Electric field on a surface

1. The problem statement, all variables and given/known data
A metallic surface is placed in an electrostatic field. Derive an expression for the electric field on the metal surface.

2. Relevant equations
[itex]\oint \underline{E} \cdot d\underline{A}=\dfrac{q_{enc}}{\epsilon_0}[/itex]

3. The attempt at a solution
My initial thought was to set up a cylindrical Gaussian surface, I’ve tried to show it in the picture attachment.

Then the dA facing downwards will be 0 since the electric field within the metal is 0, and the ‘side’ of the cylinder dA is perpendicular to the electric field and also 0. So the only contribution is the upward facing circular area. Letting Δh→0, it gives the electric field on the surface, EA=q/ε, and q/A=σ the surface charge density. So E=σ/ε.

Is this method correct, I’m not entirely sure to be honest.


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