Electric Dipole Moment - Potential and Field of An Electric Dipole

Potential and Field of An Electric Dipole

An ideal dipole consists of two opposite charges with infinitesimal separation. The potential and field of such an ideal dipole are found next as a limiting case of an example of two opposite charges at non-zero separation.

Two closely spaced opposite charges have a potential of the form:

with charge separation, d, defined as

The radius to the center of charge, R, and the unit vector in the direction of R are given by:

Taylor expansion in d/r (see multipole expansion and quadrupole) allows this potential to be expressed as a series.

where higher order terms in the series are vanishing at large distances, R, compared to d. Here, the electric dipole moment p is, as above:

The result for the dipole potential also can be expressed as:

which relates the dipole potential to that of a point charge. A key point is that the potential of the dipole falls off faster with distance R than that of the point charge.

The electric field of the dipole is the negative gradient of the potential, leading to:

Thus, although two closely spaced opposite charges are not an ideal electric dipole (because their potential at close approach is not that of a dipole), at distances much larger than their separation, their dipole moment p appears directly in their potential and field.

As the two charges are brought closer together (d is made smaller), the dipole term in the multipole expansion based on the ratio d/R becomes the only significant term at ever closer distances R, and in the limit of infinitesimal separation the dipole term in this expansion is all that matters. As d is made infinitesimal, however, the dipole charge must be made to increase to hold p constant. This limiting process results in a "point dipole".