Ncert Exercises & Solutions

A total charge $-Q$ is uniformly spread along the length of a ring of radius $R.$ A small test charge $+q$ of mass $m$ is kept at the center of the ring and is given a gentle push along the axis of the ring. What type of motion will the test charge exhibit, and what will be its time period if it oscillates?

1.	Uniform motion; Time period: Not applicable
2.	Uniformly accelerated motion; Time period: Not applicable
3.	Simple harmonic oscillation; Time period: $T=2\pi \sqrt{\dfrac{4\pi \epsilon_0 m R^3}{Qq}} $
4.	Simple harmonic oscillation; Time period: $T=2\pi \sqrt{\dfrac{4\pi \epsilon_0 m R}{Qq}} $

Let us draw the figure according to question,

A gentle push on q along the axis of the ring gives rise to the situation shown in the figure below.

Taking line elements of charge at A and B, having unit length, then charge on each elements.

$dF = 2 (- \frac{Q}{2 πR}) q \times \frac{1}{4 {πε}_{0}} \frac{1}{r^{2}} \cos θ$

Total force on the charge q, due to entire ring

$F = - \frac{Qq}{πR} (πR) . \frac{1}{4 {πε}_{0}} \frac{1}{r^{2}} . \frac{2}{r}$
$F = - \frac{Qqz}{4 {πε}_{0} {(Z^{2} + R^{2})}^{3 / 2}}$

Here, Z<<R,

$F = - \frac{Qqz}{4 {πε}_{0} R^{3}} = - Kz$

where

$\frac{Qq}{4 {πε}_{0} R^{3}} = constant$

$\Rightarrow F \propto - Z$

Clearly, force on q is proportional to negative of its displacement. Therefore, motion of q is simple harmonic.

$ω = \sqrt{\frac{K}{m}} and T = \frac{2 π}{ω} = 2 π \sqrt{\frac{m}{K}}$
$T = 2 π \sqrt{\frac{m 4 {πε}_{0} R^{3}}{Qq}}$
$\Rightarrow T = 2 π \sqrt{\frac{4 {πε}_{0} {mR}^{3}}{Qq}}$

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