Question

The energy of a photon is \[E = h\nu \] and the momentum of the photon is \[p = \dfrac{h}{\lambda }\], then the velocity of photon will be,
A. \[\dfrac{E}{p}\]
B. \[Ep\]
C. \[{\left( {\dfrac{E}{p}} \right)^2}\]
D. \[3 \times {10^8}m/s\]

Answer 1

Hint: A photon has zero rest mass. A quantization of an electromagnetic wave's energy is a photon. A particle's characteristic of motion is its momentum. A photon is a unit of energy that is carried by an electromagnetic wave as it moves across space.

Formula used:
\[p = \dfrac{h}{\lambda }\]
where p is the momentum of the particle, h is the Plank’s constant and \[\lambda \]is the wavelength of the wave.
\[E = pc\]
where E is the energy of the particle of momentum p and c is the velocity of light.

Complete step by step solution:
It is given that the energy of the photon is \[E = h\nu \] and the momentum of the photon is given as,
\[p = \dfrac{h}{\lambda }\]
We need to find the velocity of the photon. Using the relation between the energy of the photon, the momentum of the photon and the velocity of light, we get;
\[E = pc\]
By replacing the expression for energy and momentum from the given expression, we get
\[c = \nu \lambda \]
The photon is the energy packet of the light, so it travels with the velocity of light, i.e. c.

From the expression of energy and momentum,
\[E = h\nu \]
\[\Rightarrow \nu = \dfrac{E}{h}\]
From the expression of momentum of the photon, we get
\[p = \dfrac{h}{\lambda }\]
\[\Rightarrow \lambda = \dfrac{h}{p}\]
So, the velocity of the photon will be,
\[c = \left( {\dfrac{E}{h}} \right)\left( {\dfrac{h}{p}} \right)\]
\[\Rightarrow c = \left( {\dfrac{E}{p}} \right)\left( {\dfrac{h}{h}} \right)\]
\[\therefore c = \left( {\dfrac{E}{p}} \right)\]
Hence, the velocity of the photon is \[\dfrac{E}{p}\].

Therefore, the correct option is A.

Note: The velocity of light is an absolute constant which is equal to \[3 \times {10^8}\,m/s\]. The velocity of the photon depends on the refractive index of the medium in which it is travelling. If the photon is travelling in a vacuum then the velocity of the photon will be equal to \[3 \times {10^8}\,m/s\] but when the medium is of refractive index greater than 1, then the velocity of the photon will be the ratio of the energy and the momentum of the photon.