Electrostatics
Gauss's law

Gauss's law states that "the total electric flux through any closed surface in free space of any shape drawn in an electric field is proportional to the total electric charge enclosed by the surface." Many numerical problems can be solved by considering a Gaussian surface around a body. Mathematically, Gauss's law takes the form of an integral equation:

where d 3 r = d x d y d z {\displaystyle \mathrm {d} ^{3}r=\mathrm {d} x\ \mathrm {d} y\ \mathrm {d} z} is a volume element. If the charge is distributed over a surface or along a line, replace ρ d 3 r {\displaystyle \rho \,\mathrm {d} ^{3}r} by σ d A {\displaystyle \sigma \,\mathrm {d} A} or λ d ℓ {\displaystyle \lambda \,\mathrm {d} \ell }. The divergence theorem allows Gauss's Law to be written in differential form:

where ∇ ⋅ {\displaystyle \nabla \cdot } is the divergence operator.
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