What Are the Mathematical Foundations of the Dipole Approximation?

[center o bass]

Homework Statement

I'm referring to http://hyperphysics.phy-astr.gsu.edu/hbase/electric/dipole.html and the approximation $$r_1 - r_2 \approx = d \cos \theta$$. I see that it is correct if I draw it up, but I wondered if there were any "more mathematical" ways to see this?

Where does these kind of approximations come from? Is this just a special case or is it often used? I don't think I've seen this kind before.

 

[gabbagabbahey]

Homework Statement

I'm referring to http://hyperphysics.phy-astr.gsu.edu/hbase/electric/dipole.html and the approximation $$r_1 - r_2 \approx = d \cos \theta$$. I see that it is correct if I draw it up, but I wondered if there were any "more mathematical" ways to see this?

Where does these kind of approximations come from? Is this just a special case or is it often used? I don't think I've seen this kind before.

The only approximation you need is that $d \ll r$, i.e. the dipole is very small compared to the distance(s) from its center at which you are interested in measuring the field or potential.

To see how this approximation implies that $|\textbf{r}_{+}-\textbf{r}_{-}|\approx d\cos\theta$, you simply use the standard rules for vector addition/subtraction and calculate the magnitude of $|\textbf{r}_{+}-\textbf{r}_{-}|$, then Taylor expand it for small $\frac{d}{r}$.