JEE Main 2024 — 1 February, Shift 2 — Question 55

$E=-13.6 z^{2}\left(\frac{1}{n_{i}^{2}}-\frac{1}{n_{f}^{2}}\right)$

$E=C\left(\frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}}\right)$

$\mathrm{h} v=\mathrm{C}\left[\frac{1}{\mathrm{n}_{\mathrm{f}}^{2}}-\frac{1}{\mathrm{n}_{\mathrm{i}}^{2}}\right]$

$\frac{v_{1}}{v_{2}}=\frac{\left[\frac{1}{\mathrm{n}_{\mathrm{f}}^{2}}-\frac{1}{\mathrm{n}_{\mathrm{i}}^{2}}\right]_{2-1}}{\left[\frac{1}{\mathrm{n}_{\mathrm{f}}^{2}}-\frac{1}{\mathrm{n}_{\mathrm{i}}^{2}}\right]_{3-1}}$

$=\frac{\left[\frac{1}{1}-\frac{1}{4}\right]}{\left[\frac{1}{1}-\frac{1}{9}\right]}=\frac{3 / 4}{8 / 9}$

$=\frac{3}{4} \times \frac{9}{8}$ $\frac{v_{1}}{v_{2}}=\frac{27}{32}$

$v_{2}=\frac{32}{27} v_{1}=\frac{32}{27} \times 3 \times 10^{15} \mathrm{~Hz}=\frac{32}{9} \times 10^{15} \mathrm{~Hz}$