JEE MAIN - Physics (2023 - 31st January Morning Shift - No. 19)
For hydrogen atom, $$\lambda_{1}$$ and $$\lambda_{2}$$ are the wavelengths corresponding to the transitions 1 and 2 respectively as shown in figure. The ratio of $$\lambda_{1}$$ and $$\lambda_{2}$$ is $$\frac{x}{32}$$. The value of $$x$$ is __________.
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Answer
27
Explanation
$\frac{1}{\lambda}=\mathrm{RZ}^{2}\left[\frac{1}{\mathrm{n}_{1}^{2}}-\frac{1}{\mathrm{n}_{2}^{2}}\right]$
$$ \begin{aligned} & \frac{1}{\lambda_{1}}=\mathrm{RZ}^{2}\left[\frac{1}{1^{2}}-\frac{1}{3^{2}}\right]=\frac{8}{9} \mathrm{RZ}^{2} ........(1)\\\\ & \frac{1}{\lambda_{2}}=\mathrm{RZ}^{2}\left[\frac{1}{1^{2}}-\frac{1}{2^{2}}\right]=\frac{3}{4} \mathrm{RZ}^{2} ........(2) \end{aligned} $$
$$ \begin{gathered} 1 / 2 \Rightarrow \frac{\lambda_{2}}{\lambda_{1}}=\frac{8}{9} \times \frac{4}{3}=\frac{32}{27} \\\\ \frac{\lambda_{1}}{\lambda_{2}}=\frac{27}{32} \end{gathered} $$
$$ \begin{aligned} & \frac{1}{\lambda_{1}}=\mathrm{RZ}^{2}\left[\frac{1}{1^{2}}-\frac{1}{3^{2}}\right]=\frac{8}{9} \mathrm{RZ}^{2} ........(1)\\\\ & \frac{1}{\lambda_{2}}=\mathrm{RZ}^{2}\left[\frac{1}{1^{2}}-\frac{1}{2^{2}}\right]=\frac{3}{4} \mathrm{RZ}^{2} ........(2) \end{aligned} $$
$$ \begin{gathered} 1 / 2 \Rightarrow \frac{\lambda_{2}}{\lambda_{1}}=\frac{8}{9} \times \frac{4}{3}=\frac{32}{27} \\\\ \frac{\lambda_{1}}{\lambda_{2}}=\frac{27}{32} \end{gathered} $$
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