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  4. A particle of charge q and mass m is subjected to an electric field E = E 0(1- ax 2) in the x -direction, where a and E0 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 :

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Q. 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 :

JEE MainJEE Main 2020Electric Charges and Fields

Solution:

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