An electron (mass \(m\)) with initial velocity \(\overrightarrow{\mathrm{v}}=\mathrm{v}_{0} \hat{\mathrm{i}}+\mathrm{v}_{0} \hat{\mathrm{j}}\) is in an electric field \(\overrightarrow{\mathrm{E}}=-\mathrm{E}_{0} \hat{\mathrm{k}} .\) If \(\lambda_{0}\) is initial de-Broglie wavelength of electron, its de-Broglie wave length at time \(t\) is given by

1. A \(\frac{\lambda_{0} \sqrt{2}}{\sqrt{ {\Large{1}}+\cfrac{\mathrm{e}^{2} \mathrm{E}^{2} \mathrm{t}^{2}}{\mathrm{m}^{2} \mathrm{v}_{0}^{2}}}}\)
2. B \(\frac{\lambda_{0}}{\sqrt{{\Large{2}}+\cfrac{\mathrm{e}^{2} \mathrm{E}^{2} \mathrm{t}^{2}}{\mathrm{m}^{2} \mathrm{v}_{0}^{2}}}}\)
3. C \(\frac{\lambda_{0}}{\sqrt{{\Large{1}}+\cfrac{\mathrm{e}^{2} \mathrm{E}^{2} \mathrm{t}^{2}}{2 \mathrm{m}^{2} \mathrm{v}_{0}^{2}}}}\)
4. D \(\frac{\lambda_{0}}{\sqrt{{\Large{1}}+\cfrac{\mathrm{e}^{2} \mathrm{E}_{0}^{2} \mathrm{t}^{2}}{\mathrm{m}^{2} \mathrm{v}_{0}^{2}}}}\)