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. 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 O and continues to move along the negative z - axis towards $z=-\infty$

Answer 1

Answer: (A), (C)

The correct options are
A Periodic for all values of $z_0$ satisfying $0<z_0<\infty$
C Approximately simple harmonic provided $z_0<<R$

Here \(E=\frac{1}{4 \pi \epsilon_0} . \frac{Q z_0}{(R^2 + z_0^2)^{3/2}}
where Q is the charge on ring and $z_0$ is the distance of the point from origin.
Then $F=qE=\frac{-Qq{z}_0}{4\pi {ϵ}_0(R^2+z_0^2{)}^{3/2}}$
When charge – q crosses origin, force is again towards centre i.e.,motion is periodic.
Now if $z_0<<R$
$\therefore F=-\frac{1}{4\pi {ϵ}_0}.\frac{Qq{z}_0}{R^2}\Rightarrow F\infty -z_0$ i.e., motion is S.H.M.