PlanetPhysics/Gauss's Law

Introduction

Gauss's law, one of [[../MaxwellsEquations/|Maxwell's equations]], gives the [[../Bijective/|relation]] between the electric or gravitational [[../AbsoluteMagnitude/|flux]] flowing out a closed surface and, respectively, the [[../Charge/|Electric Charge]] or [[../Mass/|mass]] enclosed in the surface. It is applicable whenever the inverse-square law holds, the most prominent examples being electrostatics and Newtonian gravitation.

If the [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] in question lacks symmetry, then Gauss's law is inapplicable, and integration using [[../CoulombsLaw/|Coulomb's law]] is necessary.

Definition (Integral form)

In its integral form, Gauss's law is \begin{displaymath} \Phi = \oint_S \vec{E} \cdot \,\vec{dA} = \frac{1}{\epsilon_0}\int_V \,dV = \frac{q_{enc}}{\epsilon_0} \end{displaymath} where $\mathrm{\Phi }$ is electric flux, $S$ is some closed surface with outward normals, $\overrightarrow{E}$ is the [[../ElectricField/|Electric Field]], $\overrightarrow{dA}$ is a differential area element, ${ϵ}_0$ is the permittivity of free space, $q_{enc}$ is the charge enclosed by $S$, and $V$ is the [[../Volume/|volume]] enclosed by $S$.

Definition (Differential form)

In its differential form, Gauss's law is \begin{displaymath} \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \end{displaymath} where $\mathrm{\nabla }$ is the [[../DivergenceOfAVectorField/|divergence]] [[../QuantumOperatorAlgebra4/|operator]], and $\rho$ is the charge density.

Gauss's Law with Electric Displacement

When dielectrics or other polarizable media enter the system, we must modify Gauss's law accordingly. However, we rescind the mathematical perfection of the above formulation of Gauss's law in favor of a more accurate approximation of the real world.

Polarizable media can contain two [[../Bijective/|types]] of charge - free and bound. Free charge can move around, while bound charge results from the induced dipoles within the dielectric. Replacing the electric field with the electric displacement [[../CosmologicalConstant/|field]], and the charge density with, specifically, the free charge density, we have a new form of Gauss's Law: \begin{displaymath} \nabla \cdot \vec{D} = \rho_{\mathrm{free}} \end{displaymath}