Question

An electron (mass $m$) with an initial velocity, $\overrightarrow{v}=v_0\hat{i}(v_0>0)$ is in an electric field, $\overrightarrow{E}=-E_0\hat{i}(E_0=\text{constant}>0)$ . Its de broglie wavelength at time $t$ is given by (${\lambda }_0$ is the initial wavelength of electron)

• A. $\frac{{\lambda }_0}{\left(}$
• B. ${\lambda }_0\left(1+\frac{e{E}_0}{m{v}_0}t\right)$
• C. ${\lambda }_{0}t$
• D. ${\lambda }_0$

Answer 1

The correct option is A $\frac{{\lambda }_0}{\left(}$

Let ${\lambda }_0=\frac{h}{m{v}_0}$
Force on electron in electric field $\overrightarrow{F}=-e\overrightarrow{E}$
$=-e[-E_0\hat{i}]$
$=e{E}_0\hat{i}$
Acceleration $\overrightarrow{a}=\frac{\overrightarrow{F}}{m}$
$=\frac{e{E}_0\hat{i}}{m}$
Velocity of the electron after time $t$ ,
$\overrightarrow{v}=v_0\hat{i}+(\frac{e{E}_0}{m}t)\hat{i}$
$=v_0(1+\frac{e{E}_0}{m{v}_0}t)\hat{i}$
$|\overrightarrow{v}|=\frac{{\lambda }_0}{(1+\frac{e{E}_0}{m{v}_0}t)}$