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12.3 Gauss's Theorem

We will now review these laws and make the conversions. We will simplify matters by ignoring polarization produced by fields in media which require using both E and D as electric fields and B and H as magnetic fields. We will identify these pairs, setting E = D and B = H in the following discussion.

Electrostatics: Gauss's Theorem

You probably encountered electrostatics through Coulomb's Law, which gives the electric force between two point charges, and hence the electric field of a point charge. An equivalent formulation of elctrostatics is that the field is derivable from a potential (so that curl E = 0) and Gauss's Theorem holds.

This Theorem is the statement that the integral of the flux of electric field over the boundary, V, of a region V, is the amount of electrical charge within V:

Combining this statement with the Divergence Theorem applied to E:

yields for any volume V: