Two point charges, + q and - q are located at points (0, 0, - a) and (0, 0, a), respectively in a uniform external electric field \( \vec{E} \) . The potential energy of this system of charges in its orientation of unstable equilibrium will be
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 2aqE
A pair of equal and opposite charges forms an electric dipole. Its dipole moment points from −q to +q, here from (0, 0, a) to (0, 0, −a), and has magnitude
\( p = q \times 2a = 2aq \)
In a uniform field the dipole’s potential energy depends on the angle \( \theta \) between \( \vec{p} \) and \( \vec{E} \):
\( U = -\vec{p}\cdot\vec{E} = -pE\cos\theta \)
The equilibrium is stable when \( \vec{p} \) is along \( \vec{E} \) (\( \theta = 0 \), U = −pE) and unstable when \( \vec{p} \) is opposite to the field (\( \theta = 180^\circ \)), where a small turn makes the torque swing the dipole further away.
\( U = -pE\cos 180^\circ = +pE \)
\( U = 2aqE \)
This is the energy of the dipole in the external field; the mutual potential energy of the two charges is constant and is not included.