Obtain an expression for electric field at the surface of a charged conductor.

Explanation

We consider a Gaussian surface of a pill box extremely small length and extremely small cross-section as `sigma`.
A fraction of it ls inside the swface and tht remaining part is outside the surface.
The total charge enclosed by this pill box is q =` sigma` ds
where `sigma` = surface charge density of conductor.
At every point on the surface of the conductor `vecE` is perpendicular to the surface. Hence, it is parallel to the surface vector `vecE||d vecs`
But inside the surface `vecE=0` . Hence, the flux coming out from the cross-section of pill box inside the surface=0
The flux coming out from the cross-section of pill box outside the surface,
`phi= vecE.d vecs` = E ds cos `0^(@)` = Eds
According to Gauss.s therorem
`phi = Eds `
`:. (q)/(epsilon_(0))=` Eds
`:. (sigmads)/(epsilon_(0))`= Eds
`:. E = (sigma )/(epsilon_(0))`
In the vector form `vecE= (sigma)/(epsilon_(0)). hatn`
If `sigma` is positive `vecE` is in the direction of normal coming out from the surface . If `sigma` is negative `vecE` is in direction of normal entering into the surface.