MOTIONS OF ELECTRONS 193 



be changed to produce for such a conductor a given 

 change in current. 



If the average velocity is doubled the same number 

 of electrons will pass a cross section of the conductor 

 in half the time previously required ; that is, the cur- 

 rent, or rate of transfer of electricity, is doubled. The 

 kinetic energy of the electrons is, however, quadrupled, 

 since it varies as the square of the velocity. This 

 energy is dissipated in heat in the circuit, and thus we 

 see that the energy required to force a current through 

 a given circuit varies as the square of the current. 

 This is known as Joule's Law and is usually expressed as 



W=Ri 2 (5) 



where R is a factor of proportionality, known as the 

 resistance. 



Returning to the numerical problem we see that the 

 energy has been quadrupled by doubling the current, 

 but that the quantity of electricity transferred has 

 only been doubled. The energy per unit quantity, 

 that is, the potential difference, has thus been doubled. 

 The current and potential difference are therefore 

 directly proportional. 1 



The fact that the current is directly proportional to 

 the difference of potential is usually expressed hi sym- 

 bols as 



E = Ri (6) 



where R may be shown to be the same factor of pro- 

 portionality as was introduced into equation (5) above. 



1 This is also evident from equation (3), namely, W = EQ, since 

 if W is made four times as large, while Q is made twice as large, 

 then E must be twice as large, 

 o 



