Derivation of displacement current.
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The flux is only has a nonzero part where there is a nonzero electric field. Next if we assume the σσ is the charge density on the plate, then when you replace the EE integral with a σσ integral then the area there is just over the area of the capacitor CC because that is where the electric field is nonzero, and equals σ/ϵ0.σ/ϵ0.
ΦE=∫SE⋅n^da=∫Cσϵ0da
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displacement current is equal to rate of change of electric flux with respect to time
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