Question

A positively charged thin metal ring of radius $R$ is fixed in the $xy-$plane with its centre at the origin $O$. A negatively charge particle $P$ is released from rest from 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 $0<z_0\le R$
• C. approximately simple harmonic, provided $z_0\ll R$
• D. such that $P$ crosses $O$ and continues to move along negative $z-$axis towards $z=-\infty$

Answer 1

Answer: (A), (C)

approximately simple harmonic, provided $z_0\ll R$
Periodic, for all values of $Z_0$ satisfying $0<z_0<\infty$

Ring will attract the (-ve)ly charged particle so, it will come down and when it will go below the ring then its velocity will decrease as force will be in opposite direction with the velocity and then finally it will stop and then will go up and a so, it will have a periodic motion.

Now, if $z_0<<R,$ then $\left|F\right|\simeq \frac{kQq\left(z_0\right)}{{\left(R^2\right)}^{3/2}}$

$\left|F\right|\simeq \frac{kQq\left(z_0\right)}{R^3}$

$\left|F\right|\alpha z$ and it is opposite in direction of displacement .

So, it will be nearly a simple harmonic motion.

Hence, this question has multiple answers.