Question

A sensor is exposed for time t to a lamp of power P placed at a distance l. The sensor has an 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
A. $N=\dfrac{P\lambda {{d}^{2}}t}{hc{{l}^{2}}}$
B. $N=\dfrac{4P\lambda {{d}^{2}}t}{hc{{l}^{2}}}$
C. $N=\dfrac{P\lambda {{d}^{2}}t}{4hc{{l}^{2}}}$
D. $N=\dfrac{P\lambda {{d}^{2}}t}{16hc{{l}^{2}}}$

Answer 1

The number of photons entering the surface is proportional to the area of that surface. The basic relation between power, energy and time is a very necessary and important part of this question. Obtain the energy of light in terms of the wavelength of the light. Then we can find the number of photons entering the sensor.

Complete step by step solution:
Given that, a sensor is exposed for time t to a lamp of power P placed at distance l. We consider the energy of the lamp to be E.
Now we solve this equation step by step to reach the final expression.
We know the formula power as, $P=\dfrac{E}{t}$
The wavelength of light is $\lambda$ , then we can write above equation as,
$P=\dfrac{nhc}{\lambda t}$
$n=\dfrac{P\lambda t}{hc}\text{ }\to \text{ 1}$
where n is the total number of photons emitted by the source, h is Planck’s constant and c is the speed of light.
The sensor has an opening that is 4d in diameter. So, the area of that opening is $4\pi {{d}^{2}}$
The number of photons entering or leaving the surface is proportional to the area of that surface. So, we can write.
$\dfrac{N}{n}=\dfrac{4\pi {{d}^{2}}}{4\pi {{l}^{2}}}$
where N is the number of photons entering the sensor.
Putting the value from equation (1) in above equation, we get,
$\begin{aligned} & N=\dfrac{P\lambda t}{hc}\times \dfrac{4\pi {{d}^{2}}}{4\pi {{l}^{2}}} \\\ & N=\dfrac{P\lambda {{d}^{2}}t}{hc{{l}^{2}}} \\\ \end{aligned}$

Hence, the number of photons entering the sensor is, $N=\dfrac{P\lambda {{d}^{2}}t}{hc{{l}^{2}}}$
Correct option is A.

Note: In this question, the diameter of the opening is given, so don’t forget to convert it in radius. Otherwise, the answer will be option B which is incorrect.
We can define the energy of photons in terms of frequency as, $E=nh\nu$. Again, frequency can be expressed in terms of wavelengths as, $\nu =\dfrac{c}{\lambda }$.
So, energy can be given as, $E=\dfrac{nhc}{\lambda }$