An electron of mass m and magnitude of charge $|e|$ initially at rest gets accelerated by a constant electric field E.
Question:

An electron of mass $m$ and magnitude of charge $|e|$ initially at rest gets accelerated by a constant electric field $\mathrm{E}$. The rate of change of de-Broglie wavelength of this electron at time $t$ ignoring relativistic effects is:

  1. (1) $-\frac{h}{|e| \mathrm{E} \sqrt{t}}$

  2. (2) $\frac{|e| \mathrm{Et}}{h}$

  3. (3) $-\frac{h}{|e| \mathrm{Et}}$

  4. (4) $\frac{-h}{|e| \mathrm{Et}^{2}}$


Correct Option: , 4

Solution:

(4) Acceleration of electron in electric field, $a=\frac{e E}{m}$

Using equation

$v=u+a t$

$\Rightarrow v=0+\frac{e E}{m} t$

$\Rightarrow v=\frac{e E t}{m}$      …(i)

De-broglie wavelength $\lambda$ is given by

$\lambda=\frac{h}{m v}=\frac{h}{m\left(\frac{e E t}{m}\right)}$         [using (i)]

$\Rightarrow \lambda=\frac{h}{e E t}$

Differentiating w.r.t. $t$

$\frac{d \lambda}{d t}=\frac{d\left(\frac{h}{e E t}\right)}{d t}$          $\Rightarrow \frac{d \lambda}{d t}=\frac{-h}{e E t^{2}}$

 

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