310 Steady Currents in Linear Conductors [OH. ix 



of amount V P V Q . The energy represented by the work shows itself in 

 a heating of the conductor. 



On the electron theory, we may imagine that the electrons are driven through the 

 conductor by the forces of the electric field, their motion being accompanied by in- 

 numerable collisions between the moving electrons and the atoms or molecules of 

 which the conductor is formed. If it were not for these collisions, the electrons would 

 continually gain in momentum under the action of the electric forces : the effect of 

 the collisions is to check this growth of momentum, so that through the agency of these 

 collisions all the excess momentum is transferred from the electrons to the atoms and 

 molecules of the conductor. The agitation of these latter is accordingly increased, 

 and this, according to the kinetic theory of matter, will result in a rise in the temper- 

 ature of the conductor, the energy of this rise of temperature being the exact equivalent 

 of the energy yielded up by the electrons. 



We are supposing that x units of electricity pass per unit time from 

 P to Q. Hence the work done by the electric field per unit time within the 

 region PQ is x(V P VQ), and this again, by equation (283) is equal to Rx 2 . 



Thus in unit time, the heat generated in the section PQ of the con- 

 ductor represents Rx 2 units of mechanical energy. Each unit of energy is 



equal to -, units of heat, where J is the " mechanical equivalent of heat." 

 J 



Thus the number of heat-units developed in unit time in the conductor PQ 

 will be 



It is important to notice that in this formula x and R are measured in 

 electrostatic units. If the value of the resistance and current are given in 

 practical units, we must transform to electrostatic units before using formula 

 (284). 



Let the resistance of a conductor be R' ohms, and let the current flowing through it 

 be of amperes. Then, in electrostatic units, the values of the resistance R and the current 

 x are given by 



R = 9x^10" and * =3x 10V ' 

 Thus the number of heat-units produced per unit time is 



Rx* _ (3x10^ 

 J "GxlOU.,/ 



and on substituting for J its value 4'2 x 10 7 in c.G.s. -centigrade units, this becomes 



0-24 R'i' 2 . 



Generation of Heat a minimum. 



356. In general the solution of any physical problem is arrived at by the 

 solution of a system of equations, the number of these equations being equal 

 to the number of unknown quantities in the problem. The condition that 

 any function in which these unknown quantities enter as variables shall be a 



