Question

How many photons of a radiation of wavelength $\lambda =5\times 10^{-7}$ m must fall per second on a blackened plate in order to produce a force of $6.62\times 10^{-5}$ N?
A. $3\times 10^{19}$
B. $5\times 10^{22}$
C. $2\times 10^{22}$
D. $1.67\times 10^{18}$

Answer 1

Hint: To find out the number of photons per second, we first need to find the momentum exhibited by each photon. After that as the plate is blackened, whatever photon falls on it will get absorbed. So, we will take force as the change in momentum that will help us in getting the number of photos required per second.

Formulae used:
Momentum of photons, $p=\dfrac{h}{\lambda}$, where $h$ is the Planck’s constant having value $6.62\times 10^{-34}$ and $\lambda$ is the de Broglie wavelength of light.
Force, $F=\dfrac{\triangledown p}{\triangledown t}$, where p is the momentum of photons.

Complete step by step solution:
We have been given that wavelength of radiation $\lambda =5\times 10^{-7}$ and the force needed to be produced is $6.62\times 10^{-5}$ N.

We have to first find out the momentum of one photon which is given by $p=\dfrac{h}{\lambda}$, where $h$ is the Planck’s constant having value $6.62\times 10^{-34}$ and $\lambda$ is the de Broglie wavelength of light.

So, momentum of $n$ photons will be $n\dfrac{h}{\lambda}$..........$(i)$.

According to Newton’s second law of motion, we know that force is the change in momentum of an object, so we can write $F=\dfrac{\triangledown p}{\triangledown t}$, where p is the momentum of photons.

As we need to find the number of photons falling in one second, $\triangledown t=1$ and so putting the value of p from equation (i) in the equation of force, we get $F=n\dfrac{h}{\lambda}$ and on rearranging terms, we will get $\implies n=F\dfrac{\lambda}{h}=\dfrac{6.62\times 10^{-5}\times 5\times 10^{-7}}{6.62\times 10^{-34}}=5\times 10^{22}$ photons per second.

Hence option b is the correct answer.

Additional information:
The radiation pressure force or sometimes also called the Force of light is the force associated with the mechanical pressure that is exerted on a surface because of momentum exchange between the electromagnetic field and the object. This includes the momentum of any wavelength electromagnetic radiation or light which is emitted, reflected or absorbed.

Note: Newton’s second law of motion is also given as F = ma, where F is the force applied on an object with mass m such that it starts to move with a constant acceleration a, which is also derived from the formula F =$\dfrac{\triangledown p}{\triangledown t}$ when the mass of the object remains same after application of force. So, we must make use of this law in solving such problems in case of any confusion.