An electron (mass n) with an initial velocity $\mathrm{v}={\mathrm{v}}_0\widehat{\mathrm{i}}$ is in an electric field $\mathrm{E}={\mathrm{E}}_0\widehat{\mathrm{j}}\cdot \text{ If }{\mathrm{\lambda }}_0=\frac{\mathrm{h}}{{\mathrm{mv}}_0}$, its de-Broglie wavelength at time t is given by

According to the problem de-Broglie wavelength 'of electron at time t = 0 is
${\mathrm{\lambda }}_0=\frac{\mathrm{h}}{{\mathrm{mv}}_0}$
Electrostatic force on electron in electric field is ${\overrightarrow{\mathrm{F}}}_{\mathrm{e}}=-\mathrm{e}\overrightarrow{\mathrm{E}}=-{\mathrm{eE}}_0\widehat{\mathrm{j}}$
The acceleration of electron,$\overrightarrow{\mathrm{a}}=\frac{\overrightarrow{\mathrm{F}}}{\mathrm{m}}=-\frac{{\mathrm{eE}}_0}{\mathrm{m}}\widehat{\mathrm{j}}$
It is acting along negative y-axis.
The initial velocity of electron along r-axis,${\mathrm{v}}_{{\mathrm{x}}_0}={\mathrm{v}}_0\widehat{\mathrm{i}}$
This component of velocity will remain constant as there is no force on electron in this direction. Now considering y-direction, initial velocity of electron along y-axis,${\mathrm{v}}_{\mathrm{y}0}=0$
Velocity of electron after time t along y-axis ${\mathrm{v}}_{\mathrm{y}}=0+\left(-\frac{{\mathrm{eE}}_0}{\mathrm{m}}\mathrm{I}\right)\mathrm{t}=-\frac{{\mathrm{eE}}_0}{\mathrm{m}}\mathrm{t}\widehat{\mathrm{j}}$
Magnitude of velocity o{ electron after time t is
$\begin{array}{l}\mathrm{v}=\sqrt{{\mathrm{v}}_{\mathrm{x}}^2+{\mathrm{v}}_{\mathrm{y}}^2}={\sqrt{{\mathrm{v}}_0^2+\left(\frac{-{\mathrm{eE}}_0}{\mathrm{m}}\mathrm{t}\right)}}^2\\ \Rightarrow ={\mathrm{v}}_0\sqrt{1+\frac{{\mathrm{e}}^2{\mathrm{E}}_0^2{\mathrm{t}}^2}{{\mathrm{m}}^2{\mathrm{v}}_0^2}}\end{array}$
de-Broglie wavelength, ${\mathrm{\lambda }}_0=\frac{\mathrm{h}}{{\mathrm{mv}}_0}$
$\begin{array}{l} =\frac{\mathrm{h}}{{\mathrm{mv}}_0\sqrt{1+\frac{{\mathrm{e}}^2{\mathbf{E}}_0^2{\mathrm{t}}^2}{{\mathrm{m}}^2{\mathrm{v}}_0^2}}}=\frac{{\mathrm{\lambda }}_0}{\sqrt{1+\frac{{\mathrm{e}}^2{\mathbf{E}}_0^2{\mathrm{t}}^2}{{\mathrm{m}}^2{\mathrm{v}}_0^2}}}\\ \Rightarrow {\mathrm{\lambda }}^{\mathrm{\prime }}=\frac{{\mathrm{\lambda }}_0}{\left(1+\frac{{\mathrm{e}}^2{\mathrm{E}}_0^2{\mathrm{t}}^2}{{\mathrm{m}}^2{\mathrm{v}}_0^2}\right)}\end{array}$