Question

A sensor is exposed for time $t$ to a lamp of power $P$ at distance $l$. The sensor has an opening that is 4$d$ in diameter. Assuming all of the energy is given off as light, the number of photons entering the sensor if the wavelength of the light is $\lambda$ is

• A. $N=\frac{P{\lambda }^{}{d}^{2}t}{hc{l}^2}$
• B. $N=\frac{4P{\lambda }^{}{d}^{2}t}{hc{l}^2}$
• C. $N=\frac{P{\lambda }^{}{d}^{2}t}{4hc{l}^2}$
• D. $N=\frac{P{\lambda }^{}{d}^{2}t}{16hc{l}^2}$

Answer 1

The correct option is A $N=\frac{P{\lambda }^{}{d}^{2}t}{hc{l}^2}$

$E=\frac{hc}{\lambda }$
Number of photons emitted is $\frac{Pt}{\frac{hc}{\lambda }}=n_0$
$n_0=\frac{P\lambda t}{hc}$
Since the radiation is spherically symmetric, so the number of photons entering the sensor is $n_0$ times the ratio of aperture area to the area of sphere of radius $l$
$N=n_0\frac{\pi \left(2^{}{d}^2\right)}{4\pi l^2}=\frac{P\lambda t^{}{d}^2}{hc{l}^2}$