Electric Field due to a Dipole

The field of an arbitrary localized charge distribution can be expanded as a Laurent series in inverse powers of the distance: $1/r^2$, $1/r^3$, $1/r^4$, etc. This is called a multipole expansion.

The total charge or “monopole moment” gives the strength of the $1/r^2$ term. The “dipole moment” is related to the strength of the $1/r^3$ term. The “quadrupole moment” is related to the strength of the $1/r^4$ term. The “octupole moment” is related to the strength of the $1/r^5$ term. Etc.

What introductory treatments call “a dipole” — two equal and opposite charges, in the limit that their charge goes to infinity while their separation goes to zero — is simply an idealized charge distribution where only the $1/r^3$ term happens to be nonzero.

The multipole expansion is usually expressed on terms of the potential rather than the field. That expansion starts with a $1/r$ monopole term and a $1/r^2$ dipole term.

The key mathematics behind it is the identity

$$frac{1}{|mathbf{r}-mathbf{r}’|}=frac1rsum_{ell=0}^inftyleft(frac{r’}{r}right)^ell P_ell(cosgamma)$$

where $P_ell(x)$ is a Legendre polynomial and $gamma$ is the angle between the vectors $mathbf{r}$ and $mathbf{r}’$. (Legendre just expanded the left side for $rgg r’$ and found that the terms involved polynomials in $hat{mathbf{r}}cdothat{mathbf{r}}’$, which have come to be named after him.)

Using this expansion in the integral for the potential of an arbitrary charge distribution,

$$varphi(mathbf{r})=frac{1}{4piepsilon_0}intfrac{rho(mathbf{r}’)d^3mathbf{r}’}{|mathbf{r}-mathbf{r}’|},$$

gives one kind of multipole expansion for the potential.

The physical significance of the dipole moment is that if the net charge (i.e., the monopole moment) is zero, then the dipole term in the multipole expansion is the dominant term at a large distance away: the additional terms terms are negligible in comparison as $rtoinfty$. In other words, you can approximate any charge distribution that is overall neutral as a dipole at large distances.