Electrostatics 7 Question 2

2. An electric dipole is formed by two equal and opposite charges q with separation d. The charges have same mass m. It is kept in a uniform electric field E. If it is slightly rotated from its equilibrium orientation, then its angular frequency ω is

(a) 2qEmd

(b) 2qEmd

(c) qEmd

(d) qE2md

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Solution:

  1. Key Idea When an electric dipole is placed in an electric field E at some angle θ, then two forces equal in magnitude but opposite in direction acts on the tve and ve charges, respectively. These forces forms a couple which exert a torque, which is given as

τ=p×E

where, p is dipole moment.

Torque on the dipole is given as

τ=Iα=pEsinθ

where, I is the moment of inertia and α is angular acceleration.

For small angles, sinθθ

α=pEIθ

Moment of inertia of the given system is

I=md22+md22=2md24=md22

Substituting the value of I in Eq. (i), we get

α=2pEmd2θ

The above equation is similar to the equation for a system executing angular SHM.

Comparing Eq. (ii), with the general equation of angular SHM, i.e.

α=ω2θ

where, ω is the angular frequency,

we get

ω2=2pEmd2 or ω=2pEmd2 As, p=qdω=2qdEmd2=2qEmd



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