Animated Solution for Physics - Electrostatics: Two identical electric point dipoles have dipole moments ${\mathbf{p}}_1=p\hat{\mathbf{i}}$ and ${\mathbf{p}}_2=-p\hat{\mathbf{i}}$ are held on the X-axis at distance $a$ from each other. When released, they move along the X-axis with the direction of their dipole moments remaining unchanged. If the mass of each dipole is $m$, their speed when they are infinitely far apart is

The interaction between electric dipoles is one of the most fascinating topics in electromagnetism. It bridges the gap between simple point charges and complex charge distributions. In this problem, we are presented with a beautiful scenario: two identical electric point dipoles placed on the same axis, facing each other head-on.

Imagine the X-axis stretching out before you. At the origin, we place our first dipole, with its dipole moment ${\mathbf{p}}_1=p\hat{\mathbf{i}}$ pointing proudly to the right. At a distance $a$ away, we place the second dipole. But this one is rebellious; its dipole moment ${\mathbf{p}}_2=-p\hat{\mathbf{i}}$ points to the left.

What does this mean physically? A dipole moment points from the negative charge to the positive charge. So, for the first dipole, its positive charge is on the right. For the second dipole, its positive charge is on the left. This means the two positive charges are staring directly at each other across the gap! As we know from the fundamental laws of electrostatics, like charges repel. Therefore, these two dipoles will experience a strong repulsive force, pushing them apart.

When the dipoles are released from rest, they begin to accelerate away from each other. Because there are no external forces acting on the system (like friction or an external push), the total linear momentum of the system must remain conserved.

Initially, both dipoles are at rest, so the total momentum is zero. As they move apart, their momenta must perfectly cancel each other out at every instant:

Since both dipoles have the exact same mass $m$, this implies that their velocities are equal in magnitude but opposite in direction. Let's call this common speed $v$.

Furthermore, the electrostatic force is a conservative force. This is a massive advantage for us! Instead of trying to integrate the complex, distance-varying repulsive force over time, we can simply use the principle of conservation of mechanical energy. The total energy at the start must equal the total energy at the end:

Initially, the dipoles are at rest, so the initial kinetic energy $K_i$ is zero. But what about the initial potential energy $U_i$?

The potential energy of a dipole in an external electric field is given by the elegant dot product:

Let's calculate the energy of the second dipole sitting in the electric field created by the first dipole. The first dipole creates an axial electric field at a distance $a$. The formula for the axial field of a dipole is:

Now, we plug this field into our potential energy formula for the second dipole:

The two negative signs cancel out, and the dot product of $\hat{\mathbf{i}}$ with itself is 1. This leaves us with:

This positive potential energy confirms our earlier physical intuition: the system is in a state of high energy due to repulsion, eager to convert this stored energy into kinetic energy!

As the dipoles repel and move infinitely far apart, the distance between them approaches infinity. Since the potential energy depends on $1/r^3$, the final potential energy $U_f$ drops to zero.

At this infinite distance, all that stored potential energy has been fully converted into kinetic energy. The final kinetic energy is the sum of the kinetic energies of both dipoles:

Now, we bring our master equation back and substitute everything we've found:

Isolating $v^2$, we get:

Taking the square root gives us the speed:

Finally, we substitute the electrostatic constant $k=\frac{1}{4\pi {ϵ}_0}$ to match the options provided in the question:

And there we have it! A beautiful synthesis of electrostatics, kinematics, and conservation laws leading us directly to the correct answer.