Question

If the rest mass of an electron or positron is $ 0.51{\text{ }}MeV $, then the kinetic energy of each particle, in the electron-positron pair produced by a $ \gamma $ photon of $ 2.42{\text{ MeV}} $, will be?
A) $ 0.3\,MeV $
B) $ 1.9\,MeV $
C) $ 0.7\,MeV $
D) $ 1.5\,MeV $

Answer 1

Hint: In this solution, we will use the law of conservation of energy which tells us that the net energy of a system cannot be created or destroyed but it can only transform from one form to another. The energy of the $ \gamma $ photon will be used in creating the electron-positron pair as well as giving kinetic energy to both of them.

Complete step by step answer
In electron-positron pair production, a $ \gamma $ photon spontaneously converts into an electron and a positron pair. We’ve been given that the energy of a $ \gamma $ photon is $ 2.42{\text{ MeV}} $.
When the $ \gamma $ photon converts into an electron-positron pair, the energy of the $ \gamma $ photon will be converted into creating the electron-positron pair as well as the departing kinetic energy to the electron and the positron.
Since the mass of the electron and the positron is $ 0.51{\text{ }}MeV $, creating the electron-positron pair will require an energy of $ 0.51 + 0.51 = 1.02\,MeV $ energy. The remaining energy left over will be $ 2.42 - 1.02 = 1.4\,MeV $.
This energy will be given to the electron and the positron in the form of kinetic energy. In a pair production mechanism, the electron and the positron are ejected in opposite directions to satisfy the law of momentum of conservation of momentum. Hence this kinetic energy will be provided equally to the electron and the positron.

So, they both will have a kinetic energy of $\dfrac{1.4}{2} = 0.7\,MeV $ which corresponds to option (C).

Note
The rest mass of an electron and a positron is often given in terms of energy and since the dimensions of rest mass given to us in the question are all in the forms of energy, we can directly use the law of conservation of energy. Usually, pair production can only happen if the energy of the $ \gamma $ photon is more than $ 1.02MeV $ since the electron and positron will recombine if they are not given kinetic energy to move away from each other.