An electron of mass $m$ with an initial velocity $\vec{V} = V_0 \hat{i}$ ($V_0 0$) enters an electric fie

Question

An electron of mass $m$ with an initial velocity $\vec{V} = V_0 \hat{i}$ ($V_0 > 0$) enters an electric field $\vec{E} = -E_0 \hat{i}$ ($E_0 = 	ext{constant} > 0$) at $t = 0$. If $\lambda_0$ is de-Broglie wavelength initially, then its de-Broglie wavelength at time $t$ is

Accepted Answer

Initial de-Broglie wavelength $\lambda_0 = \frac{h}{mV_0}$. The acceleration of the electron is $a = \frac{eE_0}{m}$. Velocity after time $t$ is $V = V_0 + \frac{eE_0}{m}t$. The new wavelength $\lambda = \frac{h}{mV} = \frac{h}{m(V_0 + \frac{eE_0}{m}t)} = \frac{h}{mV_0(1 + \frac{eE_0}{mV_0}t)} = \frac{\lambda_0}{1 + \frac{eE_0}{mV_0}t}$.

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