Last updated on Thursday, January 05, 2023

Table of Contents

Maxwell's Equations

Deriving the Equations

First Equation

Gauss's Law (Electric flux through a closed surface)

Using the Divergence Theorem :

Second Equation

Gauss's Law for Magnetism (Magnetic flux through a closed surface)

Using the Divergence Theorem :

This is equivalent to saying :

• Magnetic monopoles / charges do not exist (base entity is the dipole)
• Magnetic field lines have neither a beginning nor an end

Third Equation

Faraday's Law (electromotive force, emf)

Using Stokes Theorem :

Faraday's Law :

The electromotive force around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path.

Fourth Equation

Ampere's Law

Maxwell's equation has the following component added to it :

Using Stoke's Theorem :

Ampère's Law :

The original law states that magnetic fields relate to electric current, Maxwell's addition states that they also relate to changing electric fields

Note :

Differential Forms

Integral Forms

First Equation

Second Equation

Third Equation

Fourth Equation

In Empty Space

Notation

Here are some remarks on the notation used that may be useful :

is

is the divergence of

is the curl (rotationel) of

Divergence Theorem

In 2 dimensions (useless here)

In 3 dimensions

is the outwards pointing unit normal at each point on the boundary

If we use a surface such that the normal to the surface is either perpendicular or parallel to :

• The perpendicular parts have zero flux through the surface
• The parallel parts have a flux through the surface simply equal to their value ( becomes )

Stoke's Theorem

Constants and Variables

: electric field

: magnetic field

: electric charge density (total charge per unit volume)

: current density (total current per unit area)

: total electric charge

: net electric current

: emf (electromotive force)

: total current through the loop

: permittivity of free space

: permeability of free space

: Coulomb constant

: Magic constant

: any volume with closed boundary surface

: any surface with closed boundary curve