## Electric dipoles

A system consisting of positive and a negative charge of equal magnitude q, separated by a distance d is called an electric dipole.  For an electric dipole we define a new vector, called the electric dipole moment.  The magnitude of the dipole moment vector p is the magnitude of the charge q times the distance d between them, p = qd.    The vector points from the negative towards the positive charge.

The dipole moment is a useful concept when effects of microscopic charge separation are observable, but the actual distance between the charges is too small to be measured.  Molecules can have permanent dipole moments and atoms and molecules without a permanent dipole moment can acquire one when placed in an external electric field.

The potential produced by an electric dipole is calculated summing the potential of the two point charges that produce it.  For a point r whose distance from the negative charge is r- and from the positive charge is r+ we have

V(r) = kq[1/r+  - 1/r-] = kq(r- - r+)/(r+r-).
If a the negative charge is placed on the z-axis at z = -d/2 and the positive charge is placed on the z-axis at z = +d/2 then for large r this becomes

V(r) = k q d cosθ/r2 = k p cosθ/r2,
where r- - r+ = ∆r = d cosθ, and 1/(r+r-) is nearly equal to 1/r2. since r >> ∆r.

The dipole potential does not have spherical symmetry and decreases as 1/distance2, much faster than the Coulomb potential, which decreases as 1/distance.  The dipole field decreases as 1/distance3, and dipole effects become quickly negligible as the distance increases.  The figure to the right shows the equipotential lines and field lines of an electric dipole.

In a uniform electric field the net force on an electric dipole is zero.

But if the dipole moment p is not aligned with the electric field, then a torque acts on the dipole, trying to align p with E.
The magnitude of this torque if τ = pE sinθ.

To rotate the dipole away from alignment, you have to apply an external torque and do work.  The work done by this external torque is stored as potential energy and can be converted into other forms of energy.  The potential energy of a dipole in an external field is

PEdipole = -pE cosθ.

The potential energy is lowest when the dipole is aligned with E and highest if it is anti-aligned.

If the field is not uniform, then the magnitude of the electric force acting on the positive charge can be different from that acting on the negative charge, and there can be a net force acting on the dipole.