Note : the below is a simplified version of the Maxwell equations in their microscopic form, i.e. in vacuum, as they can be found in an introductory course on electrostatics.

Before we get into it though a quick explanation of the symbols used ( bold face denotes vector quantities ) :

: Electric field in [V/m]

: Magnetic field in [T]

: Total charge enclosed in volume V

: Vacuum permittivity

: Vacuum permeability

: Electric and magnetic flux, respectively

: Vector surface element

: Vector line element

: Boundary of a volume of space, i.e. a surface

: Boundary of a surface, i.e. a line

GAUSS's LAW FOR ELECTRIC FIELDS

Plain text meaning : The electric field through a closed surface is directly proportional to the amount of electrostatic charge enclosed within that surface. Equivalently stated : all electric field lines begin or end at electric charges.

GAUSS's LAW FOR MAGNETIC FIELDS

Plain text meaning : In the presence of a magnetic field, the same amount of magnetic field lines enters a given volume of space than exits it. Equivalently : Magnetic field lines form closed loops, or : magnetic field lines do not end. Or, even more generally : there are no magnetic monopoles.

AMPERE's CIRCUIT LAW

Plain text meaning : Both circuit current and displacement current act as sources of the magnetic field.

FARADAY's LAW

Plain text meaning : A changing magnetic flux induces an electric field.

LORENTZ FORCE

A charge q moving through an electromagnetic field with velocity v experiences a force along electric, but perpendicular to magnetic field lines.

SAMPLE CALCULATION

Let's give a very simple, if not to say trivial, example of a field calculation using Maxwell's equations.

Let's say we want to find the electric field around a stationary, constant, isolated point charge Q in free space. The simplest relation for the electric field is Gauss's law :

Since this takes place in free space, and the charge is isolated and stationary, it is intuitive that this problem possesses spherical symmetry - this means that we can choose the uniform surface of a sphere centred on the charge Q as our Gaussian surface :

Since the charge is stationary and constant, the electric field will be constant as well, and the integral simply becomes :

and thus

which is just the familiar Coulomb's law for a single point charge in free space.