An electron and a positron, both essentially at rest, undergo...

In pair annihilation, an electron and a positron are completely converted into energy.

The mass of a positron is exactly equal to the mass of an electron.

Total mass converted into energy = $m_{e} + m_{e} = 2m_{e}$.

Given the rest mass energy of an electron is $0.511 \text{ MeV}$, the total energy released is:
$E = 2 \times 0.511 \text{ MeV} = 1.022 \text{ MeV}$.

The option with Joules is incorrect because the numerical value corresponds to the energy in MeV, not Joules.

Answer: $1.022 \text{ MeV}$