Derive the following equation for the Compton shift.

$[10$ Marks$]$

When a sodium metal is illuminated with light of wavelength $4\mathrm{.}20{10}^{2}nm,$ the stopping potential is found to be $0\mathrm{.}65$ V $,$ when the wavelength changes to $3\mathrm{.}10{10}^{2}nm,$ the stopping potential is $1\mathrm{.}69$ V $.$ Determine the work function of sodium.

$[3$ Marks$]$