Question

Monochromatic light of wavelength 632.8 nm is produced by a helium-neon laser. The power emitted is 9.42 mW. Find the energy and momentum of each photon in the light beam.

Answer 1

Hint: In this question, we are asked to find the energy and momentum of each photon in the light beam. By making use of the equation for the photon energy and momentum, we will be computing the energy and momentum of each photon.

Formula used: \[E=\dfrac{hc}{\lambda }\]
\[p=\dfrac{h}{\lambda }\]

Complete step by step answer:
From the data,
The wavelength of the monochromatic light, \[\lambda =632.8\,nm\]
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 value of the energy of the photon emitted by the monochromatic light.
The formula for calculating the energy of the photon emitted by the monochromatic light is given as follows.
\[E=\dfrac{hc}{\lambda }\]
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 substitute the given value and the constant values in the above equation to find the value of the energy of the photon emitted.
\[\begin{align}
  & E=\dfrac{6.63\times {{10}^{-34}}\times 3\times {{10}^{8}}}{632.8\times {{10}^{-9}}} \\
 & \Rightarrow E=3.14\times {{10}^{-19}}J \\
\end{align}\]
Now, we will compute the power emitted by a helium-neon laser.
 The formula for calculating the power emitted by a helium-neon laser is given as follows.
\[p=\dfrac{h}{\lambda }\]
Where h is the Planck's constant and \[\lambda \]is the wavelength of the light.
Now substitute the given value and the constant values in the above equation to find the value of the power emitted by a helium-neon laser.
\[\begin{align}
  & p=\dfrac{6.63\times {{10}^{-34}}}{632.8\times {{10}^{-9}}} \\
 & \Rightarrow p=1.05\times {{10}^{-27}}\,kg\,m\,{{s}^{-1}} \\
\end{align}\]

Therefore, the energy and momentum of each photon in the light beam are \[3.14\times {{10}^{-19}}J\]and \[1.05\times {{10}^{-27}}\,kg\,m\,{{s}^{-1}}\]respectively.

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