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Electric field due to charged conductor

In our previous post we have discussed that electric field inside a conductor is zero and any charge the conductor may carry shall be distributed on the surface of the conductor. For our discussion consider a conductor carrying charge on its surface again consider a small surface element ds over which we can consider surface charge density σ to be approximately constant.
For positive charge distributed over the surface of the conductor , electric field E would be directed at right angels to the surface pointing in outwards direction. Now E due to charge carrying conductor can be calculated using Gauss's law. For this draw a Gaussin cylendrical surface as shown below in the figure

Now S is the area of cross-section of the surface. The flux due to cylendrical surface is zero because electric field and the normal to the surface are perpandicular to each other. Since electric field inside the conductor is zero hence only contribution to the flux is due to the chare on area S lying outside the  surface of the conductor. So total flux through the surface would be

From Gauss's law,
ES=q/ε0=σS/ε0
or,
E=σ/ε0
and this is the required relation for the field of charged conductor