Consider a metallic cube of edge length $L$. Its resistance, $R$, measured across its opposite faces is $R =\frac{ m _{ e } v }{ ne ^2 L ^2}$, where $n$ is the number density and $v$ is the drift speed of electrons in the cube, and $e$ and $m _{ e }$ are the charge and mass of an electron respectively. Assuming the de-Broglie wavelength of the electron to be $L$, the maximum resistance of the sample is closest to ............. $\,\Omega$ $\left(e=1.60 \times 10^{-19} \,C ; m _{ e }=9.11 \times 10^{-31} \,kg\right.$; Planck's constant, $h=6.63 \times 10^{-34} \,Js$ )
KVPY 2021, Advanced
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(B)
$L =\frac{ h }{ mv }$
$R =\frac{ m _{ e } v }{ ne ^2 L ^2}$
$\therefore R =\frac{ h }{ L ^3 n e ^2}$
$R$ maximum when $n$ is minimum that is at least
$1 e ^{-}$in the cube
$R =\frac{ h }{ L ^3 \frac{1}{ L ^3} e ^2}$
$=\frac{10^{-34}}{10^{-38}}=10^4 \,\Omega$
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