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The gas between the capacitor plates separated by a distance $d$ is uniformly ionized by ultraviolet radiation so that $n_{0}$ electrons per unit volume per second are formed. A constant potential difference is applied across the capacitor. These electrons moving in the electric field of the capacitor ionize gas molecules each electron producing $α$ new electrons per unit length of its path. Ionization by ions is neglected. Plate 1 is at higher potential compared to plate 2. Which of the following are correct?

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The number of electrons per unit volume at a distance of $x$ from plate 2 is $n_{0} e^{α x}$

The number of electrons per unit volume at a distance of $x$ from plate 2 is $n_{0} (1 + α x)$

The electronic current density at plate 1 is $e n_{0} d e^{α d}$

The electronic current density at plate 1 is $\frac{e n_{0}}{α} (e^{α d} - 1)$

answer is A, D.

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Detailed Solution

$n = n_{0} e^{α x} I = \int_{0}^{d} e n (A d x) ∴ J = \frac{I}{A} = \frac{e n_{0}}{α} (e^{α d} - 1)$

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