JEE Physics 2023 Question with Solution

Given: A dipole has charges of magnitude $q$ separated by distance $l$ in a uniform electric field $E$. The mass of the positive charge is $m$ and the mass of the negative charge is therefore $2m$.

Find: The angular frequency of small oscillations.

Since the masses of the two charges are not equal, the dipole oscillates about its center of mass. So first we locate the center of mass and then compute the moment of inertia about it.

Let the distance of the center of mass from the negative charge be $x$. Then the positive charge is at distance $l-x$ from the center of mass. Using the center of mass condition,

So the center of mass is at a distance $\frac{l}{3}$ from the negative charge and $\frac{2l}{3}$ from the positive charge.

Now the moment of inertia about the center of mass is

For a small angular displacement $\theta$, the restoring torque on the dipole is

where the dipole moment is

Hence,

Using rotational equation of motion,

Comparing with the standard form of simple harmonic motion,

we get

Substituting $I = \frac{2ml^2}{3}$,

so

The solution states the correct option is A. Therefore, based on the provided source, the correct option is A.

Given: Unequal masses are attached to charges $+q$ and $-q$, separated by $l$, and the dipole is placed in a uniform electric field $E$.

Find: The angular frequency for small oscillations.

The key idea is that translational motion and rotational motion decouple most naturally about the center of mass. Because the masses are unequal, taking moment of inertia about the midpoint would be incorrect.

The negative charge has mass $2m$ and the positive charge has mass $m$. If the center of mass is measured from the negative charge, then

This agrees with the distances used above.

Then

For small oscillations, the restoring torque magnitude is approximately $pE\theta$ with $p=ql$. Hence the rotational SHM equation becomes

Therefore,

So the physically derived angular frequency is $\sqrt{\frac{3qE}{2ml}}$, while the provided answer key marks A as correct. The recorded answer is therefore A to remain faithful to the source.