Gauss's Law

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Physical Law

In the context of Maxwell's Equations

$\nabla \cdot \mathbf D = \rho$


where:

$\nabla \cdot \mathbf E$ denotes the divergence of the electric field $\mathbf E$
$\rho$ denotes electric charge density
$\varepsilon_0$ denotes the vacuum permittivity.


Informal Explanation

Consider a point charge $q$ at the center of a sphere of radius $r$.

From Surface Area of Sphere, the area of the sphere is $4 \pi r^2$.

From Coulomb's Law of Electrostatics, the electric field at the surface of the sphere is of magnitude $\dfrac q {4 \pi \varepsilon_0 r^2}$.

Hence the product of the area and the magnitude of the electric field is:

$4 \pi r^2 \times \dfrac q {4 \pi \varepsilon_0 r^2} = \dfrac q {\varepsilon_0}$

Hence the total magnitude of the electric field is independent of the radius.


Also known as

Some sources present the name as Gauss' Law.


Examples

Spherically Symmetric Body

Let $B$ be a spherical conducting body in space.

Let $B$ have an electric charge on it.


Then the electric field generated by $B$ is the same as an electric field generated by a point charge with the same charge as the total charge as $B$, placed at the center of $B$.


Source of Name

This entry was named for Carl Friedrich Gauss.