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Oh since this got moved: It's not for any course or class so I'm not following any book.

I'm not sure if anyone will be able to answer this but I'll ask anyway:

I've solved laplaces equation in cyclindrical coordinates for a disc of radius 'a' and constant potential V on the disk (disc in z=0 plane, centered on z axis) and got this as the potential everywhere

[tex]\phi(r,z) = \frac{2V}{\pi} \arcsin{\frac{2 a}{\sqrt{z^2 + (r + a)^2} + \sqrt{z^2 + (r-a)^2}}}[/tex]

I want to now find the charge density on the disc. From Gauss and the fact that at the surface the electric field will be perpendicular to the surface we have [tex]E_{z} = 4 \pi \sigma[/tex] where sigma is the charge density.

So I differentiate Phi w.r.t z to get the z component of the electric field and take the limit as z->0. I get zero as my answer.

What am I doing wrong? I can only get this to work in oblate spheroidal coordinates. They are very convenient for this problem but not so when I try to generalize to more discs of different radii and potential.

I'm not sure if anyone will be able to answer this but I'll ask anyway:

I've solved laplaces equation in cyclindrical coordinates for a disc of radius 'a' and constant potential V on the disk (disc in z=0 plane, centered on z axis) and got this as the potential everywhere

[tex]\phi(r,z) = \frac{2V}{\pi} \arcsin{\frac{2 a}{\sqrt{z^2 + (r + a)^2} + \sqrt{z^2 + (r-a)^2}}}[/tex]

I want to now find the charge density on the disc. From Gauss and the fact that at the surface the electric field will be perpendicular to the surface we have [tex]E_{z} = 4 \pi \sigma[/tex] where sigma is the charge density.

So I differentiate Phi w.r.t z to get the z component of the electric field and take the limit as z->0. I get zero as my answer.

What am I doing wrong? I can only get this to work in oblate spheroidal coordinates. They are very convenient for this problem but not so when I try to generalize to more discs of different radii and potential.

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