HALL. — ELECTRIC CONDUCTION AND THERMOELECTRIC ACTION. 89 



Conditions of Equilibrium at a Junction of Two Metals at the same 



Temperature. 



Let the two metals be called Mi and Mi, and let each be throughout 

 at the temperature T. As the thermo-electric force of an isothermal 

 circuit is zero, whatever the arrangement of the metals, we may 

 assume the transition from Mi to Mi to be as gradual as we please to 

 make it, by way of an intermediate stretch of an alloy of the metals, 

 this alloy changing by any law from pure Mi at one end to pure Mo 

 at the other end. 



If we take the ordinary view of an alloy and think of it as a physical 

 mixture of the component metals in small particles, much larger, 

 however, than the atoms or molecules, we shall have the mean n and 

 the mean R, in the successive cross-sections of the bridge reaching 

 from M\ to pure Mo, changing gradually, by the increasing proportion 

 of Mo particles, from in and Ri to no and Ro. It seems reasonable, 

 if not inevitable, to assume that for any given cross-sectional slice of 

 this bridge we shall have 



\ R\ — Ri 



and 



ni — no 

 Ri - Ri 



(24) 



(25) 



ni — no 



For equilibrium of condition of the slice in. question, the thickness 

 of which we shall call dl, we must, if we disregard thermal effusion 

 and assume that the free electrons, as a gas, tend to uniformity of 

 pressure, have the difference between the electron gas-pressures at the 

 two faces equal to the pull of the virtual potential gradient on all the 

 free electrons in the slice. This consideration, with the reflection 

 that p, the gas pressure of the electrons, is n R T, leads to the equation 



dp=T ( ^dl=-n ( ^dl, (26) 



dl dl 



whence 



dF= - TR— - TdR. (27) 



Substituting for R and dR from (24) and (25), we get 



dp = _ 2T Ri- R 2 ^ _ f r _ Ri - RA dn^ 



ni — no \ iii — no J n 



