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For a charged conductor, there are “other” charges around (there must be some to make E= 0 in the conductor). The charges in the immediate neighborhood of a point P, considering an arbitrary point P, on the surface do give a field E local = σ/(2*?o) both inside and outside the surface. But all the rest of the charges on the conductor conspire to produce an additional field at the point P equal in magnitude to E local. The total field inside goes to zero and the field outside to 2* E local(=σ/?o).
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