An electron of mass ' m ' with an initial velocity \(\vec{v}={v}_{0}\overset{^}{i}\left({v}_{0}>0\right)\) enters an ele…
An electron of mass ' m ' with an initial velocity \(\vec{v}={v}_{0}\overset{^}{i}\left({v}_{0}>0\right)\) enters an electric field \(\vec{E}=-{E}_{0}\overset{^}{k}\). If the initial de Broglie wavelength is \({\lambda }_{0}\), the value after time t would be
[JEE Main 2025, 24 Jan (Shift 1)]
\(\frac{{\lambda }_{0}}{\sqrt{1+\frac{{e}^{2}{E}_{0}{{}^{2}t}^{2}}{{m}^{2}{{v}_{0}}^{2}}}}\)
The electron's velocity changes over time due to the electric field, therefore,
\(\vec{v}={\vec{v}}_{0}+\left(\frac{e{E}_{0}t}{m}\right)\overset{^}{k}\)
The new wavelength is:
\({\lambda }^{'}=\frac{h}{mv}\\ {\lambda }^{'}=\frac{h}{m\sqrt{{v}_{0}^{2}+{\left(\frac{e{E}_{0}t}{m}\right)}^{2}}}\\ {\lambda }^{'}=\frac{{\lambda }_{o}}{\sqrt{1+{\left(\frac{e{E}_{0}t}{m{v}_{o}}\right)}^{2}}}where,{\lambda }_{o}=\frac{h}{m{v}_{o}}\\ {\lambda }^{'}=\frac{{\lambda }_{o}}{\sqrt{1+\left(\frac{{e}^{2}{E}_{o}^{2}{t}^{2}}{{m}^{2}{v}_{o}^{2}}\right)}}\)
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