Question 4.7: Show that the energy of an ideal dipole p in an electric fie...

If the potential is zero at infinity, the energy of a point charge Q is (Eq. 2.39) W = QV (r). For a physical dipole, with –q at r and +q at r + d,

\left.U=q V( r + d )-q V( r )=q[V( r + d )-V( r )]=q\left[-\int_{ r }^{ r + d } E \cdot d\right]\right] .

For an ideal dipole the integral reduces to E · d, and

U=-q E \cdot d =- p \cdot E ,

since p = qd. If you do not (or cannot) use infinity as the reference point, the result still holds, as long as you bring the two charges in from the same point, r _{0} (or two points at the same potential). In that case W=Q\left[V( r )-V\left( r _{0}\right)\right] , and 

U=q\left[V( r + d )-V\left( r _{0}\right)\right]-q\left[V( r )-V\left( r _{0}\right)\right]=q[V( r + d )-V( r )] ,

as before.