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 



TT _ T>f<2 _ _ ^ & 



~(f+W 



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

 equation : 



R R ___ 1 



(r + R) 2 ~ R 2 + 2Rr + r 2 ~ R + 2r + ' 



R 



r 2 



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



Jbv 



r being constant, the problem amounts to finding the minimum 



ofE + i- 



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

 that there is, between R and r a difference a, positive or negative, 



the equation R 4- 73 w ^ become r + a -\ -- , a quantity 



Xii Ct ~T~ Y 



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

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

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

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



r * 

 + -=r- which, as we have seen, corresponds to the maximum 



1C 



capacity, will be obtained when the external resistance R 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 : 



which amounts to saying that when the battery produces the 

 greatest possible amount of external work, it consumes internally 



