98 Systems of Conductors [OH. iv 



This being true for all values of E z , E 3 , ... we must have 



Pl2=P&> Pl3=P23, etC. 



Next let us put unit charge on 1, leaving the other conductors uncharged. 

 The energy is ^pu. If we join 1 and 2 by a wire, the conductors 1 and 2 

 form a single conductor, so that the electricity will all flow to the outer 

 surface. The wire may now be removed, and the energy in the system is \p^. 

 Energy must, however, have been lost in the flow of electricity, so that p 22 

 must be less than p u . 



Since we have already seen that p l2 = p 22 and p n -pu cannot be negative, 

 it is clear that p^ cannot be greater than p u . The foregoing argument, 

 however, goes further and enables us to prove that pup& is actually 

 positive. 



Let us next suppose that conductor 2 is put to earth, so that F 2 = 0. 

 Then if E 1 = 0, it follows that F x = 0. Hence from the equations 



E 1 = q ll V l + q l2 V 2 + ...+?i W 7 n (44) 



we obtain in this special case that 



quV* + quV*+..-+qinV n = 0. 

 This is true, whatever the values of F 3 , F 4 , ..., so that 



#13 = #14 = = qm = 0. 



Suppose that conductor 1 is raised to unit potential while all the other 

 conductors are put to earth. The aggregate strength of the tubes of force 

 which go to infinity, namely q u + q iz + ... +q ln ( 112), is in this case zero, so 

 that q l2 =-qu. 



The system of equations (44) now reduces, when F 2 = 0, to 



E, = q u V, (45), 



^ 2 = ^i 2 F 1 + g 23 F 3 + g 24 F 4 + (46), 



Equations (47) shew that the relations between charges and potential 

 outside 2 are quite independent of the electrical conditions which obtain 

 inside 2. So also the conditions inside 2 are not affected by those outside 2, 

 as is obvious from equation (45). These results become obvious when we 

 consider that no lines of force can cross conductor 2, and that there is no way 

 except by crossing conductor 2 for a line of force to pass from the conductors 

 outside 2 to those inside 2. 



An electric system which is completely surrounded by a conductor at 

 potential zero is said to be "electrically screened" from all electric systems 



