Electric Dipole Field

When the separation distance "d" between two opposite point charges in the previous page is small compared to the distance from the center of the two charges to the observation point P, the system is called an electric dipole. The field it produces is called the electric dipole field and it has many applications - for example, in the dielectric properties of matter.

Potential from a Dipole

Figure 1


Assume that the electric dipole is located as shown in the above figure, and we want to calculate the potential and the electric field it produces at point P. We specify the location of P in polar coordinates as (r, q)

(15)

where

(16)

(17)

Using the Law of Cosines, we can rewrite equation 16:

(16a)

Notice that the quantity in the square brackets is in the form [1+x]-1/2, where

X is a quantity much smaller than 1, because the separation distance is much smaller than the distance r to the observation point. We can then expand [1+x]-1/2 by using a Taylor Series:

Then equation 16a becomes

It is in the nature of dipole approximations that Therefore, it is a good approximation to retain up to the linear term in d/r:

(18)

Similarly,

(19)

Substituting equations 18 and 19 into equation 15, we obtain

The product of charge and the separation distance is called the Electric Dipole Moment. It is given by P = qd. This quantity appears often in the study of dipoles and dielectric materials.

(20)

Electric Field from a Dipole

In polar coordinates, the gradient operator is

Therefore,

(21a)

(21b)

The equipotential lines and the electric field lines due to a dipole can be seen in figure 2 below, where the separation distance d is small compared to the observation distance.

Next, you should move on the explore an electric dipole using our Field Applet. You may also take a look at one of the many applications of electric dipoles.


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