3.4: Potential Energy of a Dipole in an Electric Field
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Refer again to Figure III.3. There is a torque on the dipole of magnitude pEsinθ. In order to increase θ by δθ you would have to do an amount of work pEsinθδθ. The amount of work you would have to do to increase the angle between p and E from 0 to θ would be the integral of this from 0 to θ, which is pE(1−cosθ), and this is the potential energy of the dipole, provided one takes the potential energy to be zero when p and E are parallel. In many applications, writers find it convenient to take the potential energy (P.E.) to be zero when p and E perpendicular. In that case, the potential energy is
P.E=−pEcosθ=−p⋅E.
This is negative when θ is acute and positive when θ is obtuse. You should verify that the product of p and E does have the dimensions of energy.