p dxdydz, 



while the rate at which electricity flows into the surface will, as in 375, be 



(lu + mv + n w) dS, 



where u, v, w are the components of current and I, m, n are the direction 

 cosines of the normal drawn into the surface. Since this rate of flow into 

 the surface must be equal to the rate at which the charge inside the surface 

 increases, we must have 



1 1 (lu + mv + nw) dS = -r Ml P dxdydz 



^ t dxdydz. 



-SSI 



The integral on the left may, by Green's Theorem, be transformed into 



du dv 



M 



and this again is equal, by equations (310), to 



8 2 F 8 2 F 



