Monopole & Dipole Homework: Sphere + Line Charge

[Varnson]

Homework Statement

A sphere of radius R and uniform charge density 'row' is situated at the origin. A uniformly charged line with length L and charge density 'lamda' (for simplicity assume L>2R) is a distance D from the origin in the y=0 plane and orientated so as to be parallel to the x-axis with is center on the z axis. What is the monpole moment of the line charge distribution? What is the dipole moment of the line charge distribution?

Homework Equations

The Attempt at a Solution

I need a little help visuallysing and getting started. Thanks for the help in advance!

 

[StatusX]

The monopole moment should be easy, it's just the total charge. What is the definition of the dipole moment?

 

[Varnson]

The dipole moment is the } r(s) * row * dtau Where } is the integral and r(s) is the vector pointing from source to the dipole moment. Would the charge of the sphere have any influence on the totla charge of the line?

 

[StatusX]

No, the charges are as given. I'm sure you can assume all the appropriate forces are holding them in place, otherwise this would be a complicated dynamical problem.

 

[Varnson]

That is what I was thinking. What vector would you choose for your source vector?

 

[StatusX]

It's probably easiest to use superposition and the formula for changing the origin. Specifically, say you want to calculate the dipole moment with the origin at the center of the sphere. The part that comes from the sphere should be easy. If you get the part from the line, you can just add this to get the total moment. But rather than calculating this directly, you can calculate the moment of the line about the center of the line, and then shift from this origin to the original one at the center of the sphere.

 

[Varnson]

Am I correct if I choose two arbitrary points at each end of the line to figure out the dipole for the line?

 

[Varnson]

Also, if the line is moved a distance 'D' which is greater than 'R' would the force on it be zero? I am thinking no, because the force from the more top of the sphere is greater than the force on the bottom so they won't cancel out. I am assuming the same for the other two direction as well.

 

[StatusX]

I don't know what you mean. If you choose two arbitrary points for what? And as far as the forces, as I said, you don't have to worry about them for this problem, but if you're just curious, there will always be a net force between the line and the sphere, unless they have charges of opposite signs and their centers coincide. There will also be sturcurual forces required to maintain their shapes, but these are more complicated.