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## Homework Statement

To derive Potential Energy for dipole p in Electric Field E.

2. Homework Equations

2. Homework Equations

Potential Energy is the work done by the external agent in turning the angle of the dipole from the U=0 position to another position against the influence of the electric field applied right?

## The Attempt at a Solution

So if Torque exerted by field for a particular θ is given by $$ \tau = pE\sin\theta $$

then when working out potential energy, should we not take the following:

τ

_{app}will act in same sense as dθ and opposite sense of τ

_{field}right?

So $$ \tau_{app} = - pE\sin\theta $$

And the potential energy is just

$$ U = \int_{\theta_1}^{\theta_2}\tau_{app}\,d\theta $$

$$ U = \int_{\theta_1}^{\theta_2}-pE\sin\theta\, d\theta $$

$$U=-pE(-\cos(\theta_2)+\cos(\theta_1))$$

Now if θ

_{1}=π/2 and θ

_{2}=θ

$$U=pE\cos(\theta)$$

$$U=p\cdot E$$

But the traditional derivation outputs $$-p\cdot E$$ and takes τ

_{field}and not τ

_{app}in the first step. Why is this the case?