Work $dW = -\tau d\phi$ done by an electric dipole

Question

How would you explain the intuition behind the equation $dW = -\tau d\phi$ for an electric dipole?

Answer 1

We normally define work something along the lines of "force applied over a distance," but for the case of work done on a rotating object, it would be force applied over an arc-length.

So, for some small amount of work $dW$, need to have a force at some distance form the center of rotation (actually a torque), $\tau$, which goes over some small angle, $d\theta$. The presence of the negative sign is determined if the dipole is doing work on something, or if work is being done on the dipole.

If you're concerned about the arc-length, which is the radius multiplied by some angle, you can get to this by via the quoted math below.

$dW = -F*r*d\theta$

$F*r = \tau$

$dW = -\tau * d\theta$

Answer 2

By analogy to $ \mathbf F = - \nabla U = - \nabla ( - \mathbf p \cdot \mathbf E ) $.