A small plate of a metal having work function of $1.17eV$ is placed at a distance of $2m$ from a monochromatic light source of wavelength $4.8\times {10}^{-7}m$ and power $1W$. The light falls normally on the plate. Find the number of photons striking the metal plate per $m^2$ per second. If a constant uniform magnetic field of strength ${10}^{-4}T$ is applied parallel to the metal surface, find the radius of the largest circular path followed by the emitted photoelectrons. 

(Given $h=6.6\times {10}^{-4} \mathrm{Js}$, $c=3\times {10}^8 {\mathrm{ms}}^{-1}$, $e=1.6\times {10}^{-19} \mathrm{C}$ and electron mass $m=9.1\times {10}^{-31}kg$.)

Energy of photons of wavelength $4.8\times {10}^{-7} \mathrm{m}$ is

$E=\frac{hc}{\lambda }=\frac{6.6\times {10}^{-34}\times 3\times {10}^8}{4.8\times {10}^{-7}}=4.125\times {10}^{-19}J$

Power of source $=1W=1J{s}^{-1}$

So, rate of emission of photons from the source is

$n=\frac{1J{s}^{-1}}{4.125\times {10}^{-19}J}=2.424\times {10}^{18}{s}^{-1}$

These photons move in all directions randomly. At a distance $r$ from the source, the photons fall normally over a spherical surface of area $4\pi r^2$. The plate is a distance $r=2m$. Hence the number of photo striking the surface per $m^2$ per second is

$n=\frac{2.424\times {10}^{18}}{4\times 3.14\times (2{)}^2}=4.82\times {10}^{16}$

The maximum $KE$ of a photoelectron emitted from the plate is

$K_{\max }=\frac{hc}{\lambda }-W_0$

$\Rightarrow K_{\max }=4.125\times {10}^{-19}-1.17\times 1.6\times {10}^{19}$

$\Rightarrow K_{\max }=2.253\times {10}^{-19}J$

Hence the maximum velocity of the photoelectrons

$v_{\max }=\sqrt{\frac{2K_{\max }}{m}}$

$\Rightarrow v_{\max }=\sqrt{\frac{2\times 2.253\times {10}^{-19}}{9.1\times {10}^{-31}}}=7.036\times {10}^{5}m{s}^{-1}$

Radius of the largest circular path of the photoelectrons in the magnetic field is

$r=\frac{m{v}_{\max }}{eB}=\frac{9.1\times {10}^{-31}\times 7.036\times {10}^5}{1.6\times {10}^{-19}\times {10}^{-4}}$

$\Rightarrow r=4\times {10}^{-2}m=4cm$