A uniformly charged ring of radius 3a and total charge q is placed in… | JEE Main 2019 (10 Apr, Shift 1)

Q54JEE Main 2019 · 10 APR · SHIFT 1Difficult

Electrostatic Potential Energy

A uniformly charged ring of radius 3a and total charge q is placed in xy-plane centred at origin. A point charge q is moving towards the ring along the z-axis and has speed u at z = 4a. The minimum value of u such that it crosses the origin is:

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Answer: (a)

On the axis of a ring: $V(z) = \frac{kq}{\sqrt{9a^2 + z^2}}$ , $k = \frac{1}{4\pi\varepsilon_0}$ .
$V(4a) = \frac{kq}{5a}$ , $V(0) = \frac{kq}{3a}$ ; $V$ is largest at the origin, so the charge must reach it.
$\frac{1}{2}mu^2 = q\left(\frac{kq}{3a} - \frac{kq}{5a}\right) = \frac{2}{15}\frac{kq^2}{a}$
$u = \sqrt{\frac{2}{m}}\left(\frac{2}{15}\frac{q^2}{4\pi\varepsilon_0 a}\right)^{1/2}$ , answer (a).

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