Question

A gamma-ray photon creates an electron-positron pair. If the total kinetic energy of the electron-positron pair is 0.78 MeV and the rest mass energy of the electron is 0.5 MeV, find the energy of the gamma-ray photon.
A) 0.78 MeV
B) 1.78 MeV
C) 1.28 MeV
D) 0.28 MeV

Answer 1

Hint: The gamma-ray photon converts into an electron-positron pair moving with a total kinetic energy of 0.78 MeV. The energy of this reaction is conserved. Also, the rest mass energy of the electron is equal to the rest mass energy of the positron.


Complete step by step answer:
Step 1: List the parameters known from the question.
The creation of the electron-positron pair can be expressed as a reaction equation.
${\text{gamma - ray photon}} \to e + {e^ + } + K.E$
The rest mass energy of the electron is ${E_{e0}} = 0.5{\text{MeV}}$.
The total kinetic energy of the electron-positron pair is $K.E = 0.78{\text{MeV}}$.
Step 2: Find the energy of the gamma-ray photon using the energy conservation theorem.
According to the energy conservation theorem, the energy before creation must be equal to the energy after creation.
i.e., ${E_b} = {E_a}$
The energy before creation is the energy of the gamma-ray photon.
i.e., ${E_b} = {E_\gamma }$
The energy after creation is the sum of the rest mass energy of the electron ${E_{e0}}$, the rest mass energy of the positron ${E_{p0}}$ and the total kinetic energy of the electron-positron pair $K.E$.
i.e., ${E_a} = {E_{e0}} + {E_{p0}} + K.E$
Hence, the energy of the gamma-ray photon is given by, ${E_\gamma } = {E_{e0}} + {E_{p0}} + K.E$ ------- (1)
Substituting values for ${E_{e0}} = {E_{p0}} = 0.5{\text{MeV}}$ and $K.E = 0.78{\text{MeV}}$ in equation (1) we get, ${E_\gamma } = 0.5 + 0.5 + 0.78 = 1.78{\text{MeV}}$
Thus the energy of the gamma-ray photon is ${E_\gamma } = 1.78{\text{MeV}}$
So, the correct option is B.

Note: Positron is the antiparticle of the electron and is also known as antielectron. It has the same spin and same mass as that of the electron but with a charge $ + 1e$. So, it will have the same rest mass energy as that of the electron. The energy of the gamma-ray photon is used to create the electron-positron pair and to provide them with kinetic energy.