An electron of mass ' m ' with an initial velocity $\overrightarrow{\mathrm{v}}=\mathrm{v}_0…

An electron of mass ' m ' with an initial velocity $\overrightarrow{\mathrm{v}}=\mathrm{v}_0 \hat{i}\left(\mathrm{v}_0 \gt 0\right)$ enters an electric field $\overrightarrow{\mathrm{E}}=-\mathrm{E}_{\mathrm{o}} \hat{\mathrm{k}}$. If the initial de Broglie wavelength is $\lambda_0$, the value after time t would be
  1. $\frac{\lambda_0}{\sqrt{1+\frac{\mathrm{e}^2 \mathrm{E}_0{ }^2 \mathrm{t}^2}{\mathrm{~m}^2 \mathrm{v}_0{ }^2}}}$
  2. $\lambda_{\mathrm{o}} \sqrt{1+\frac{\mathrm{e}^2 \mathrm{E}_0^2 \mathrm{t}^2}{\mathrm{~m}^2 \mathrm{v}_{\mathrm{o}}^2}}$
  3. $\frac{\lambda_0}{\sqrt{1-\frac{\mathrm{e}^2 \mathrm{E}_{\mathrm{o}}{ }^2 \mathrm{t}^2}{\mathrm{~m}^2 \mathrm{v}_{\mathrm{o}}^2}}}$
  4. $\lambda_0$

Solution

$\begin{aligned} & \overrightarrow{\mathrm{v}}=\mathrm{v}_0 \hat{\mathrm{i}}-\frac{\mathrm{E}_0 \mathrm{e}}{\mathrm{m}} t \hat{\mathrm{k}} \\ & |\overrightarrow{\mathrm{v}}|=\sqrt{\mathrm{v}_0^2+\frac{\mathrm{E}_0^2 \mathrm{e}^2 \mathrm{t}^2}{\mathrm{~m}^2}} \\ & \lambda_0=\frac{\mathrm{h}}{\mathrm{mv}_0} \\ & \lambda^{\prime}=\frac{\mathrm{h}}{\mathrm{mv}_0 \sqrt{1+\frac{\mathrm{E}_0^2 \mathrm{e}^2 \mathrm{t}^2}{\mathrm{v}_0^2 \mathrm{~m}^2}}} \\ & \lambda^{\prime}=\frac{\lambda_0}{\sqrt{1+\frac{\mathrm{E}_0^2 \mathrm{e}^2 \mathrm{t}^2}{\mathrm{v}_0^2 \mathrm{~m}^2}}}\end{aligned}$

Asked in: JEE Main 2025 (24 Jan Shift 1)

Practice more Dual Nature of Matter questions on Aicharya