(a) A \({\bf{\gamma }}\)-ray photon has a momentum of 8.00(10^-21)kg.m/s .What is its wavelength?

(b) Calculate its energy in MeV.

To evaluate the wavelength, of the value \({\rm{\lambda }}\), of a photon with a given momentum. Consider the equation of the wavelength.

\({\bf{\lambda = }}\frac{{\bf{h}}}{{\bf{\rho }}}\)

(a)

Considering the given information,

\(\begin{array}{l}p = 8 \times {10^{ - 21}}\;\;{\rm{kg}} \cdot {{\rm{m}}^{ - 1}}\\h = 6.63 \times {10^{ - 34}}\;{\rm{J}} \cdot {{\rm{s}}^{ - 1}}\end{array}\)

Apply the formula and determine the wavelength of a photon.

\(\begin{array}{c}\lambda \approx \frac{{6.63 \times {{10}^{ - 34}}\;\;{\rm{J}}}}{{8 \times {{10}^{ - 21}}\;\;{\rm{k}}{{\rm{g}}_{{\rm{g}} \times {{\rm{m}}_{\rm{s}}}}}{{\rm{s}}^{ - 1}}}}\\\lambda \approx 8.27 \times {10^{ - 15}}\;\;{\rm{m}}\end{array}\)

Therefore, the required wavelength of a photon is \({\rm{8}}{\rm{.27 \times 1}}{{\rm{0}}^{{\rm{ - 15}}}}{\rm{\;m}}\).

(b)

Considering the given information,

\(\begin{array}{l}c = 3 \times {10^8}\;{\rm{m}} \cdot {{\rm{s}}^{ - 1}}\\p = 8 \times {10^{ - 21}}\;\;{\rm{kg}} \cdot {\rm{m}} \cdot {{\rm{s}}^{ - 1}}\end{array}\)

Apply the formula,

The relationship between energy and wavelength in photons is:

\(E \approx pc\)

The value of the relation is as follows:

\(1\;{\rm{eV}} = 1.6 \times {10^{ - 19}}\;{\rm{J}}\)

Currently, the photon's energy is,

\(\begin{array}{l}E \approx 8 \times {10^{ - 21}} \times 3 \times {10^8}\;{\rm{J}}\\E \approx 24 \times {10^{ - 21}} \times \frac{1}{{1.6 \times {{10}^{ - 19}}}}{\rm{eV}}\\{\rm{E}} \approx 15\;{\rm{MeV}}\end{array}\)

Therefore, the required energy in MeV of photons is\({\rm{15}}\;{\rm{MeV}}\).