Question

A positively charged thin metal ring of radius R is fixed in the xy - plane with its centre at the o. A negatively charged particle p is released from rest at the point $(0,0,z_0)$ , where $z_0>0$ . Then the motion of p is

• A. not periodic for all values of $z_0$ satisfying $0<z_0<\infty$
• B. Simple harmonic for all values of satisfying $0<z_0<R$
• C. Approximately simple harmonic provided $z_0<<R$
• D. Such that p crosses oand continues to move along the negative z - axis towards $z=-\infty$

Answer 1

The correct option is C Approximately simple harmonic provided $z_0<<R$

Hence $E=\frac{1}{4\pi {ϵ}_0}.\frac{Q{z}_0}{(R^2+z_0^2{)}^{\frac{3}{2}}}$
where Q is the charge on ring and $z_0$ is the distance of the point from origin.
Then $F=qE=\frac{-Qq{z}_0}{4\pi {ϵ}_0(R^2+z_0^2{)}^{\frac{3}{2}}}$
When charge – q crosses origin, force is again towards centre i.e., motion is periodic.
Now if $z_0<<R$
$\therefore F=-\frac{1}{4\pi {ϵ}_0}.\frac{Qq{z}_0}{R^2}$$\Rightarrow F\propto -z_0$ i.e., motion is S.H.M.