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Outward Flux of the Electric Field Intensity

The total outward flux of the electric field intensity is given by:


and using Gauss's theorem ( Divergence Theorem ) we have:


, where  and Q is the total charge contained in volume V.

It states that the total outward flux of the electric field intensity over any closed surface in free space is equal to the total charge enclosed in the surface divided by ε0.

Using Stokes's Theorem we also have:


, which asserts that the scalar line integral of the static electric field intensity around any closed path vanishes.  E · dl is the voltage alone that path, therefore it must be 0, when the path is closed. If there are 100 different paths between two points, the voltage between these two points is always the same no matter which path we choose. It only depends on the start and end points.
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