An electron of mass ' m ' with an initial velocity $\overrightarrow{\mathrm{v}}=\mathrm{v}_0 \hat{i}\left(\mathrm{v}_0>0\right)$ enters an electric field $\overrightarrow{\mathrm{E}}=-\mathrm{E}_{\mathrm{o}} \hat{\mathrm{k}}$. If the initial de Broglie wavelength is $\lambda_0$, the value after time t would be

<p><strong>Derivation of the de Broglie wavelength after time $t$</strong></p> <p>The de Broglie wavelength is given by</p> <p>$\lambda = \frac{h}{p},$</p> <p>where $h$ is Planck's constant and $p$ is the momentum of the particle.</p> <p><p><strong>Initial Conditions:</strong> </p> <p>Initially, the electron has a velocity</p> <p>$\vec{v} = v_0\,\hat{i},$</p> <p>so its momentum is</p> <p>$p_0 = m v_0.$</p> <p>The initial de Broglie wavelength is therefore</p> <p>$\lambda_0 = \frac{h}{m v_0}.$</p></p> <p><p><strong>Effect of the Electric Field:</strong> </p> <p>The electron enters an electric field</p> <p>$\vec{E} = -E_0\,\hat{k}.$</p> <p>For an electron with charge $-e$, the force is given by</p> <p>$\vec{F} = -e\,\vec{E} = -e\left(-E_0\,\hat{k}\right) = e E_0\,\hat{k}.$</p> <p>This force causes an acceleration in the $\hat{k}$ direction:</p> <p>$a_z = \frac{F_z}{m} = \frac{e E_0}{m}.$</p></p> <p><p><strong>Momentum Components After Time $t$:</strong> </p></p> <p><p>In the $\hat{i}$ direction, there is no force, so the momentum remains constant:</p> <p>$p_x = m v_0.$</p></p> <p><p>In the $\hat{k}$ direction, starting from rest, the momentum becomes:</p> <p>$p_z = m v_z = m \left(a_z t\right) = e E_0 t.$</p></p> <p><p><strong>Total Momentum of the Electron:</strong> </p> <p>The total momentum is the vector sum of the components:</p> <p>$p = \sqrt{p_x^2 + p_z^2} = \sqrt{(m v_0)^2 + (e E_0 t)^2}.$</p></p> <p><p><strong>New de Broglie Wavelength:</strong> </p> <p>Substituting the expression for the total momentum into the de Broglie relation gives:</p> <p>$\lambda = \frac{h}{\sqrt{m^2 v_0^2 + e^2 E_0^2 t^2}}.$</p> <p>Expressing this in terms of the initial de Broglie wavelength $\lambda_0 = \frac{h}{m v_0}$:</p> <p>$\lambda = \frac{\lambda_0}{\sqrt{1+\frac{e^2 E_0^2 t^2}{m^2 v_0^2}}}.$</p></p> <p>This final expression matches the option:</p> <p>$\frac{\lambda_0}{\sqrt{1+\frac{e^2 E_0^2 t^2}{m^2 v_0^2}}}.$</p>