How Does a Neutral Metal Ball Interact with a Nearby Point Charge?

[funoras]

Homework Statement

Find the attractive force between a neturally charged metal ball of radius $r$ and a point charge $q$, located a distance $l$ from the center of the ball. Also find the work needed to move the charge to infinity. The ball is not grounded.

Homework Equations

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The Attempt at a Solution

I'm stuck on this one. I know that potential must be constant and the electric field must vanish inside the conductor (ball) but that's it.

 

[pcm]

have you learned about the method of electrical images?
how does that apply to the situation of your problem if the sphere is grounded?
what is the relation between net induced charge and the image charge?

 

[funoras]

Yes, i know how to solve this problem using method of images if the sphere is grounded. The induced charge would be $q_i=-qr/l$. But in this case there is no charge induced,since the sphere is not grounded, the charges just move inside the sphere, but the net charge still remains zero.

 

[pcm]

the boundary condition obviously is a constant potential on the sphere surface.
that you can achieve by the image charge(say, Q) at the same location as that in the case of grounded sphere.
but the net image charge must be equal to total induced charge(a consequence of gauss's law).so you need to put another charge -Q inside sphere such that total image charge is zero.
you should now be able to figure out the location of charge -Q.

 

[pcm]

A useful article for this problem.
http://www.ipho2012.ee/physicscup/physics-solvers-mosaic/5-images-or-roulette/

 

[funoras]

it should be at the center of the sphere i guess ? in that case the potential at any point on the sphere will be $q/4πε_0d$ , as expected
edit: didn't see that article you posted. it helped a lot. thanks! The rest is piece of cake.