1. Tardigrade
  2. Question
  3. Physics
  4. An electric dipole of moment vecP is placed in a uniform electric field vecE such that vecP points along vecE . If the dipole is slightly rotated about an axis perpendicular to the plane containing vecE and vecP and passing through the centre of the dipole, the dipole executes simple harmonic motion. Consider I to be the moment of inertia of the dipole about the axis of rotation. What is the time period of such oscillation?

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Q. An electric dipole of moment $\vec{P}$ is placed in a uniform electric field $\vec{E}$ such that $\vec{P}$ points along $\vec{E}$ . If the dipole is slightly rotated about an axis perpendicular to the plane containing $\vec{E}$ and $\vec{P}$ and passing through the centre of the dipole, the dipole executes simple harmonic motion. Consider I to be the moment of inertia of the dipole about the axis of rotation. What is the time period of such oscillation?

Electric Charges and Fields

Solution:

The dipole experiences a torque $p_E \, \sin \, \theta$ tending to bring itself back in the direction of field.
Therefore, on being released (i.e. rotated) the dipole oscillates about an axis through its centre of mass and perpendicular to the field. If I is the moment of inertia of the dipole about the axis of rotation, then the equation of motion is
$I. d^2 \theta / dt^2 = - pE \, \sin \, \theta$
For small amplitude $\sin \, \theta \approx \theta$
Thus $d^2 \theta / dt^2 = -(pE / I). \theta = -w^2 \theta$
where $\omega = \sqrt{(pE / I)}$ .
This is a S.H.M., whose period of oscillation is $T = 2 \pi / \omega = 2 \pi \sqrt{ (I / pE)}$ .