E/M question?

Question

A piece of Styrofoam maving a mass m carries a net charge of -q and floats above the center of a very large horizontal sheet of plastic that has a uniform charge density on its surface. What is the charge per unit area on the plastic sheet?

The force of gravity must be equal to the electric force for the Styrofoam to "float"..

Fg = Fe
mg = qE

I'm at a loss as to what to do next. I think there's a relationship that ties the field strength to the charge density, but I have no clue what it is. Some help?

Answer 1

I do not agree with doug_donaghue's approach: It is inappropriate to model the field due to the sheet as if it were a point charge.

The right way to do this: Build a gaussian "pill box" around the sheet, so that the E-field flux comes out of the top and the bottom. The amount of charge contained within is sigma * area, so the since the amount of area of the box is 2 x as great, you find:
E*2*A = sigma*A/epsilon0, so:
E = sigma/(2*epsilon0).

This field is what provides the upward force:
Fe = q*sigma/(2*epsilon0)
must = Fg = mg

So sigma = 2*epsilon0*m*g/q

I should point out that sigma must be negative, because it has to repulse the negatively charged styrofoam upwards. So correct the sign accordingly.

Answer 2

On an infinite sheet, the charge 'seen' by the floating charge will appear to be at a point directly under it, so calculate the charge needed to exactly counteract gravitational force at some distance s above the plastic. Then 'distribute' the charge across the entire surface by dividing by its area A.

HTH

Doug