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Check Your Understanding What is the potential on the x -axis? The z -axis?
The x -axis the potential is zero, due to the equal and opposite charges the same distance from it. On the z -axis, we may superimpose the two potentials; we will find that for , again the potential goes to zero due to cancellation.
Now let us consider the special case when the distance of the point P from the dipole is much greater than the distance between the charges in the dipole, for example, when we are interested in the electric potential due to a polarized molecule such as a water molecule. This is not so far (infinity) that we can simply treat the potential as zero, but the distance is great enough that we can simplify our calculations relative to the previous example.
We start by noting that in [link] the potential is given by
where
This is still the exact formula. To take advantage of the fact that we rewrite the radii in terms of polar coordinates, with and . This gives us
We can simplify this expression by pulling r out of the root,
and then multiplying out the parentheses
The last term in the root is small enough to be negligible (remember and hence is extremely small, effectively zero to the level we will probably be measuring), leaving us with
Using the binomial approximation (a standard result from the mathematics of series, when is small)
and substituting this into our formula for , we get
This may be written more conveniently if we define a new quantity, the electric dipole moment ,
where these vectors point from the negative to the positive charge. Note that this has magnitude qd . This quantity allows us to write the potential at point P due to a dipole at the origin as
A diagram of the application of this formula is shown in [link] .
There are also higher-order moments, for quadrupoles, octupoles, and so on. You will see these in future classes.
We have been working with point charges a great deal, but what about continuous charge distributions? Recall from [link] that
We may treat a continuous charge distribution as a collection of infinitesimally separated individual points. This yields the integral
for the potential at a point P . Note that r is the distance from each individual point in the charge distribution to the point P . As we saw in Electric Charges and Fields , the infinitesimal charges are given by
where is linear charge density, is the charge per unit area, and is the charge per unit volume.
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