Source: High school physics (Chinese)
Problem Sets:
Problem
An electric dipole consists of two point charges, $+q$ and $-q$, separated by a distance $l$. It is placed in a uniform external electric field of magnitude $E$. The axis of the dipole makes an angle $\theta$ with the direction of the electric field, as shown in the figure.
- Find the torque experienced by the electric dipole.
- How will this torque cause the electric dipole to rotate?
[Q1] $\tau = qlE\sin\theta$ [Q2] The torque will cause the dipole to rotate until it aligns with the electric field.
The electric field exerts a force $\vec{F}_+ = q\vec{E}$ on the positive charge and a force $\vec{F}_- = -q\vec{E}$ on the negative charge. These two forces are equal in magnitude and opposite in direction, forming a couple. The net force on the dipole is zero.
The magnitude of the torque is the product of the force magnitude $F = qE$ and the perpendicular distance (lever arm) $d$ between the lines of action of the forces. From the diagram, the lever arm is $d = l\sin\theta$.
The magnitude of the torque $\tau$ is:
$$\tau = F \cdot d = (qE)(l\sin\theta)$$ $$\tau = qlE\sin\theta$$The torque acts to align the dipole with the electric field. It will cause the dipole to rotate until its axis is parallel to the electric field lines, which is the position of stable equilibrium ($\theta=0$).