# Multipole expansion

1. Feb 6, 2012

### shikhapunia

given a dipole on z-axis(+q at z=a and -q at z= -a) , find out the non vanishing multipoles using spherical harmonics.
can somebody tell me how to do this problem using spherical harmonics..because when we write charge density using dirac delta function in spherical polar coordinates. then we get
phi = tan^-1 (0/0) because x and y coordinates for both the charges are zero.

2. Feb 6, 2012

### vela

Staff Emeritus

3. Feb 6, 2012

### shikhapunia

In spherical polar coordinates charge density can be written as Ʃqi*δ(r-ri)*δ(θ-θi)*δ(∅-∅i).
where ∅=tan^-1(y/x) . since its a dipole on z-axis therefore ∅'=tan^-1(0/0) . i dont know how to deal with this form of ∅.

4. Feb 6, 2012

### vela

Staff Emeritus
You can pick an arbitrary angle since it doesn't matter anyway when $\theta = 0\text{ or }\pi$.

5. Feb 7, 2012

### shikhapunia

does that mean its ∅ independent? i.e. it has azimuthal symmetry.

6. Feb 7, 2012

### vela

Staff Emeritus
Yes, the charge distribution is azimuthally symmetric.

7. Feb 7, 2012

### shikhapunia

ok..thanks a lot

8. Feb 7, 2012

### shikhapunia

Generally a surface which is symmetric about z-axis is s.t.b azimuthally symmetric like a sphere as we can easily see its symmetric. How can a dipole along z-axis be azimuthally symmetric?

9. Feb 8, 2012

### vela

Staff Emeritus
Does the dipole look different if you rotate the system about the z-axis?

10. Feb 8, 2012

### shikhapunia

ok..now its clear.
thankyou.

11. Feb 8, 2012

### sunjin09

a point charge in spherical coordinates is given by $\rho(r,\theta,\phi)=q\delta(r-r_q)\delta(\theta-\theta_q)\delta(\phi-\phi_q)/(r^2\sin\theta)$

12. Feb 9, 2012

### shikhapunia

oh! yeah..thanks