Question #42905

This question tests the student's ability to handle an extended dipole in a non-uniform field with a spatially varying gradient.

The primary trap is the 'Point Dipole Approximation'.

Students commonly misapply the standard formula $U = -\vec{p} \cdot \vec{E}$ or $\vec{\tau} = \vec{p} \times \vec{E}$, evaluating the electric field only at the pivot $(a,0)$.

This yields an incorrect work done of $p \alpha a^2 = 2q\alpha a^3$.

However, because the field gradient $\partial E/\partial x$ is not constant, the force varies non-linearly across the dipole's length.

To find the correct work done, one must either integrate the exact variable torque over the rotation angle or calculate the exact change in potential energy using $V(x) = -\int E dx$ for the individual point charges.