An electron of mass m with an initial velocity \(\overrightarrow{V} = V_0 \hat{i} (V_0 \gt 0)\) enters an electric field \(\overrightarrow{E} = -E_0 \;\hat{i} (E_0=constant \gt 0)\) at t = 0. If \(\lambda_0\) is its de-Broglie wavelength initially, then its de-Broglie wavelength at time t is
- \(\frac{\lambda_0}{\left(1+\frac{eE_0}{mV_0}t\right)}\)
- \(\lambda_0t\)
- \(\lambda_0\left(1+\frac{eE_0}{mV_0}t\right)\)
- \(\lambda_0\)
The Correct Option is A
Solution and Explanation
To determine the de-Broglie wavelength of an electron in the given electric field at time \( t \), we start by analyzing how its velocity changes due to the electric field.
Initially, the electron has a velocity \( \overrightarrow{V} = V_0 \hat{i} \) and a de-Broglie wavelength \(\lambda_0\). The electric field \(\overrightarrow{E} = -E_0 \hat{i}\) exerts a force on the electron, where the electron charge \( e \) is negative. The force on the electron is given by:
\( \overrightarrow{F} = e \overrightarrow{E} = - e E_0 \hat{i} \)The acceleration \( \overrightarrow{a} \) of the electron is:
\( \overrightarrow{a} = \frac{\overrightarrow{F}}{m} = - \frac{e E_0}{m} \hat{i} \)Over time \( t \), the velocity \( \overrightarrow{v}(t) \) of the electron changes according to:
\( \overrightarrow{v}(t) = V_0 \hat{i} + \overrightarrow{a} t = \left(V_0 - \frac{e E_0}{m} t\right) \hat{i} \)The de-Broglie wavelength \(\lambda(t)\) of the electron at time \( t \) is related to its momentum \( p(t) = m v(t) \) by the de-Broglie equation:
\( \lambda(t) = \frac{h}{p(t)} \)Initially at \( t = 0 \):
\( \lambda_0 = \frac{h}{m V_0} \)At time \( t \):
\( \lambda(t) = \frac{h}{m \left(V_0 - \frac{e E_0}{m} t\right)} \)Using the relationship at \( t = 0 \), we can express \(\lambda(t)\) as:
\( \lambda(t) = \frac{\lambda_0 V_0}{V_0 - \frac{e E_0}{m} t} \)Rewriting the denominator:
\( \lambda(t) = \frac{\lambda_0}{1 + \frac{e E_0}{m V_0} t} \)This matches option:
\( \frac{\lambda_0}{\left(1+\frac{eE_0}{mV_0}t\right)} \)Therefore, the de-Broglie wavelength of the electron at time \( t \) is given by this formula, which also justifies why this option is correct.
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