Equivalent point charge, dipole, and quadrupole

[Alicia6]

There is a big sphere with charges distributed on the surface. N charges are distributed on a patch of the sphere (all the information of the charges such as their locations are known). Suppose the rest of the sphere has similar charge distribution as the patch, how to approximately calculate the equivalent point charge, dipole and quadruple of the charges on the rest of the spheres based on the charges on the patch? (the actual distribution of charges on the rest of the sphere is not needed, because I am only calculating the electric field somewhat far away).
Really appreciate it if anyone can help!

 

[kuruman]

Suppose the rest of the sphere has similar charge distribution as the patch ...

What does that mean?

... the actual distribution of charges on the rest of the sphere is not needed ...

It is needed if you need to find the moments higher than monopole. Otherwise, you might as well find the monopole charge by adding all the charges to get Q_total in which case the leading term in the electric field is E = kQ_total/r².

 

[Alicia6]

What does that mean?

It is needed if you need to find the moments higher than monopole. Otherwise, you might as well find the monopole charge by adding all the charges to get Q_total in which case the leading term in the electric field is E = kQ_total/r².

Hi， thank you very much for your reply! I only simulate the charge distribution on a patch to reduce the computing time, so I don't know the charge distribution on the rest of the sphere surface and assume it has similar charge distribution as the patch (i.e., charge density). I can calculate the monopole,dipole and quadripole of the patch, can I use them to approximately infer the monopole,dipole and quadripole of the rest of the sphere?

 

[kuruman]

I can calculate the monopole,dipole and quadripole of the patch, can I use them to approximately infer the monopole,dipole and quadripole of the rest of the sphere?

Not unless you know how many other patches there are and where they are placed, i.e. you know at least the distribution of patches. Say you have only two patches. It will make a difference if you place one patch at the pole and the second at the other pole as opposed to at the equator. Different distributions give rise to different symmetries of the potential.

 

[Alicia6]

Thank you so much for the reply! Suppose I know the monopole, dipole and quadripole of a patch at the pole, how can I calculate the dipole and quadripole of a patch (with same size and charge distribution as the one at the pole) located at (r, theta, phi), where theta is the polar angle and phi is the azimuthal angle? Thank you in advance!