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

The Correct Option is D

To solve this problem, we need to determine the position where the kinetic energy of the particle becomes zero after it starts at rest from the origin. The electric field given is E = E_{0}(1 - ax^2).

1. First, consider the force experienced by the particle due to the electric field. The force F on a charge q in an electric field E is given by:
   F = qE = qE_{0}(1 - ax^2)
2. Using Newton's second law, F = ma, where a is the acceleration, we equate the force to ma to find the acceleration:
   ma = qE_{0}(1 - ax^2)
   a = \frac{qE_{0}}{m}(1 - ax^2)
3. The particle starts from rest, so using the work-energy principle, the work done on the particle is transformed into potential energy because the kinetic energy becomes zero at the turning point:
   W = \Delta KE = 0 at the point where kinetic energy is zero, thus all initial kinetic energy is turned into potential energy.
4. The work done by the electric field from x = 0 to a point x is:
   W = \int_{0}^{x} qE dx = \int_{0}^{x} qE_{0}(1 - ax^2)dx
5. Calculate the integral:
   W = qE_{0} \left[ \int_{0}^{x} 1 dx - \int_{0}^{x} ax^2 dx \right]
   W = qE_{0} \left[ x - \frac{a}{3}x^3 \right]_{0}^{x} = qE_{0} \left[ x - \frac{a}{3}x^3 \right]
6. The potential energy conservation implies:
   qE_{0} \left[ x - \frac{a}{3}x^3 \right] = 0
7. Solving the equation x - \frac{a}{3}x^3 = 0 gives:
   x(1 - \frac{a}{3}x^2) = 0
   The solution is x = 0 or 1 - \frac{a}{3}x^2 = 0
   x^2 = \frac{3}{a} giving x = \pm \sqrt{\frac{3}{a}}

Therefore, other than the initial position, the particle's kinetic energy becomes zero at a distance \sqrt{\frac{3}{a}} from the origin. Thus, the correct answer is \sqrt{\frac{3}{a}}.