Question

According to Einstein’s theory of relativity, mass can be converted into energy and vice-versa. The lightest elementary particle, taken to be the electron, has a mass equivalent to $0.51\text{MeV}$ of energy. Which of the following statements are true?
A) The minimum amount of energy available through the conversion of mass into energy is $1.02\text{MeV}$.
B) The least energy of a $\gamma$ - photon that can be converted into mass is $1.02\text{MeV}$.
C) The minimum energy released by the conversion of mass into energy is $1.02\text{MeV}$, it is a $\gamma$ - photon of energy $0.51\text{MeV}$ and energy above that can be converted into mass.
D) The minimum energy released by the conversion of mass into energy is $0.51\text{MeV}$, it is a $\gamma$ - photon of energy $1.02\text{MeV}$ and energy above that can be converted into mass.

Answer 1

Hint:The conversion of the mass of the electron into energy refers to the annihilation of an electron-positron pair. The annihilation of the pair generates a $\gamma$ - photon whose least energy equals the sum of the rest mass energies of the electron and the positron.

Complete step by step answer:
Step 1: Express the reaction involved in the conversion of the mass of the electron into energy.
Here, it is mentioned that the mass of the electron is converted into energy. This refers to the annihilation of the electron-positron pair. The reaction produces a $\gamma$ - photon. The annihilation can be represented as $e^{-}+e^{+}\to \gamma$ .
The rest mass-energy of the electron is given to be $0.51\text{MeV}$. The positron has the same rest mass-energy. So, the minimum energy produced by the above reaction is $0.51+0.51=1.02\text{MeV}$ .
The first statement is in correspondence with the obtained minimum energy.
The fourth statement mentions the minimum energy produced in the conversion to be $0.51\text{MeV}$ .
Thus the first statement is correct but the fourth statement is not.
Step 2: Give the least energy of the $\gamma$ - photon to see if the second and third statements hold.
The minimum energy produced by the conversion serves as the least energy of the $\gamma$ - photon. Thus the least energy of the $\gamma$ - photon will be $1.02\text{MeV}$ .
The second statement is in correspondence with the obtained least energy of the photon.
The third statement suggests that out of the minimum energy produced in the conversion, $0.51\text{MeV}$ will be the energy of the $\gamma$ - photon and the rest can be converted into mass.
So, the second statement is true but the third statement is false.

Therefore, the correct options are A and B.

Note:In the annihilation of an electron-positron pair to produce a $\gamma$ - photon, it is the total mass of the particles and not a part of it that gets converted into energy. Similarly, in the production of an electron-positron pair by a$\gamma$ - photon, it is the total energy of the photon that gets converted into mass. In the reaction, $e^{-}+e^{+}\to \gamma$ the charge is conserved as the electron has a charge $-1e$, the positron has a charge $+1e$ and the $\gamma$ - photon has no charge.