When photons of wavelength  ${\lambda }_1$  are incident on an isolated sphere, the corresponding stopping potential is found to be V. When photons of wavelength ${\lambda }_2$   are used, the corresponding stopping potential was thrice that of the above value. If light of wavelength ${\lambda }_3$   is used then find the kinetic energy for this case

Let the threshould wavelength for sphere be ${\lambda }_0$  . According to Einstein’s photoelectric equation
 $$\begin{gathered} e{V}_s=\frac{hc}{\lambda }-\frac{hc}{{\lambda }_0}\therefore eV=\frac{hc}{{\lambda }_1}-\frac{hc}{{\lambda }_0}\mathrm{....}\left(i\right) \\ 3eV=\frac{hc}{{\lambda }_2}-\frac{hc}{{\lambda }_0}\mathrm{...}\left(ii\right);eV\text{'}=\frac{hc}{{\lambda }_3}-\frac{hc}{{\lambda }_0}\mathrm{...}\left(iii\right) \end{gathered}$$
 
From eqns. (i) and (ii)  

$\frac{2hc}{{\lambda }_0}=\frac{3hc}{{\lambda }_1}-\frac{hc}{{\lambda }_2}or\frac{hc}{{\lambda }_0}=\frac{3hc}{2{\lambda }_1}-\frac{hc}{2{\lambda }_2}$
Substituting in eqn. (iii)    

$eV\text{'}=\frac{hc}{{\lambda }_3}-\frac{3hc}{2{\lambda }_1}+\frac{hc}{2{\lambda }_2}$

$\left(or\right)V\text{'}=\frac{hc}{e}[\frac{1}{{\lambda }_3}-\frac{3}{2{\lambda }_1}+\frac{1}{2{\lambda }_2}]$