Animated Solution for Physics - Electric Charges and Fields: An electric dipole is formed by two equal and opposite charges $q$ with separation $d$. The charges have same mass $m$. It is kept in a uniform electric field $E$. If it is slightly rotated from its equilibrium orientation, then its angular frequency $\omega$ is

Imagine you are observing a tiny, invisible world. In this world, an electric dipole—a pair of equal and opposite charges, $+q$ and $-q$, separated by a rigid distance $d$—is resting peacefully. It is immersed in a uniform electric field $\mathbf{E}$, which acts like a steady, invisible river flowing across space.

When the dipole is perfectly aligned with the electric field, it is in a state of stable equilibrium. The positive charge is pulled downstream, and the negative charge is pulled upstream. The forces cancel out perfectly, and the dipole feels no urge to move.

But what happens if we disturb this peace? What if we gently nudge the dipole, rotating it by a small angle $\theta$?

The moment the dipole is rotated, the electric field fights back. The force on the positive charge is $q\mathbf{E}$, and the force on the negative charge is $-q\mathbf{E}$. Because the dipole is no longer aligned with the field, these two forces no longer share the same line of action. They form a couple, creating a torque that tries to twist the dipole back to its original alignment.

The magnitude of this restoring torque is given by the cross product of the dipole moment $\mathbf{p}$ and the electric field $\mathbf{E}$:

The negative sign is crucial here. It is nature's way of saying, "I will oppose your rotation." If you rotate the dipole clockwise, the torque acts counter-clockwise.

Now, dealing with $\sin \theta$ in differential equations can be a nightmare. It leads to complex, non-linear motion. But we have a secret weapon: the small angle approximation.

Because we only nudged the dipole slightly, the angle $\theta$ is very small. For small angles measured in radians, the sine of the angle is practically equal to the angle itself:

This beautiful mathematical simplification transforms our torque equation into a linear form:

Before we can find out how fast the dipole oscillates, we need to know how much it resists being rotated. This resistance is called the moment of inertia ($I$).

Our dipole consists of two masses, each of mass $m$. They are rotating about the center of mass, which lies exactly in the middle of the dipole. Therefore, each mass is at a distance of $d/2$ from the axis of rotation.

The total moment of inertia is the sum of the individual moments of inertia:

Now, we bring in Newton's Second Law for rotation, which states that the net torque equals the moment of inertia times the angular acceleration ($\alpha$):

Substituting our expressions for torque and moment of inertia, we get:

Rearranging this to solve for $\alpha$:

Take a step back and look at this equation. It is a masterpiece. It has the exact mathematical signature of Simple Harmonic Motion (SHM), which is defined by:

By comparing the two equations, we can instantly see the square of the angular frequency ($\omega$):

We are almost there. We just need to remember the definition of the electric dipole moment $p$. It is simply the magnitude of one of the charges multiplied by the separation distance:

Let's substitute this back into our frequency equation:

One of the $d$'s in the numerator gracefully cancels out with one in the denominator, leaving us with our final, elegant result:

And there we have it! The dipole oscillates back and forth, executing simple harmonic motion with an angular frequency determined by the charge, the electric field, the mass, and the separation distance. It is a perfect harmony of electromagnetism and mechanics.