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(A) The electric field $E$ due to an infinite non-conducting sheet is $E = \frac{\sigma}{2\varepsilon_0}$.Since the electron is released from rest,the

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$2$

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(A) The electric field $E$ due to an infinite non-conducting sheet is $E = \frac{\sigma}{2\varepsilon_0}$.Since the electron is released from rest,the force on it is $F = eE = \frac{e\sigma}{2\varepsilon_0}$.The acceleration $a$ of the electron is $a = \frac{F}{m} = \frac{e\sigma}{2m\varepsilon_0}$,which is constant.The velocity of the electron at time $t$ is $v = at = \left(\frac{e\sigma}{2m\varepsilon_0}\right)t$.The momentum of the electron is $p = mv = m(at) = \left(\frac{e\sigma}{2\varepsilon_0}\right)t$.The de Broglie wavelength is $\lambda = \frac{h}{p} = \frac{h}{(e\sigma / 2\varepsilon_0)t}$.The rate of change of wavelength is $\frac{d\lambda}{dt} = \frac{d}{dt} \left[ \frac{2h\varepsilon_0}{e\sigma t} \right] = -\frac{2h\varepsilon_0}{e\sigma} \cdot \frac{1}{t^2}$.Thus,the magnitude of the rate of change of wavelength varies inversely as $t^2$,so $n = 2$.

An electron is released from rest near an infinite non-conducting sheet of uniform charge density $-\sigma$. The rate of change of de Broglie wavelength associated with the electron varies inversely as the $n^{\text{th}}$ power of time. The numerical value of $n$ is . . . . . . .

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