First slide

Electric field

Question

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 the initial position the kinetic energy of the particle becomes zero when the distance of the particle from the origin is:

Moderate

A

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

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

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

a

Solution

From work-energy theorem
$\begin{array}{l} dK = dW = qEdx = {qE}_{0} (1 - {ax}^{2}) dx \\ ∴ Δ K = {qE}_{0} \int_{0}^{x} (1 - {ax}^{2}) dx = {qE}_{0} (x - \frac{{ax}^{3}}{3}) = 0 \\ \Rightarrow x (1 - \frac{a}{3} x^{2}) = 0 \\ \Rightarrow x = 0, \pm \sqrt{\frac{3}{a}} \end{array}$

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