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

Analytical notations:

Moment of inertia is l

Dipole moment is$\overrightarrow{p}$

Magnitude of electric dipole is E

Using the concept of torque, we can get the small oscillations of the body by using the frequency and torque relation.

Formulae:

The torque acting on a dipole tends to rotate the dipole p (hence the dipole) into the direction of field, E is given by$\tau =-pEsin\theta$:(i)

where, θ is the angle between p and E.

The angular frequency of the oscillations, $\omega =\sqrt{\frac{\kappa }{I}}$ (ii)

The frequency of the oscillations, $f=\omega /2\pi$ (iii)

The torsion constant of an oscillation, $\kappa =pE$ (iv)

Equation (1) captures the sense as well as the magnitude of the effect. That is, this is a restoring torque, trying to bring the tilted dipole back to its aligned equilibrium position. If the amplitude of the motion is small, we may replace sin θ with θ in radians. Thus,$\tau =pE\theta$.

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. The angular frequencyusing equation (iv) in equation (ii) is given by:

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

where, I is the rotational inertia of the dipole.

Now, the frequency of oscillation using the above value in the equation (iii) is given as:

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

Hence, the value of the frequency of the oscillations is.role="math" $\frac{1}{2\pi }\sqrt{\frac{pE}{I}}$