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Pressure caused by the repulsion of charges

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A balloon made of light conducting material could be kept approximately spherical by connecting it to a high voltage supply. The balloon has a diameter of 100 millimeters, and the maximum breakdown field in air is 3 megavolts/meter.
a) What is the maximum permissible voltage?
b) What gas pressure, in atmospheres, inside the balloon would have the same effect?
c) How large could the surface mass density of the balloon be?

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Let's denote the radius of 50 millimeters of the balloon by R.

If the total charge on the balloon (assumed to be spherical) is Q then the electric field at distance r from the center of the balloon is:

E(r) = Q/(4 pi epsilon_{0} r^2) r-hat (1)

Here it is assumed that r >= R and that r-hat is the unit vector that points radially away from the center. The electric potential relative to infinity is given by:

V(r) = Q/(4 pi epsilon_{0} r) (2)

Minus the gradient of V must yield E. This ...

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