Let us consider two point charges of charge –*q* and *q* (*q*>0) separated by a distance *d*. Let **d** be the vector from the negative to the positive charge. Choosing the point halfway between the charges as the origin of our coordinate system, then the electric potential is given by:

and the electric field is

.

First, let us examine the field far from the charges (*r*≫*d*). To simplify things, let *θ* denote the angle between **d** and **r**. Then the law of cosines tells us that

and

. Thus, for *r*≫*d*, we see

and

So we have

.

Using , we see that for distant points, the field is approximated by

.

Defining , and using the unit vector in the direction of **r** , we have

, or, in terms of *θ*,

, so we see the magnitude of the distant field goes as the inverse cube of distance. The vector **p** is the dipole moment, and if one lets *d* go to zero while holding **p** constant, the field approaches the one above at all points, and one has a point dipole.

Now, let us instead consider the potential and field near the point between the charges (*r*≪*d*). In this case, , and

Thus

and so

, where is the unit vector in the direction of **d**. Note that this approximation is a constant vector. Thus, if we have while holding constant, our electric field appoaches a uniform electric field.

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Tags: Dipole, Elecrostatics, Friday Physics, physics

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October 3, 2008 at 12:04 am |

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