JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Modern Physics

Hydrogen energy levels are $E_n=-\frac{13.6}{n^2}\text{eV}$, with the atoms initially in the ground state ($n=1$). Exciting to $n=2$ needs $10.2\text{eV}$, to $n=3$ needs $12.09\text{eV}$, and to $n=4$ needs $12.75\text{eV}$. A beam of $12.5\text{eV}$ can raise the atom to $n=3$ at most, since the jump to $n=4$ needs more energy than is available. The number of spectral lines emitted on de-excitation from $n=3$ is $\frac{n(n-1)}{2}=\frac{3\times 2}{2}=3$, corresponding to the transitions $3\to 2$, $3\to 1$ and $2\to 1$. The number of spectral lines emitted is $3$.