A small electric dipole p0, having a moment of inertia I about its center, is kept at a distance r from the center of a spherical shell of radius R. The surface charge density is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle as shown in the figure. While staying at a distance r, the dipole is free to rotate about its center. [figure] If released from rest, then which of the following statement(s) is(are) correct? [ 0 is the permittivity of free space.]
- The dipole will undergo small oscillations at any finite value of r.
- The dipole will undergo small oscillations at any finite value of r > R.
- The dipole will undergo small oscillations with an angular frequency of 2 p0 0 I at r = 2R.
- The dipole will undergo small oscillations with an angular frequency of p0100 0 I at r = 10R.
Answer
(2) The dipole will undergo small oscillations at any finite value of r > R., (4) The dipole will undergo small oscillations with an angular frequency of p0100 0 I at r = 10R.
Solution
Inside the shell (r<R) E = 0: no torque, no oscillation — (A) wrong.
Outside: E = R2 0 r2 (radial), so for every r > R there is a restoring torque = -p0E — (B) correct.
I = -p0E = p0EI = p0R2 0 I r2
r = 2R: = p04 0 I — (C) wrong.
r = 10R: = p0100 0 I — (D) correct.
Answer: (B), (D)