The Maxwell equations stand at the very basis of the whole edifice of classical physics. Where Newton's laws tell us how particles move once the forces are specified, the Maxwell equations allow us to specify the electromagnetic forces. They determine the electric and magnetic fields, E and B respectively, caused by a given distribution of charges, or generally by a charge density and a density of electrical currents j. Knowing the fields the resulting Lorentz force on charges can be calculated.

Maxwell's equations describe in full generality the electromagnetic phenomena. From these equations follows for instance the existence of electromagnetic radiation, which – depending on its wavelength – we know as radio waves, visible light or X-rays. It also follows that this radiation is emitted by charges if they get accelerated. This theory is a magnificent example of what is called a field theory, exhibiting the crucial property that fields of force like the electromagnetic fields represent physical degrees of freedom, which carry energy and momentum and which can propagate in space and time (in the form of radiation).

The equations at first look contrived, but in fact the form they are presented in is both natural and efficient, wonderfully pairing convenience with transparency. The expressions and derivative operators which feature in these equations are quite intricate, but basically follow naturally from the fact that we are considering vector fields. The electric and magnetic fields that appear in the Maxwell equations are vector fields, meaning that they depend on space and time, and furthermore they have three components each. So at any point in space there exist an electric and a magnetic field vector, each pointing in some direction in space. Time derivatives of the fields appear in two of the equations determining the dynamics of the electric and magnetic fields; those are the ‘equations of motion’.

The first Maxwell equation determines the electric field caused by the presence of charges ρ and is known as Gauss’ law. The form of the equation is very similar to the continuity equation discussed before. The charge density ρ is a source (or sink) of electric field lines.