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 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 $z_0$ satisfying 0 < $z_0$ ≤ R
• C. approximately simple harmonic, provided $z_0$<< R
  
• D. such that P crosses O & continues to move along the −ve z-axis towards z = −$\infty$
  

Answer 1

Answer: (A), (C)

approximately simple harmonic, provided $z_0$<< R


Let the negative charged particle is at P.

Let OP = $z_0$
Field at P, E $=\frac{KQ{z}_0}{(R^2+z_{0^2}{)}^{\frac{3}{2}}}$
E is always directed away from O. Hence a negatively charged particle is accelerated towards O and undergoes periodic motion.
For $z_0$<< R, acceleration is
proportional to the z co-ordinate and the particle undergoes approximate SHM.