Exercise 100320

Problem

Find an expression for the oscillation frequency of an electric dipole of dipole moment ${\vec {p}}$ and rotational inertia $I$ for small amplitudes of oscillation about its equilibrium position in a uniform electric field of magnitude $E$.

Solution

$\tau =pE\sin \theta$

is a restoring torque, trying to bring the tilted dipole back to its aligned equilibrium position.

small angle approximation:

$\sin \theta \rightarrow \theta$.

Thus,

$\tau \approx -pE\theta$.

Since this exhibits a simple negative proportionality to the angle of rotation, the dipole oscillates in simple harmonic motion, like a torsional pendulum with torsion constant $\kappa =pE$. The angular frequency $\omega$ is given by

$\omega ^{2}={\frac {\kappa }{I}}={\frac {pE}{I}}$

where I is the rotational inertia of the dipole. The frequency of oscillation is

${\mathcal {f}}={\frac {\omega }{2\pi }}={\frac {1}{2\pi }}{\sqrt {\frac {pE}{I}}}$