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A sensor is exposed for time $t$ to a lamp of power $P$ placed at a distance $l$. The sensor has a circular opening that is $4d$ in diameter. Assuming all energy of the lamp is given off as light, the number of photons entering the sensor if the wavelength of light is $\lambda $ is $(l >> d)$
$\frac{{P\lambda {d^2}t}}{{hc{l^2}}}$
$\frac{{4P\lambda {d^2}t}}{{hc{l^2}}}$
$\frac{{P\lambda {d^2}t}}{{4hc{l^2}}}$
$\frac{{P\lambda {d^2}t}}{{16hc{l^2}}}$
Solution
Energy emitted by the $\operatorname{lamp}$ in time $t=P t$ where $P$ is the power of the $\operatorname{lamp}$ If $2 \mathrm{d}$ is radius of the sphere and $l$ is the distance of source, then the energy reaching the sphere.
${\rm{E}} = \frac{{{\rm{Pt}}}}{{4\pi {l^2}}}\pi {(2{\rm{d}})^2}$
${\rm{E}} = \frac{{{\rm{Pt}}{{\rm{d}}^2}}}{{{l^2}}}$
$\frac{{{\rm{nhc}}}}{\lambda } \Rightarrow \frac{{{\rm{Pt}}{{\rm{d}}^2}}}{{{l^2}}}$
${\rm{n}} = \frac{{{\rm{Pt}}{{\rm{d}}^2}}}{{{l^2}}} \times \frac{\lambda }{{{\rm{hc}}}}\quad \Rightarrow {\rm{n}} = \frac{{{\rm{Pt}}\lambda {{\rm{d}}^2}}}{{{\rm{hc}}{{\rm{l}}^2}}}$
