90 ELECTROLYSIS. 



Calling K the capacity, E the electromotive force, r the 

 internal resistance, E the external resistance, and C the intensity, 

 we have : 



(r + E) 2 



E being constant, we have to find out the maximum of the 

 equation : 



BE 1 



(r + E) 2 " E 2 + 2Br + r 2 ~ B + 2r + v* ' 



B 



r 2 



The maximum will occur when B + 2r +^ will be minimum ; 



K 



r being constant, the problem amounts to finding the minimum 



r 2 



of B + =j . 

 Jtl 



That quantity is minimum when B = r. For if we suppose 

 that there is, between Band r a difference a, positive or negative, 



r 2 r * 



the equation B + TT will become r + a -\ ; , a quantity 



B a + r 



which will always be greater than 2 r, as if we subtract r + a 

 from these two equations, multiplying the results by r -j- a we 

 obtain respectively r 2 and r 2 a?. The second expression 1r is 

 therefore inferior to the first one, and the minimum of B + 2r 



r 2 



+ which, as we have seen, corresponds to the maximum 

 B 



capacity, will be obtained when the external resistance B is equal 

 to the internal resistance r. 



ELECTKICAL EFFICIENCY OF A BATTERY. The electrical 

 efficiency of a battery is greater in proportion to the external 

 resistance. 



In the case of the maximum capacity it is : 



(, + B)&-i&- 5 ' 



which amounts to saying that when the battery produces the 

 greatest possible amount of external work, it consumes internally 



