- #1

yoghurt54

- 19

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## Homework Statement

There are three charges arranged on the z-axis. Charge [tex]+Q_2[/tex] at the origin, [tex]-Q_1[/tex] at [tex](0,0,a)[/tex]

and [tex]-Q_1[/tex] at [tex](0,0,-a)[/tex].

Using spherical polar coordinates (i.e the angle [tex]\vartheta[/tex] is between [tex]r[/tex] and the positive z-axis), find the potential at a point with a distance [tex]r[/tex] from the origin, and in the case [tex]a<<r[/tex], expand the potential up to terms including [tex](a/r)^2[/tex]. Identify terms due to a charge, a dipole and a quadrupole.

## Homework Equations

Well, I found that before the expansion, we find that the potential V is:

[tex] V = 1 / 4 \pi \epsilon ( Q_2 / r -Q_1 (1 / \sqrt{r^2 + a^2 - 2*a*r*cos\vartheta} + 1 / \sqrt{r^2 + a^2 + 2*a*r*cos\vartheta}) )[/tex]

The denominators of the [tex]Q_1[/tex] charges are derived from the cosine rule, and the fact that for the bottom charge, the angle made with the z-axis is [tex] \pi - \vartheta [/tex] which makes the cosine of that angle the negative of the cosine of theta.

## The Attempt at a Solution

Right, after taking out a factor of [tex]r[/tex] and expanding the square root denominators to the [tex] ((a/r)^2 - 2(a/r)cos \vartheta)^2 [/tex] term and ignoring terms greater that the degree 2 we get this:

[tex] V = 1 / (4 \pi \epsilon r) (Q_2 - Q_1(2 + (a/r)^2(3cos^2 \vartheta- 1)) [/tex]

I have a term for the charge and a term for the quadrupole, but no term for the dipole, as those terms canceled when summing up terms in the expansion.

Have I done this right? Should there be no dipole term? I've been stuck on this for a couple of months.