Question

Two identical electric point dipoles have dipole moments \(\vec{p}_1=p\hat{i}\) and \(\vec{p}_2=-p\hat{i}\) and 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 :

• A. \(\dfrac{p}{a}\sqrt{\dfrac{3}{2\pi\epsilon_0 ma}}\)
• B. \(\dfrac{p}{a}\sqrt{\dfrac{1}{\pi\epsilon_0 ma}}\)
• C. \(\dfrac{p}{a}\sqrt{\dfrac{1}{2\pi\epsilon_0 ma}}\)
• D. \(\dfrac{p}{a}\sqrt{\dfrac{2}{\pi\epsilon_0 ma}}\)

Hint:

Find the interaction energy of two collinear antiparallel dipoles, \(U=\frac{2p^2}{4\pi\epsilon_0a^3}\), and equate it to the total kinetic energy at infinity.

Answer 1

The Correct Option is C

Field of dipole 1 on its axis at distance \(a\): \(E=\dfrac{2p}{4\pi\epsilon_0a^3}\hat{i}\).
Interaction energy of dipole 2 in this field: \(U=-\vec p_2\cdot\vec E=-(-p\hat i)\cdot\dfrac{2p}{4\pi\epsilon_0a^3}\hat i=\dfrac{2p^2}{4\pi\epsilon_0a^3}=\dfrac{p^2}{2\pi\epsilon_0a^3}\).
\(U\gt0\), so the dipoles repel, move apart symmetrically and the energy at infinity is zero.
Energy conservation: \(2\times\dfrac12mv^2=\dfrac{p^2}{2\pi\epsilon_0a^3}\Rightarrow v^2=\dfrac{p^2}{2\pi\epsilon_0ma^3}\).
\(v=\dfrac{p}{a}\sqrt{\dfrac{1}{2\pi\epsilon_0ma}}\).
Answer: (3) \(\dfrac{p}{a}\sqrt{\dfrac{1}{2\pi\epsilon_0 ma}}\)