An electron (mass $\mathrm{m}$) with an initial velocity $\vec{v}=v_{0} \hat{i}\left(v_{0}>0\right)$ is moving in an electric field $\vec{E}=-E_{0} \hat{i}\left(E_{0}>0\right)$ where $E_{0}$ is constant. If at $\mathrm{t}=0$ de Broglie wavelength is $\lambda_{0}=\frac{h}{m v_{0}}$, then its de Broglie wavelength after time t is given by
Solution
$\text { At } t=0, \lambda_0=\frac{h}{m v_0}$
<br/><br/>Since $\vec{v}=v_0 \hat{i}$ and $\overrightarrow{\mathrm{E}}=\mathrm{E}_0 \hat{i}$
<br/><br/>its velocity $v$ at any time $t$ is given by
<br/><br/>$v=v_o+\frac{\varepsilon \mathrm{E}_{\mathrm{o}}}{m} t$
<br/><br/>De Broglie wavelength $\lambda$ at any time $t$ is given by
<br/><br/>$$
\begin{aligned}
\lambda & =\frac{h}{m v}=\frac{h}{m\left(v_0+\frac{e \mathrm{E}_0}{m} t\right)} \\\\
& =\frac{h}{m v_0\left(1+\frac{e \mathrm{E}_0}{m v_0} t\right)} \\\\
& =\frac{\lambda_0}{1+\frac{e \mathrm{E}_0}{m v_0} t}
\end{aligned}
$$
About this question
Subject: Physics · Chapter: Dual Nature of Matter and Radiation · Topic: de Broglie Hypothesis
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