Question 6: Electrostats & Capacitance

Question

Positive charge $$Q$$ is distributed uniformly over a circular ring of radius $$R$$. A point particle having a mass $$(m)$$ and a negative charge $$-q$$ is placed on its axis at a distance $$x$$ from the centre. Assuming $$x < R$$, find the time period of oscillation of the particle, if it is released from there [neglect gravity].

Accepted Answer

When the negative charge is shifted at a distance $$x$$ from the centre of the ring along its axis, then force acting on the point charge due to the ring.
$$\begin{aligned}
F & =q E ~(	ext {towards centre}) \\
& =q \cdot \frac{k Q x}{\left(R^2+x^2ight)^{3 / 2}}
\end{aligned}$$
If $$R >> x$$, then $$R^2+x^2 \simeq R^2$$
and $$\quad F=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{Q q x}{R^3}$$ (towards centre)
$$\Rightarrow \quad a=\frac{F}{m}=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{Q q x}{m R^3}$$
Since, restoring force $$F_E \propto x$$, therefore motion of charge particle will be SHM.
Time period of SHM,
$$T=\frac{2 \pi}{\omega}=\left[\frac{16 \pi^3 \varepsilon_0 R^3 m}{Q q}ight]^{1 / 2}\left[\because \omega^2=\frac{Q q}{4 \pi \varepsilon_0 R^3}ight]$$

Answer

$$\left[\frac{16 \pi^3 \varepsilon_0 R^3 m}{Q q}\right]^{1 / 2}$$

Answer

$$\left[\frac{8 \pi^2 \varepsilon_0 R^3}{q}\right]^{1 / 2}$$

Answer

$$\left[\frac{2 \pi^3 \varepsilon_0 R^3}{3 q}\right]^{1 / 2}$$

Answer

None of these

Electrostats & Capacitance

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