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A particle of charge q and mass m is subjected to an electric field $E = E_{0} (1 - a x^{2})$ in the x-direction , where a and $E_{0}$ are constants. Initially the particle was at rest at x=0. Other than initial position the kinetic energy of the particle becomes zero when the distance of the particle from , the origin is

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$\sqrt{\frac{2}{a}}$

$\sqrt{\frac{1}{a}}$

$\sqrt{\frac{5}{a}}$

$\sqrt{\frac{3}{a}}$

answer is D.

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Detailed Solution

A particle of charge =q,

and mass “m” is subjected to an electric field ,

$E = E_{0} (1 - a x^{2})$

According to work –energy theorem

$W_{e x t} = Δ K . E = 0$

$w = \int_{0}^{x} F d x = \int_{0}^{x} q E d x = q \int_{0}^{x} E_{0} (1 - a x^{2}) d x = 0$

$E_{0} {(x - \frac{a x^{3}}{3})}_{0}^{x} = 0$

$\Rightarrow x - \frac{a x^{3}}{3} = 0 \Rightarrow 3 x = a x^{3}$

$\Rightarrow \frac{3}{a} = x^{2} \Rightarrow x = \pm \sqrt{\frac{3}{a}}$

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