An electron of mass ' $m$ ' with initial velocity $\vec{v}=v_0 \hat{i}\left(v_0\gt0\right)$ enters in an…

An electron of mass ' $m$ ' with initial velocity $\vec{v}=v_0 \hat{i}\left(v_0\gt0\right)$ enters in an electric field $\overrightarrow{\mathrm{E}}=-\mathrm{E}_0 \hat{i} \quad\left[\mathrm{E}_0\right.$ is constant $\left.\gt0\right]$ at $t=0$. If $\lambda$ is its de-Brogli wavelength initially, then the de-Brogli wave length after time ' $t$ ' is
  1. $\frac{\lambda}{1+\frac{e \mathrm{E}_o t}{m v_0}}$
  2. $\frac{\lambda}{\left(1-\frac{e \mathrm{E}_o t}{m v_0}\right)^2}$
  3. $\left(1-\frac{e \mathrm{E}_o t}{m v_0}\right) \lambda$
  4. $\left(1+\frac{e \mathrm{E}_o t}{m v_0}\right)^2 \lambda$

Solution

$\overrightarrow{v_1}=v_0 \hat{i}, \vec{E}=-E_0 \hat{i}, \lambda_1=\lambda$ $\therefore \quad$ Acceleration of electron, $\vec{a}=\frac{\vec{F}}{m}=\frac{-\mathrm{e} \vec{E}}{m}=\left(\frac{\mathrm{eE}_{\mathrm{o}}}{\mathrm{m}}\right) \hat{\mathrm{i}}$ $\therefore \quad$ Velocity after time t is $\overrightarrow{v_2}=\overrightarrow{v_1}+\vec{a} t=v_0 \hat{i}+\left(\frac{e E_0 t}{m}\right) \hat{i}=\left(v_0+\frac{e E_0 t}{m}\right) \hat{i}$ $\therefore$ de-Broglie wavelength of electron, $\lambda=\frac{\mathrm{h}}{\mathrm{mv}} \Rightarrow \lambda \propto \frac{1}{\mathrm{v}} \therefore \frac{\lambda_2}{\lambda_1}=\frac{\mathrm{v}_1}{\mathrm{v}_2}=\frac{\mathrm{v}_{\mathrm{o}}}{\mathrm{v}_{\mathrm{o}}+\frac{\mathrm{eE}_0 \mathrm{t}}{\mathrm{m}}}$ $\therefore \lambda_2=\frac{\lambda_1}{\left(1+\frac{\mathrm{eE}_0 \mathrm{t}}{\mathrm{mv}_0}\right)}=\frac{\lambda}{\left(1+\frac{\mathrm{eE}_0 \mathrm{t}}{\mathrm{mv}_{\mathrm{o}}}\right)}$

Asked in: AP EAMCET 2024 (20 May Shift 1)

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