MaxwellEquations

Maxwell's equations can be summarized by four differential equations, each of which relates electric and magnetic fields to their sources (charge and current). They are named after the physicist James Clerk Maxwell who consolidated them into a coherent theory.

Gauss's Law for Electricity: This law states that the electric flux out of any closed surface is proportional to the total charge enclosed within the surface.

$$ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} $$

where:

• $\nabla \cdot \mathbf{E}$ is the divergence of the electric field $\mathbf{E}$,
• $\rho$ is the electric charge density, and
• $\varepsilon_0$ is the permittivity of free space.

Gauss's Law for Magnetism: This law states that there are no magnetic monopoles in nature, i.e., the total magnetic flux out of any closed surface is zero.

$$ \nabla \cdot \mathbf{B} = 0 $$

where $\nabla \cdot \mathbf{B}$ is the divergence of the magnetic field $\mathbf{B}$.

Faraday's Law of Induction: This law states that a time-varying magnetic field induces an electromotive force, which appears as an electric field in space.

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

where:

• $\nabla \times \mathbf{E}$ is the curl of the electric field $\mathbf{E}$, and
• $\frac{\partial \mathbf{B}}{\partial t}$ is the rate of change of the magnetic field $\mathbf{B}$.

Ampere's Law with Maxwell's Addition: This law states that magnetic fields can be induced by electric currents and by changing electric fields (the latter is Maxwell's addition).

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

where:

• $\nabla \times \mathbf{B}$ is the curl of the magnetic field $\mathbf{B}$,
• $\mu_0$ is the permeability of free space,
• $\mathbf{J}$ is the current density, and
• $\frac{\partial \mathbf{E}}{\partial t}$ is the rate of change of the electric field $\mathbf{E}$.

These equations completely describe the behavior of electromagnetic fields, which makes them the foundation of classical electrodynamics, optics, and electric circuits.