Q 12-11-049JEE MainJEE Main 2025 (24 Jan, Shift 1)Medium
An electron of mass $m$ with an initial velocity $\vec v = v_0\hat i$ ($v_0 > 0$) enters an electric field $\vec E = -E_0\hat k$. If the initial de Broglie wavelength is $\lambda_0$, the value after time $t$ would be
Answer: (A) $\dfrac{\lambda_0}{\sqrt{1 + \dfrac{e^2E_0^2t^2}{m^2v_0^2}}}$
The force on the electron is $\vec F = -e\vec E = eE_0\hat k$, perpendicular to $\vec v_0$. After time $t$:
$$v_x = v_0, \qquad v_z = \frac{eE_0t}{m}, \qquad v = v_0\sqrt{1 + \frac{e^2E_0^2t^2}{m^2v_0^2}}$$
Since $\lambda = \dfrac{h}{mv}$:
$$\lambda = \frac{\lambda_0}{\sqrt{1 + \dfrac{e^2E_0^2t^2}{m^2v_0^2}}}$$
Solution by Sreeraj P, M.Sc Physics