GATE 2026 PH – Question 58
Two atomic energy levels differ by 2.2 eV. A gas contains $8\times10^{20}$ atoms in the upper level and $5\times10^{20}$ in the lower. Ignoring spontaneous emission, the maximum energy released by stimulated emission is E joules. E (one decimal place) is _____. Given $e=1.6\times10^{-19}$ C.
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Correct answer: 52.54 to 53.06
Explanation
**Stimulated emission and absorption.** A beam can extract energy from the upper level by stimulated emission. At the same time, atoms in the lower level absorb photons (with the same probability per atom, as the Einstein coefficients $B_{12}=B_{21}$ are equal for non-degenerate levels).
The net emission stops when the populations are equal: the maximum energy that can be released is obtained by bringing the populations to the same value.
**Populations.** Upper: $8\times10^{20}$. Lower: $5\times10^{20}$. Total $13\times10^{20}$. In the equalised state each has $6.5\times10^{20}$.
**Number of atoms that make the downward transition (net):**
$$8\times10^{20}-6.5\times10^{20}=1.5\times10^{20}.$$
**Energy released:**
$$E=1.5\times10^{20}\times2.2\text{ eV}\times1.6\times10^{-19}\text{ J/eV}=1.5\times10^{20}\times3.52\times10^{-19}=\mathbf{52.8\ J}.$$