Tag Info

A set of four equations that define electrodynamics. They comprise the Gauss laws for the electric and magnetic fields, the Faraday law, and the Ampère law. Together, these equations uniquely determine the electric and magnetic fields of a physical system. Do not use this tag for the thermodynamical equations known as Maxwell's relations.

## Usage.

A set of four equations that define electrodynamics. They comprise the Gauss laws for the electric and magnetic fields, the Faraday law, and the Ampère law. Together, these equations uniquely determine the electric and magnetic fields of a physical system. Do not use this tag for the thermodynamical equations known as Maxwell's relations.

## Background.

Electrodynamics is the discipline that studies the behaviour of non-static electric and magnetic fields, $\vec E,\vec B$. These are functions of space and time, and are postulated to satisfy a system of four partial differential equations, to wit, \begin{align} \nabla\cdot\vec E&=4\pi\rho\\ \nabla\cdot\vec B&=0\\ \nabla\times\vec E&=-\frac{1}{c}\frac{\partial\vec B}{\partial t}\\ \nabla\times\vec B&=\frac{1}{c}\left(4\pi\vec j+\frac{\partial\vec E}{\partial t}\right) \end{align} where $\rho$ is the so-called charge density, and $\vec j$ the current density; these two functions are said to be the sources of $\vec E,\vec B$. Even in the absence of sources, the Maxwell equations lead to a very rich phenomenology.

The equations above, together with some appropriate boundary conditions, determine the value of the electric and magnetic fields uniquely. Given the fields $\vec E,\vec B$, one may study their effect on electrically charged objects by means of the so-called Lorentz force, $$\vec F=q\ (\vec E+\vec v\times\vec B)$$ which determines the time evolution of point particles and, by extension, to any extended body.