Suppose that a positive point charge generates an electric field . Consider a spherical
surface of radius , centred on
the charge. The normal to this surface is everywhere parallel to the direction of the
electric field , since the field always points
radially away from the charge. The area of the surface is .
Finally, the strength of the electric field at radius is
.
Hence, if we multiply the electric field-strength by the area of the surface, we obtain
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You may be wondering why it took a famous German mathematician to prove such a
trivial-seeming law.
Well, Gauss proved that this law also applies to any closed surface,
and any distribution of electric charges. Thus, if we
multiply each outward element of a general closed surface by the component
of the electric field normal to that element, and then sum over the
entire surface, the result is the total charge enclosed by the surface, divided
by . In other words,
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The electric flux through any closed surface is equal to the total charge enclosed by the surface, divided by .
Gauss' law is especially useful for evaluating the electric fields produced by charge distributions which possess some sort of symmetry. Let us examine three examples of such distributions.