Gauss's Law
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.