# Maxwell's Equations

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## Physical Laws

### Gauss's Law

- $\nabla \cdot \mathbf E = \dfrac \rho {\varepsilon_0}$

### Gauss's Law for Magnetism

- $\nabla \cdot \mathbf B = 0$

### Maxwell-Faraday Equation

- $\nabla \times \mathbf E = -\dfrac {\partial \mathbf B} {\partial t}$

### Ampère's Law with Maxwell's Addition

- $\nabla \times \mathbf B = \mu_0 \paren {\mathbf J + \varepsilon_0 \dfrac {\partial \mathbf E} {\partial t} } $

where:

- $\nabla \cdot$ denotes the divergence operator
- $\nabla \times$ denotes the curl operator
- $\dfrac \partial {\partial t}$ denotes the partial derivative with respect to time.

- $\mathbf E$ denotes the electric field strength
- $\mathbf B$ denotes the magnetic flux density
- $\mathbf J$ denotes the electric current
- $\rho$ denotes volume charge density

- $\varepsilon_0$ denotes the vacuum permittivity
- $\mu_0$ denotes the vacuum permeability.

## Also known as

**Maxwell's equations** are also known as **Maxwell's laws**.

## Source of Name

This entry was named for James Clerk Maxwell.

## Sources

- 1970: George Arfken:
*Mathematical Methods for Physicists*(2nd ed.) ... (next): Introduction: Electromagnetic Theory