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 $z_0\ll R$
  
• 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 $z_0\ll R$


Here $E=\frac{1}{4\pi {ϵ}_0}.\frac{Q{Z}_0}{(R^2+Z_0^2{)}^{\frac{3}{2}}}$
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 $Z_0\ll R$
$\therefore =-\frac{1}{4\pi {ϵ}_0}.\frac{Qq{Z}_0}{R^2}\Rightarrow F\propto -z_0$ i.e., motion is S.H.M.