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

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Correct Answer

(D) \(\sqrt{\frac{3}{a}}\)

Step-by-step Solution

Detailed explanation

\(E = E _{0}\left(1- ax ^{2}\right)\)

\(W =\int q E\, d x = q E _{0} \int_{0}^{ x _{0}}\left(1- ax ^{2}\right) dx\)

\(=q E_{0}\left[x_{0}-\frac{a x_{0}^{3}}{3}\right]\)

For \(\Delta KE =0, \quad W =0\)

Hence \(x _{0}=\sqrt{\frac{3}{ a }}\)

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