Q61P - Fundamentals Of Physics

Question

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

Step-by-Step Solution

Answer

The expression for the oscillation frequency of an electric dipole is.$\frac{1}{2\pi }\sqrt{\frac{pE}{I}}$

Step 1: The given data

Analytical notations: 

Moment of inertia is l

Dipole moment is$\overrightarrow{p}$

Magnitude of electric dipole is E

Step 2: Understanding the concept of the torque

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)

Step 3: Calculation for the expression of the oscillation frequency

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 frequency using 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. $\frac{1}{2\pi }\sqrt{\frac{pE}{I}}$