Potential of Finite Quadrupole and Zonal Harmonics

[MrCoffee2004]

Homework Statement

a) Find the potential of an axial quadrupole: point charges q, -2q, and q placed on the z-axis at distances L, 0, and -L from the origin.
b) Find the potential only at distances r>>L.
c) Show that this potential is proportional to one of the zonal harmonics.

Homework Equations

The Attempt at a Solution

Ok I am not entirely sure of where to start. I am thinking I need to do an expansion of |r-r'|^(-1) (spherical coordinates) to get to the quadrupole term. Was figuring on having to do a taylor expansion (possibly binomial)? I am not sure however if this is the right approach, and if it is, I am not sure whether to expand with respect to r or r'. It is possible that I have to involve tensors, but I hope not to as we just started talking about them in my Vector and Tensor Analysis class and haven't gone very far with them.

As far as zonal harmonics are concerned, I believe I just have to show it is proportional to one of the legendre polynomials.

I don't want anyone to flat out give me answers, just some guidance :D

Thanks guys

 

[Thaakisfox]

a) you can just use the principle of superposition, and you will get a sum.
b) What you have to do is a multipole expansion of the potential. i.e. the multipole expansion of the inverse distance.
You only have to do it till the quadrupole term (this is the term that will be non zero).

 

[MrCoffee2004]

Yea I also started doing a superposition of potentials...

[tex]\varphi[/tex] = [tex]\frac{1}{4\pi\epsilon_{0}}[/tex] [ [tex]\frac{q}{\sqrt{x^{2} + y^{2} + (z+L)^{2}}}[/tex] + [tex]\frac{q}{\sqrt{x^{2} + y^{2} + (z-L)^{2}}}[/tex] + [tex]\frac{-2q}{\sqrt{x^{2} + y^{2} + z^{2}}}[/tex] ]

Noting that r[tex]^{2}[/tex] = [tex]{\sqrt{x^{2} + y^{2} + z^{2}}}[/tex]

[tex]\varphi[/tex] = [tex]\frac{1}{4\pi\epsilon_{0}}[/tex] [ [tex]\frac{q}{\sqrt{r^{2} + 2Lz + L^{2}}}[/tex] + [tex]\frac{q}{\sqrt{r^{2} - 2Lz + L^{2}}}[/tex] + [tex]\frac{-2q}{\sqrt{r^{2}}}[/tex] ]

Little bit of algebra...

[tex]\varphi[/tex] = [tex]\frac{q}{4\pi\epsilon_{0}r}[/tex] [ [tex]\frac{1}{\sqrt{1 + \frac{2Lz + L^{2}}{r^{2}}}}[/tex] + [tex]\frac{1}{\sqrt{1 + \frac{- 2Lz + L^{2}}{r^{2}}}}[/tex] - 2 ]

Not quite sure where to go after that though.

 

[MrCoffee2004]

Forgot to ask, with respect to which variable will I be doing this expansion?..I believe that r and r' are both in the eqn.

 

[MrCoffee2004]

Ok so yea, someone give me a hand? I am basically screwed here.

 

[MrCoffee2004]

Never mind guys, solution came as a superposition of the binomial expansion of the above terms...fell apart and was easy to show to be proportional to legendre polynomial of degree n=2. May post the rest of the work if it will help anyone here.