- #1

CAF123

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## Homework Statement

A molecule with electric dipole moment ##\underline{p}## is initially aligned in an electric field ##\underline{E}## . If this molecule is perturbated from its equilbrium position by a small angle, show that it will perform simple harmonic motion.

Calculate the frequency of this motion in terms of p, E and I

## The Attempt at a Solution

What I did first was write the angular EOM for the dipole: Consider it perturbed at some angle ##\theta##. This gives a torque due to the force by the electric field about the centre of the dipole. I could also consider the torque due to gravity, however, I took gravity to be acting through the centre of mass of the dipole and since I take the torques about the centre, I can ignore it. My final expression is $$I \alpha = -pE\sin \theta\,\Rightarrow\,I \alpha = -pE \theta,$$ if I take the dipole moment p = dq, sinθ ≈ θ for small θ and the -ve because the torque acts to lower θ.

Is it enough to say from here that since this is in the form ##\tau = -k \theta,## with ##k = pE##, then the motion is simple harmonic?

If so, I can say that $$T = 2\pi \sqrt{\frac{I}{k}}\,\Rightarrow\,f = \frac{1}{2 \pi}\sqrt{\frac{k}{I}}\,\Rightarrow\,f = \frac{1}{2\pi} \sqrt{\frac{pE}{I}}.$$ The dimensions check but have I made appropriate assumptions etc and is it okay to state we have the form ##\tau = -k\theta## so SHM applies?

Many thanks