Question

The momentum of the photon in an X-ray beam of 10 meter wavelength is:
A. \[6.26 \times {10^{ - 34}}kgm/s\]
B. \[6.26 \times {10^{ - 35}}kgm/s\]
C. \[6.26 \times {10^{ - 33}}kgm/s\]
D. \[6.26 \times {10^{ - 36}}kgm/s\]

Answer 1

Hint: The rest mass of a photon is zero. The photon is a quantization of the energy of an electromagnetic wave. The momentum of a particle is the characteristic of motion. When an electromagnetic wave travels then it carries energy in the form of energy packets called photons.

Formula used:
\[p = \dfrac{h}{\lambda }\]
where p is the momentum of a photon with wavelength \[\lambda \] and $h$ is called the Plank’s constant.

Complete step by step solution:
The energy of the light (electromagnetic wave) comprises energy packets called photons. It can be absorbed by the atom or can be emitted. When an electron in an excited state moves to the ground state then it releases energy in the form of photons and when an electron absorbs energy in the form of photons then jumps to an excited state.

The wavelength of the wave is the horizontal distance of a full wave. The wavelength of the given photon in the X-ray beam is given as 10 meters. Hence, the wavelength of the given photon is,
\[\lambda = 10m\]
The value of Plank’s constant is \[h = 6.626 \times {10^{ - 34}}Js\].

Putting the values in the formula of wavelength for photon we get,
\[p = \dfrac{h}{\lambda }\]
\[\Rightarrow p = \dfrac{{6.626 \times {{10}^{ - 34}}}}{{10}}kgm/s\]
\[\therefore p = 6.626 \times {10^{ - 35}}\,kgm/s\]
When we round-off the obtained value to 3 decimal places, we get the momentum of the photon as \[6.62 \times {10^{ - 35}}kgm/s\]. From the available values in the options, the closest value is \[6.26 \times {10^{ - 35}}kgm/s\]

Therefore, the correct option is B.

Note: The momentum of the photon is the main reason because of which light can exert force on the area of incidence. But the magnitude of the force exerted by the photon is insignificant. The Newtonian laws of motion have limited the motion where speed is very less compared to the speed of light. As the photon moves with a speed which is equal to the speed of light, so we can’t use the Newtonian laws of motion to calculate the momentum, otherwise, it will be zero because as per Newtonian laws of motion the momentum of the particle is the product of the mass and the velocity. As the rest mass of the photon is zero, it will be zero.