Question

An electron (mass m) with an initial velocity $\overline{v}=v_0\hat{i}$ is in an electric field $\overline{E}=E_0\hat{j}$. If ${\lambda }_0=\frac{h}{m{v}_0}$, its de Broglie wavelength at time t is given by

• A. ${\lambda }_0$
• B. ${\lambda }_0\sqrt{1+\frac{e^2{E}_0^2{t}^2}{m^2{v}_0^2}}$
• C. $\frac{{\lambda }_0}{\sqrt{1+\frac{e^2{E}_0^2{t}^2}{m^2{v}_0^2}}}$
• D. $\frac{{\lambda }_0}{(1+\frac{e^2{E}_0^2{t}^2}{m^2{v}_0^2})}$

Answer 1

The correct option is C $\frac{{\lambda }_0}{\sqrt{1+\frac{e^2{E}_0^2{t}^2}{m^2{v}_0^2}}}$

Initial de Broglie wavelength of electron, ${\lambda }_0=\frac{h}{m{v}_0}$

Force on electron in electric field, $\overline{F}=-e\overline{E}=-e{E}_0\hat{j}$

Acceleration of electron, $\overline{a}=\frac{\overline{F}}{m}=\frac{e{E}_0}{m}\hat{j}$

It is acting along negative y-axis.

The initial velocity of electron along x-axis ${\overline{v}}_{x0}=v_0\hat{i}$. Initial velocity of electron alon y-axis ${\overline{v}}_{yo}=0$. Velocity of electron after time t along x-axis ,${\overline{v}}_x=v_0\hat{i}$

($\therefore$ there is no acceleration of electron along x-axis.)

Velocity of electron after time t along y-axis, ${\overline{v}}_y=0+(-\frac{e{E}_0}{m}\hat{j})t=-\frac{e{E}_0}{m}t\hat{j}$

Magnitude of velocity of electron after time t is $|\overline{v}|=\sqrt{v_x^2+v_y^2}=\sqrt{v_0^2+{\left(\frac{-e{E}_0}{m}t\right)}^2}=v_0\sqrt{1+\frac{e^2{E}_0^2{t}_2}{m^2{v}_0^2}}$

de Broglie wavelength associated with electron at time t is $\lambda =\frac{h}{mv}=\frac{h}{m{v}_0\sqrt{1+\frac{e^2{E}_0^2{t}^2}{m^2{v}_0^2}}}=\frac{{\lambda }_0}{\sqrt{1+\frac{e^2{E}_0^2{t}^2}{m^2{v}_0^2}}}$