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_0gt0`. Then the motion of P is

Options

A. Periodic for all the values of `Z_(0)`, satisfying `0 lt Z_(0) lt oo`
B. Simple harmonic for all the values of `Z_(0)`, satisfying `0 lt Z_(0) le R`
C. appoximately simple harmonic provided `Z_(0) lt lt R`
D. such that `P` crosses `O` and continues to move along the negative Z-axis towards `Z = - oo`

Answer

A, C

Explanation

Electric field at point `(0,0,Z_(0))`

i.e., `E = (k. QZ_(0))/((R^(2) + Z_(0)^(2))^(3//2))`
`vec(E)` acts away from `O`
Force on `(-q)` at `P`
`F = -qE = (-K Q q Z_(0))/((R^(2) + Z_(0)^(2))^(3//2))`
`vec(F)` acts towards `O`.
So, motion of particle `P` is periodic (for `0 lt Z_(0) lt oo`) for `Z_(0) lt lt R, F prop Z_(0)` and such motion is appoximately simple harmonic.