In 1910, Debye suggested that the work function W of a metal could be computed as the work performed against the electrostatic image force when an electron is removed from the interior of a finite piece of metal to a point infinitely far outside the metal. Model the metal as a perfectly conducting sphere with a macroscopic radius R and suppose that the image force only becomes operative at a microscopic distance d outside the surface of the metal.

(a) Show that

(b) Let x = d/R and take the limit R→∞ to find the Debye model prediction for the work function of a semi-infinite sample. It is understood today that the image force plays an insignificant role in the work function.

W = e 2 87o [R+d_=(R+d) R + R (R+d) - R R].