Question

A uniform charged ring of radius R and charge Q is placed in horizontal plane. If a point charge q of mass m is placed on the axis of ring in equilibrium position, then which of the following option(s) is/are correct: $[k=\frac{1}{4\pi {ϵ}_0}]$

• A. If we provide small displacement to the point charge in the vertical direction, it will execute S.H.M. .
• B. If we provide small displacement to the point charge in the vertical direction, it will execute oscillatory motion.
• C. If we provide small displacement to the point charge in the horizontal direction, it will execute S.H.M.
• D. The amplitude above the equilibrium position and below it, may be different.

Answer 1

Answer: (B), (D)

The correct options are
B If we provide small displacement to the point charge in the vertical direction, it will execute oscillatory motion.
D The amplitude above the equilibrium position and below it, may be different.

At equilibrium position.
$\frac{k{Q}_q}{(R^2+x^2{)}^{\frac{3}{2}}}x=mg$
$\frac{k{Q}_q}{mg}=\frac{(R^2+x^2{)}^{\frac{3}{2}}}{x}$
If we displace it by small distance 'y', then
$F=\frac{k{Q}_q(x+y)}{{[R^2+(x^2+y^2\left)\right]}^{\frac{3}{2}}}$
As $F_{net}$ is not in the form of F $\infty$ y. So it will not do S.H.M.
Since $F_{net}$ is opposite to the displacement, motion