Electrostatics problem: potential from CONE with surface charge density?

Question

A cone with aperture alpha, centred on the z axis and with UNIFORM charge density.
I used the general solution to Poisson's equation and got an expression for the potential at any point r, in spherical polar coordinates, so in terms of r and theta (no phi dependence because of the symmetry of the cone).

My problem is: when I set theta equal to the aperture of the cone, which corresponds to calculating the potential of the points ON the cone's surface, I do not get a constant, but something which varies with r.
Now, conductors should be equipotential surfaces shouldn't they? 
So why don't I get a constant on the surface?

Or rather: since I know that charge accumulates in the sharp regions, could this example be physically unrealistic since it is considering a UNIFORM surface charge density? Could this explain the non-constant potential on the surface of the cone?


Many thanks.

Steve4Physics · Accepted Answer

A conducting cone would have a constant potential and a non-uniform charge density which varies with the local radius of curvature.

Your problem is the 'opposite' - you have a constant charge density which will result in a nonuniform potential.  Your cone is a charged insulator, not a conductor.

So there is no problem with your result in this respect.

? · Answer

The discipline from the outside cost density is E = Qs/e0, where e0 is the permittivity of the medium. The magnitude of E is also located from the forces on the electron, that are gravity me*g and electrical E*qe. Equating these, E = (me/qe)*g. Now equate the E from each conditions Qs/e0 = (me/qe)*g Qs = e0*(me/qe)*g The field between plates with constant cost density Qs is independent of the separation as long as the separation is small compared to the scale of the plate.

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Electrostatics problem: potential from CONE with surface charge density?

2 Answers