A monochromatic light of wavelength λ is incident on an isolated metallic sphere of radius a. then threshold wavelength is  λ_0 which is larger than λ . Find the number of Photoelectron emitted before the emission of photo electrons stops.

Q: A monochromatic light of wavelength λ is incident on an isolated metallic sphere of radius a. then threshold wavelength is  λ_0 which is larger than λ . Find the number of Photoelectron emitted before the emission of photo electrons stops.

Sol: As the metallic sphere is isolated, it becomes positively charged when electrons are ejected from it.
There is an extra attractive force on the photoelectrons. If the potential of the sphere is raised to V, the
electron should have a minimum energy W + eV to be able to come out. Thus, emission of
photoelectrons will stop when.

$\large \frac{h c}{\lambda} = W_0 + e V $

$\large \frac{h c}{\lambda} = \frac{h c}{\lambda_0} + e V $

$\large V = \frac{h c}{e}(\frac{1}{\lambda}-\frac{1}{\lambda_0}) $

The charge on the sphere needed to take its potential to V is Q = (4 π εo a) V

The number of electrons emitted is, therefore,

$\large n = \frac{Q}{e} = \frac{4\pi \epsilon_0 a V}{e}$

$\large n = \frac{4\pi \epsilon_0 a h c}{e^2} (\frac{1}{\lambda}-\frac{1}{\lambda_0}) $