Question

A particle of charge $q$ and mass $m$ is subjected to an electric field $E = E _{0}\left(1- ax ^{2}\right)$ 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 :

• A. $\sqrt{\frac{2}{a}}$
• B. $\sqrt{\frac{1}{a}}$
• C. $a$
• D. $\sqrt{\frac{3}{a}}$

Answer 1

Answer: (D)

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

Answer 2

$E = E _{0}\left(1- ax ^{2}\right)$
$W =\int qE\, dx = qE _{0} \int\limits_{0}^{ x _{0}}\left(1- ax ^{2}\right) dx$
$= qE _{0}\left[ x _{0}-\frac{ ax _{0}^{3}}{3}\right]$
For $\Delta KE =0, W =0$
Hence $x _{0}=\sqrt{\frac{3}{ a }}$