A (physical) electric dipole consists of two equal and opposite charges (±q) separated by a distance d. Find the approximate potential at points far from the dipole.

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Accepted Answer

Let η− be the distance from −q and η+ the distance from +q (Fig. 3.26). Then

[latex]V(r)=\frac{1}{4\pi\epsilon _{0}}\left(\frac{q}{\eta +}-\frac{q}{\eta -} ight) , [/latex]

and (from the law of cosines)

[latex]\eta ^{2}_{\pm}=r^{2}+(d/2)^{2}\mp rd\cos heta =r^{2}\left(1\mp \frac{d}{r}\cos heta+\frac{d^{2}}{4r^{2}} ight). [/latex]

We’re interested in the régime [latex]r\gg d[/latex] , so the third term is negligible, and the binomial expansion yields

[latex]\frac{1}{\eta _{\pm }}\cong \frac{1}{r}\left(1\mp \frac{d}{r}\cos heta ight)^{-1/2}\cong\frac{1}{r}\left(1\pm \frac{d}{2r}\cos heta ight) . [/latex]

Thus

[latex]\frac{1}{\eta _\pm}-\frac{1}{\eta _{-}}\cong \frac{d}{r^{2}}\cos heta, [/latex]

and hence

[latex]V(r)\cong \frac{1}{4\pi\epsilon _{0}}\frac{qd\cos heta}{r^{2}}. [/latex] (3.90)

The potential of a dipole goes like [latex]1/r^ {2}[/latex] at large r ; as we might have anticipated, it falls off more rapidly than the potential of a point charge. If we put together a pair of equal and opposite dipoles to make a quadrupole, the potential goes like [latex]1/r^ {3}[/latex]; for back-to-back quadrupoles (an octopole), it goes like [latex]1/r^ {4}[/latex]; and so on. Figure 3.27 summarizes this hierarchy; for completeness I have included the electric monopole (point charge), whose potential, of course, goes like 1/r .

Question 3.10: A (physical) electric dipole consists of two equal and oppos...

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