142 ADVANCED ELECTRICITY AND MAGNETISM. 



(d) One may think of electric flux going out from the positively 

 charged plate and as coming into the negatively charged plate. 



(e) One of the most important theorems relating to electric 

 field is that the amount of electric flux which goes out from a 

 positive charge or which comes in to a negative charge is pro- 

 portional to the* amount of the charge. This theorem is due to 

 Gauss. We will establish the theorem only for the case of oppo- 

 sitely charged flat metal plates. 



Gauss's theorem for oppositely charged flat metal plates with 

 air between them. According to Art. 79 the capacity of a 

 parallel-plate condenser with air as the dielectric is : 



I a 

 ~B'x 



'In farads ~ r? " (0 



where D is written for the constant G so that the value of B is : 



jD 



B = 1.131 X io 13 (2) 



where a is the area of one of the plates in square centimeters, 

 and x is the thickness of the air layer in centimeters. 



Let E be the electromotive force between the plates in volts. 

 Then the positive charge on one plate (or the negative charge 

 on the other plate) in coulombs is q = CE. Therefore, using 

 the value of C from equation (i), we have: 



2-i'f- (3) 



J7 



But : - is the intensity of the electric field between the plates in 



T? 



volts per centimeter, and therefore a is the electric flux from 



x 



plate to plate, according to Art. 85. Consequently the electric 

 flux going out from + q coulombs or the electric flux coming in to 

 g coulombs is Bq volt-centimeters. 



Gauss's theorem for oppositely charged flat metal plates with 

 any dielectric between them. According to Art. 79 the capacity 

 of a parallel-plate condenser with any dielectric is : 



