Two large parallel metal plates are located in vacuum. One of these serve as cathode, a source of electrons with negligible initial speed. An electron flow directed toward the opposite plate produces a space charge causing the potential in the gap between the plate vary as $V = a x^{4 / 3}$ , where a is a +ve constant, and x is distance from cathode. Then

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Magnitude of electric field is proportional to $x^{1 / 3}$

Volume charge density is proportional to $x^{- 2 / 3}$

Current density is proportional to $x^{1 / 3}$

Current density is independent of x

answer is A, B, D.

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Detailed Solution

$v = a x^{4 / 3} υ = - ε_{0} \nabla^{2} v = - ε \frac{d}{d x} {\frac{d}{d x} (a x^{4 / 3})} = - \frac{4}{9} a ε_{0} x^{- 2 / 3} j = n e v = υ v C a l c u l a t i o n o f v : Δ U + Δ K = 0 \Rightarrow \frac{1}{2} m v^{2} = e v \Rightarrow v (x) = {\frac{2 e a}{m} x^{4 / 3}}^{1 / 2} ∴ j = \frac{4}{9} a ε_{0} x^{- 2 / 3} x^{- 2 / 3} \sqrt{\frac{2 e a}{m}} x^{2 / 3} = \frac{4}{9} ε_{0} a^{3 / 2} \sqrt{\frac{2 e}{m}}$