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Curieuse
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Homework Statement
To derive Potential Energy for dipole p in Electric Field E.
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:
τapp will act in same sense as dθ and opposite sense of τfield 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 θ1 =π/2 and θ2=θ
$$U=pE\cos(\theta)$$
$$U=p\cdot E$$
But the traditional derivation outputs $$-p\cdot E$$ and takes τfield and not τapp in the first step. Why is this the case?