Using the poissons equation to find the surface charge density and electric field in a conductor

Recently, I derived a formula for the surface charge density on the surface of a conductor( no specific shape) that is placed in an electric field from the poissons equation of electrostatics. The only condition that I considered while deriving the formula is that the outer surface of the conductor is equipotential. Now I experimented with the formula considering spheres and planes (as they are comparatively simple to deal with.). Now, I observed that the electric field outside the sphere is perfectly consistent with illustrations that I see on the internet. However, the electric field inside the sphere is fairly constant but NOT ZERO. This is an Image I found on the internet

This is what i got when I used the desmos vector field generator( I coudnt find anything better than that.). The radius of the circle is 5 units. The electric field inside the conductor isnt zero. I dont know why and how else I am supposed to solve this

EDIT:Derivation

• If your surface is an equipotential, the field inside has to be zero. Can you share your formula? – Raziman T V Jan 13 '17 at 9:43
• Just added the derivation – Chandrahas Jan 13 '17 at 10:28

• You need a third parameter for the normal direction. As such the Laplacian is incomplete. Imagine your surface is flat and lies in the $xy$ plane, you would completely ignore the $z$ derivatives, but that is what the charge causes – Raziman T V Jan 13 '17 at 11:34