- #1

Tony Hau

- 104

- 30

- Homework Statement
- A sphere of radius R, centred at the origin, carries charge density $$\rho(r,\theta) = k\frac{R}{r^{2}}(R-2r)sin\theta,$$

where ##k## is a constant. Find the approximate potential for points on the z axis, far from the sphere.

- Relevant Equations
- The multipole expansion: ##V(r)=\frac{1}{4\pi \epsilon_o}\sum_{l=0}^{\infty}\frac{1}{r^{l+1}}\int(r^{'})^{l} P_l (cos\alpha)\rho(r^{'})d\tau^{’}##

The diagram of the problem should look something like this:
,which is just the normal spherical coordinate.To calculate the potential far away, we use the multipole expansion.

##I_o## in the expansion is ok, because ##(r^{'})^{0} = 1##.

However, I am wondering how I should calculate ##I_1##, because ##(r^{'})^{1} = r^{'}##; I have to care what ##r^{'}## actually means. I know ##r## is just the normal spherical coordinate ##r##.

Can anyone kindly explain? Thanks for your answer in advance because I have learned a great deal from the forum!

##I_o## in the expansion is ok, because ##(r^{'})^{0} = 1##.

However, I am wondering how I should calculate ##I_1##, because ##(r^{'})^{1} = r^{'}##; I have to care what ##r^{'}## actually means. I know ##r## is just the normal spherical coordinate ##r##.

Can anyone kindly explain? Thanks for your answer in advance because I have learned a great deal from the forum!