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Gauss's law is a fundamental law in physics that relates the electric flux
through a closed surface to the charge enclosed within the surface. It is named
after the German mathematician and physicist Carl Friedrich Gauss, who
formulated the law in its modern form in 1835.
In its integral form, Gauss's law states that the electric flux through a
closed surface is proportional to the charge enclosed within the surface:
∮ S * E · dA = Qenc / ε0
where:
∮ S is the surface integral over a closed surface S
E is the electric field at each point on the surface S ·
indicates the dot (or inner) product dA is the differential
area element of the surface Qenc is the total charge enclosed within the surface
ε0 is the electric constant, also known as the vacuum permittivity.
This
equation implies that electric field lines originating from a positive charge
and terminating at a negative charge are closed lines, with no beginning or end,
and that the total electric flux through any closed surface is proportional to
the charge enclosed within the surface. Gauss's law is a powerful tool for
calculating electric fields in situations with high symmetry, such as spherical
and cylindrical symmetry.
An alternate form of Gauss's law is the differential form, which relates the
divergence of the electric field to the charge density at any point in space:
∇ · E = ρ / ε0
where:
∇ represents the divergence operator ·
indicates the dot (or inner) product E represents the electric field
vector ρ represents the charge density at a given point in space ε0
represents the electric constant or the permittivity of free space. This
equation states that the divergence of the electric field at any point in space
is proportional to the charge density at that point. In other words, the
electric field "flows" away from regions of high charge density, and "converges"
towards regions of low charge density. This form of Gauss's law is particularly
useful in situations where the electric field is not uniform, or where the
geometry of the charge distribution is complex. It can also be used to derive
the integral form of Gauss's law by applying the divergence theorem.
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