An electron is released from rest near an infinite non-conducting sheet of uniform charge density ''. The rate of change of de-Broglie wave length associated with the electron varies inversely as power of time. The numerical value of n is ________.

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Solution

Q.
An electron is released from rest near an infinite non-conducting sheet of uniform charge density ' $- σ$ '. The rate of change of de-Broglie wave length associated with the electron varies inversely as $n^{th}$ power of time. The numerical value of n is ________. [2025]

Ans.
(2)

Let the momentum of $e^{-}$ at any time t is p and its de-broglie wavelength is $λ$ .

Then, $p = \frac{h}{λ}$

$\frac{d p}{d t} = \frac{- h}{λ^{2}} \frac{d λ}{d t}$

$m a = F = - \frac{h}{λ^{2}} \frac{d λ}{d t}$ [m = mass of $e^{-}$ ]

Where –ve sign represents decrease in $λ$ with time

$a = - \frac{p^{2}}{m h} \frac{d λ}{d t} = - \frac{m v^{2}}{h} \frac{d λ}{d t}$

$\frac{d λ}{d t} = - \frac{a h}{m v^{2}}$ ... (i)

Here, $a = \frac{q E}{m} = \frac{e}{m} \frac{σ}{2 ε_{0}} = \frac{σ e}{2 m ε_{0}}$

and v = u + at

u = 0, v = at

Substituting values of $a$ and $v$ in equation (i)

$\frac{d λ}{d t} = - \frac{2 h ε_{0}}{σ e t^{2}}$

$\Rightarrow \frac{d λ}{d t} \propto \frac{1}{t^{2}} \Rightarrow n = 2$

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