Question

Select the correct alternative:A positively charged thin metal ring of radius $R$ is fixed in the xy-plane with its centre at the origin $O$. A negatively charged particle $P$ is released from rest at the point $(0,0,z_0)$ where $z>0$. 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 $z_0$ satisfying $0<z_0\le R$
• C. approximately simple harmonic, provided $z_0\ll R$
• D. such that $P$ crosses $O$ & continues to move along the -ve z-axis towards $x=-\infty$

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 $z_0\ll R$

Electric Field due to a charged ring at a point on the axis is given by

$E=\frac{KQx}{\sqrt{{(x^2+R^2)}^3}}$

$E=\frac{KQ{z}_0}{\sqrt{{({z_0}^2+R^2)}^3}}$

Let the test charge be -q .

Force on -q is hence given by $F=-\frac{KQq{z}_0}{\sqrt{{({z_0}^2+R^2)}^3}}$

That is , the Force is always towards the ring , hence the motion is periodic .

If $z_0<<R$

$F=-\frac{KQq{z}_0}{R^3}$

Which implies that the acceleration is proportional to the displacement from mean position and hence the motion is SHM .