Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/482

open circuit in which there is no conduction of heat or electricity (so that i and da/dx are zero). Integrating the equation

$E={\frac {m}{2e}}{\frac {a^{2}}{N}}{\frac {dN}{dx}}$

across the junction of the metals, we have

Discontinuity of potential at junction $={\frac {ma^{2}}{2e}}\log {\frac {N_{b}}{N_{a}}}$;

or since 3/4ma², which represents the average kinetic energy of an electron, is by Drude's assumption equal to qT, where q denotes a universal constant, we have

Discontinuity of potential at junction $={\frac {2}{3}}{\frac {q}{e}}T\log {\frac {N_{b}}{N_{a}}}$;

This may be interpreted as the difference of potential connected with the Peltier^[1] effect at the junction of two metals; the product of the difference of potential and the current measures the evolution of heat at the junction. The Peltier discontinuity of potential is of the order of a thousandth of a volt, and must be distinguished from Volta's contact-difference of potential, which is generally much larger, and which, as it presumably depends on the relation of the metals to the medium in which they are immersed, is beyond the scope of the present investigation.

Returning to the general equations, we observe that the flux of energy W is parallel to the axis of x, and is given by the equation

$W={\tfrac {1}{2}}m\textstyle \iiint ur^{2}f(u,v,w)du\ dv\ dw$,

where the integration is again extended over all possible values of the components of velocity; performing the integration, we have

$W={\frac {2ml}{3\pi ^{\frac {1}{2}}}}\left({\frac {2ea}{m}}NE-a^{2}{\frac {dN}{dx}}-3Na^{2}{\frac {da}{dx}}\right)$;

or, substituting for E from the equation already found,

$W={\frac {ma^{2}}{e}}i-{\frac {4ml}{3\pi ^{\frac {1}{2}}}}Na^{2}{\frac {da}{dx}}$.

1. Cf. p. 264.