Question

The formula for the kinetic mass of a photon is ______, where h is Planck’s constant and ν, λ, c are frequency, wavelength and speed of photon respectively.
\[\begin{align}
  & \text{A}\text{. }\dfrac{h\nu }{\lambda } \\
 & \text{B}\text{. }\dfrac{h}{c\lambda } \\
 & \text{C}\text{. }\dfrac{h\nu }{c} \\
 & \text{D}\text{. }\dfrac{h\lambda }{c} \\
\end{align}\]

Answer 1

Hint: We have asked the kinetic mass of photon and we know the formula of kinetic energy of photon in terms of Planck's constant and frequency. By using Einstein’s mass-energy relationship we can establish a relation between photon mass and Planck's constant h, frequency (or wavelength) and speed of photon (or light).

Formula used:
\[\begin{align}
  & E=m{{c}^{2}} \\
 & E=h\nu \\
 & \nu =\dfrac{c}{\lambda } \\
\end{align}\]

Complete step by step answer:
We know that energy of a photon is given as product of Planck’s constant h and frequency ν of the photon, so we can write
\[E=h\nu \text{ }...............\text{(i)}\]
Now from the mass energy relationship given by Einstein which says that energy is given as product of its mass and square of the speed of light, we have
\[E=m{{c}^{2}}\text{ }..............\text{(ii)}\]
We can equate the relation (i) and (ii), where m would be the mass of the photon and c is speed of light. Hence, we get
\[\begin{align}
  & m{{c}^{2}}=h\nu \\
 & m=\dfrac{h\nu }{{{c}^{2}}}\text{ }..............\text{(iii)} \\
\end{align}\]
Here we can substitute the value of frequency which is given as the ratio of speed of light and the wavelength.
\[\nu =\dfrac{c}{\lambda }\]
Substituting this in the equation (iii) we have
\[\begin{align}
  & m=\dfrac{h}{{{c}^{2}}}\left( \dfrac{c}{\lambda } \right) \\
 & m=\dfrac{h}{c\lambda } \\
\end{align}\]
We have derived an equation for the kinetic mass of a photon in terms of Planck’s constant h, speed of light (or photon) and the wavelength of the photon.

So, the correct answer is “Option B”.

Note: Photon has a speed of light therefore we can use the mass energy relationship. Also note that for any particle its velocity is given as product of frequency and wavelength, here the velocity of photon is equal to the speed of light therefore we have substituted the value of frequency in terms of speed of light.