An electron (mass $m$ ) with an initial velocity $\vec{v}=v_0 \hat{i}$ is in an electric field $\vec

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An electron (mass $m$ ) with an initial velocity $\vec{v}=v_0 \hat{i}$ is in an electric field $\vec{E}=E_0 \hat{j}$. If $\lambda_0=\frac{h}{m v_0}$, it's de Broglie wavelength at time $t$ is given by

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(c) : Initial de Broglie wavelength of electron, $$ \lambda_0=\frac{h}{m v_0} $$ Force on electron in electric field, $\vec{F}=-e \vec{E}=-e E_0 \hat{j}$ Acceleration of electron, $\vec{a}=\frac{\vec{F}}{m}=-\frac{e E_0}{m} \hat{j}$ It is acting along negative $y$-axis. The initial velocity of electron along $x$-axis $\vec{v}_{x_0}=v_0 \hat{i}$. Initial velocity of electron along $y$-axis $\vec{v}_{y_0}=0$. Velocity of electron after time $t$ along $x$-axis, $\vec{v}_x=v_0 \hat{i}$ ( ∵ there is no acceleration of electron along $x$-axis) Velocity of electron after time $t$ along $y$-axis, $ \vec{v}_y=0+\left(-\frac{e E_0}{m} \hat{j}\right) t=-\frac{e E_0}{m} t \hat{j} $ Magnitude of velocity of electron after time $t$ is $ |\vec{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}{m v}=\frac{h}{m v_0 \sqrt{1+\frac{e^2 E_0^2 t^2}{\left(m^2 v_0^2\right)}}}=\frac{\lambda_0}{\sqrt{1+\frac{e^2 E_0^2 t^2}{\left(m^2 v_0^2\right)}}} $