362 On the Thermodynamic Efficiency of the Thermopile. 



be represented approximately by 



ne(t — t Q ). 



The magnitude of the current (C) is found by dividing 

 this by the sum of the internal and external resistances 

 ( R + R) ; and the useful work done externally per second is 

 IiC 2 . It reaches a maximum when the external resistance is 

 equal to the internal ; and its amount is then 



nVQ-g 2 

 4R " 



The value of the internal resistance R depends upon the 

 dimensions and specific resistances of the bars. Denoting the 

 latter quantities by r X) r 2 , and taking cr 1? a 2 to represent the 

 areas of section, the common length being I, we have 



B =nl( r -± + r JL); 



so that the external work per second is 



ne\t-t n y 



u( r -i+ r J.) 



Wi cr 2 / 



We will now compare this with the work dissipated by 

 ordinary conduction of heat along the bars. 



If Q be the amount of heat conducted by the n pairs, r{ } 

 r 2 the thermal resistances, then 



The fraction of this heat, supplied at temperature t, which 

 might be converted into work by a perfect engine working 

 between the absolute temperatures t and t Q , is (t — t )/t; so that 

 the work dissipated per second is 



nJ(t-t ) 2 /a 1 a,\ 



tl W r 2 'J> 



where J denotes the mechanical equivalent of heat. 

 The ratio of this to the useful work is 





4J 

 te 



independent of {t — t ), of n, and of /. It is further evident 

 that the ratio in question does not depend upon the absolute 

 values of the sections, or of the electrical and thermal resist- 

 ances, but only upon the ratios of these quantities. Thus the 

 efficiency of the thermopile is independent of the absolute 



