Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/487

x-components of velocity are comprised between u and u + du is

$\pi ^{-{\frac {1}{2}}}\alpha ^{-1}Nc^{-{\frac {u^{2}}{\alpha ^{2}}}}\ du\ dx$, where ${\tfrac {3}{4}}m\alpha ^{2}=qT$,

m denoting the mass of an electron, T the absolute temperature, and q, the universal constant previously introduced.

Now, an electron whose x-component of velocity is u will arrive at the interface within an interval dt of time, provided that at the beginning of this interval it is within a distance udt of the interface. So the number of electrons whose x-components of velocity are comprised between u and u + du which arrive at unit area of the interface in the interval dt is

$\pi ^{-{\frac {1}{2}}}\alpha ^{-1}Nc^{-{\frac {u^{2}}{\alpha ^{2}}}}u\ du\ dt$.

If the work which an electron must perform in order to escape through the surface layer be denoted by φ, the number of electrons emitted by unit area of metal in unit time is therefore

$\int _{{\tfrac {1}{2}}mu^{2}=\phi }^{\inf }\pi ^{-{\tfrac {1}{2}}}\alpha ^{-1}Ne^{-{\frac {u^{2}}{\alpha ^{2}}}}u\ du$, or ${\tfrac {1}{2}}\pi ^{-{\tfrac {1}{2}}}N\alpha e^{-{\frac {2\phi }{m\alpha ^{2}}}}$.

The current issuing from unit area of the hot metal is thus

${\tfrac {1}{2}}\pi ^{-{\tfrac {1}{2}}}N\epsilon \alpha e^{-{\frac {2\phi }{m\alpha ^{2}}}}$, or $N\epsilon .(qT/3\pi m)^{\tfrac {1}{2}}e^{-{\frac {3\phi }{2qT}}}$,

where ε denotes the charge on an electron. This expression, being of the form

$AT^{b}c^{-{\frac {b}{T}}}$,

agrees with the experimental measures; and the comparison furnishes the value of the superficial discontinuity of potential which is implied in the existence of φ.^[1]

A few years after the date of this investigation, a plan was

1. This discontinuity of potential was found to be 2·45 volts for sodium, 4·1 volts for platinum, and 6·1 volts for carbon.