Question

Monochromatic light of wavelength 667 nm is produced by a helium-neon laser. The power emitted is 9 mW. The number of photons arriving per second on the average at a target irradiated by this beam is:
\[A.\,3\times {{10}^{19}}\]
\[B.\,9\times {{10}^{17}}\]
\[C.\,3\times {{10}^{16}}\]
\[D.\,9\times {{10}^{15}}\]

Answer 1

Hint: In this question, we are asked to find the number of photons arriving per second on the average at a target irradiated by a beam. We will find the value of the power emitted and the energy by making use of proper formulae, as the number of photons is equal to the ratio of power by energy.
Formula used:
\[P=\dfrac{E}{t}=\dfrac{nhc}{\lambda t}\]
\[N=\dfrac{n}{t}\]

Complete step-by-step solution:
From the data, we have,
The wavelength of the monochromatic light, \[\lambda =667\,nm\]
The power emitted by a helium-neon laser, P = 9 mW
The constant values:
The Planck’s constant is, \[h=6.63\times {{10}^{-34}}\,Js\]
The speed of light in the air is, \[c=3\times {{10}^{8}}{m}/{s}\;\]
Firstly, we will compute the expression for the power emitted by a helium-neon laser in terms of the wavelength.
The expression for calculating the power emitted by a helium-neon laser in terms of the wavelength is given as follows.
\[\begin{align}
  & P=\dfrac{energy}{time} \\
 & \Rightarrow P=\dfrac{nhc}{t\lambda } \\
\end{align}\]
Where h is the Planck's constant, c is the speed of the light in air and \[\lambda \] is the wavelength of the light.
Now, we will compute the expression for the number of photons emitted per second on the average at a target irradiated by the beam of a helium-neon laser.
The formula for calculating the number of photons by a helium-neon laser is given as follows.
\[N=\dfrac{n}{t}\]
Where t is the time taken and N is the photons emitted per second.
Substitute this expression in the above equation.
So, we get,
\[N=\dfrac{P\times \lambda }{hc}\]
Now substitute the given value and the constant values in the above equation to find the value of the number of photons emitted per second on the average at a target irradiated by the beam of a helium-neon laser.
\[\begin{align}
  & N=\dfrac{9\times {{10}^{-3}}\times 667\times {{10}^{-9}}}{6.63\times {{10}^{-34}}\times 3\times {{10}^{8}}} \\
 & \Rightarrow N=3\times {{10}^{16}}\,\,s \\
\end{align}\]
As, the number of photons arriving per second on the average at a target irradiated by the beam of a helium-neon laser is \[3\times {{10}^{16}}\,s\], thus, option (C) is correct.

Note: Even the question can be asked, as, to find the value of either the power or the number of photons emitted, giving, either of the values. In such situations, firstly find the value of the wavelength and then proceed further.