Maxwell's equations

Maxwell's equations are a set of four fundamental equations in classical electromagnetism that describe how electric and magnetic fields interact and propagate. They form the foundation of electromagnetic theory and are essential for understanding various physical phenomena, from basic electricity and magnetism to light and radio waves.

"A Dynamical Theory of the Electromagnetic Field" is a seminal paper written by the physicist James Clerk Maxwell, published in 1865. In this work, Maxwell formulated what is now known as Maxwell's equations, which describe how electric and magnetic fields interact and propagate through space. Maxwell's contributions unified previously separate laws of electricity and magnetism into a coherent theory, showing that electric fields and magnetic fields are interrelated and can influence each other.

Ampère's circuital law

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Ampère's circuital law is a fundamental principle in electromagnetism that relates the circulation of the magnetic field around a closed loop to the electric current passing through that loop.

Gauss's law

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Gauss's law is a fundamental principle in electrostatics, part of Maxwell's equations, that relates the electric field generated by a charge distribution to the charge enclosed within a closed surface.

Gauss's law for magnetism

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Gauss's law for magnetism is one of the four Maxwell's equations, which are fundamental to electromagnetism. Specifically, Gauss's law for magnetism states that the total magnetic flux passing through a closed surface is zero.

Lorentz force

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The Lorentz force is the force exerted on a charged particle moving through an electromagnetic field. It is named after the Dutch physicist Hendrik Lorentz.

Maxwell's equations in curved spacetime

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Maxwell's equations by

Ciro Santilli 37 Updated 2025-07-16

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Unified all previous electro-magnetism theories into one equation.

Explains the propagation of light as a wave, and matches the previously known relationship between the speed of light and electromagnetic constants.

The equations are a limit case of the more complete quantum electrodynamics, and unlike that more general theory account for the quantization of photon.

The equations are a system of partial differential equation.

The system consists of 6 unknown functions that map 4 variables: time t and the x, y and z positions in space, to a real number:

$E_{x} (t, x, y, z)$ , $E_{y} (t, x, y, z)$ , $E_{z} (t, x, y, z)$ : directions of the electric field $E : R^{4} \to R^{3}$
$B_{x} (t, x, y, z)$ , $B_{y} (t, x, y, z)$ , $B_{z} (t, x, y, z)$ : directions of the magnetic field $B : R^{4} \to R^{3}$

and two known input functions:

$ρ : R^{3} t o R$ : density of charges in space
$J : R^{3} \to R^{3}$ : current vector in space. This represents the strength of moving charges in space.

Due to the conservation of charge however, those input functions have the following restriction:

\frac{\partial ρ}{\partial t} + \nabla \cdot J = 0

(1)

Equation 1.

Charge conservation

Also consider the following cases:

if a spherical charge is moving, then this of course means that $ρ$ is changing with time, and at the same time that a current exists
in an ideal infinite cylindrical wire however, we can have constant $ρ$ in the wire, but there can still be a current because those charges are moving
Such infinite cylindrical wire is of course an ideal case, but one which is a good approximation to the huge number of electrons that travel in a actual wire.

The goal of finding

E

and

B

is that those fields allow us to determine the force that gets applied to a charge via the Equation "Lorentz force", and then to find the force we just need to integrate over the entire body.

Finally, now that we have defined all terms involved in the Maxwell equations, let's see the equations: