PELTIER'S PHENOMENON. 239 



Suppose, for instance, that between two points A and B, at 

 potentials Vj and V 2 , there is a series of conducting arcs (Fig. 52). 

 Let R be the resistance of one of them, and I the strength deduced 

 from Ohm's law that is to say, such that IR = V 1 -V 2 = E, and 

 suppose that by a change of conditions, the strength in this con- 

 ductor becomes ! + /. The expression for the total quantity of 

 heat developed in the new system will be 



but the product RI is a constant for each of the arcs, and on the 

 other hand i is necessarily zero, if the current which terminates 

 at the point A is not modified; the quantity of heat reduces 

 therefore to 



and it is obviously a minimum, for z = that is to say when the 

 strength divides in the branches according to Ohm's law. 



247. PELTIER'S PHENOMENON. Let us now suppose that 

 between two points A and B, always kept at the same potentials 

 Vj and V 2 , the value of the potential, instead of varying in proportion 

 to the resistances, undergoes a sudden fall Uj - U 2 = H, at a point 

 P between two adjacent surfaces, which is independent of the 

 current ; the expression for this strength will no longer be the 

 same as in the preceding case. 



If Rj and R 2 are the resistances of the two portions AP and PB, 

 we have thus (210) 



V 1 -U 1 U 2 -V 2 V 1 -V 2 -(U 1 -U 2 )E-H 



_ 

 R 2 Ri + R 2 R 



The total energy expended between the points A and B is 



W = I(V 1 -V 2 ) = IE, 

 which gives 



This energy consists then of two parts ; one which is propor- 

 tional to the square of the strength of the current, and which heats 

 the conductor throughout its entire length corresponding to Joule's 

 law ; and another, which is proportional to the current, is localised 



