There is a positively charged ring of radius R whose plane in kept vertical in gravity free space. One electron is released at rest from a point on its axis at a distance x from its centre. Assume that the positive charge is uniformly distributed over ring.

Options

A. The electron performs S.H.M about the centre if x is very small in comparison to radius of the ring.
B. The electron will attain maximum acceleration acne crossing the centre of ring if x gt R
C. Acceleration of the electron is zero at the instant it crosses the centre of ring.
D. The electron will oscillate about the centre of ring.

Answer

A, B, C, D

Explanation

We know that acceleration of electron when released at a point on the axis will be proportional (a =eE/m) to the electric field intensity of ring at the location of electron. Electric field intensity of the ring is given by the following expression :
`E=(Qx)/(4piepsilon_0(a^2+x^2)^(3//2))`
Here a is the radius of the ring.
If x < < a then `E approx (Qx)/(4piepsilon_0 a^3)`
The acceleration of electron near the centre of ring on its axis can be written as
Acceleration=`(eE)/m approx (eQx)/(4piepsilon_0ma^3)`
We can say that the magnitude of acceleration is directly proportional to the distance from the centre of the ring if the electron is very close to the centre. Acceleration is in accordance with the requirement for S.H.M and moreover electric force is always directed towards the centre of the ring on electron on both sides of ring, hence it is restoring. Now we can say that the electron will perform S.H.M if it is released on the axis of a positively charged ring at a very small distance from the centre of ring. Thus option (a) is correct.
If a is the radius of the ring then we know that electric field intensity of the ring is maximum on its axis at a distance `a/sqrt2` from its centre. We can say that the electron will attain maximum acceleration when it crosses the point of maximum electric field. So, option (b) is correct.
We can see in the expression of electric field intensity due to a charged ring that electric field at its centre is zero. The acceleration of electron becomes zero at the centre and thus option (c) is correct.
S.H.M is also a kind of idealised model of oscillatory motion but in option (a) we have discussed that it is only possible if the distance of electron from the centre of ring is lesser in comparison to radius of the ring. Here in general, the electron will oscillate about the centre even if it is released from a relatively large distance, but it will be just oscillatory motion and will not be S.H.M. So, option (d) is correct.