This is a problem out of my graduate E&M course, and I was hoping some of y'all might have some ideas on it.

We have an uncharged conducting sphere. It is floating half-submerged in a bath of oil (with dielectric constant [tex] \epsilon [/tex]). Now we put a charge Q on the sphere. The question is, does this cause the sphere to go up or down? How far does it go up or down?

Earlier in the course, we did the problem (out of Jackson, I believe) of two concentric charged shells, with only half of the region between filled with dielectric. We found that the electric field inside the region is radial and independent of [tex] \theta [/tex] and [tex] \phi [/tex]. If we let the outer shell go to infinity, this becomes our problem, and so we can conclude that in our problem the field is radial. This part I am sure about (we talked about it some in class).

From here, I am of the impression that we should use an energy approach; that is, find the energy for a virtual displacement, and determine which sign (up or down) that virtual displacement should be to minimize the energy. I am having problems incorporating this displacement into the energy, though. The energy of the fields I am fine with, using

[tex]

\int_{\mbox{all space}}\!\vec{E}\cdot\vec{D}\,dV\mbox{,}

[/tex]

but how does this displacement come into play?

Sorry for the long post, but I hope some of you have some tips.

Thanks,

Austin