## Maxwell's Equations

Electricity and magnetism were once thought to be two separate forces. They were unified into a single theory in 1864 by the Scottish physicist James Clerk Maxwell (1831-1879), and since then the theory of electromagnetism has been called Maxwell's theory, and the four key equations of electromagnetism called Maxwell's equations.

The electromagnetic force acting on a charged p[article is divided into two parts. The electric forceacts on all charged particles moving in a magnetic field, but the magnetic forceacts only on moving charged particles. The total force acting on a charged particle is given by the vector sum of the electric and magnetic forces. This is defined as the Lorentz force ** ** (1)

There are four basic laws which govern the behaviour of the electric and magnetic fields, which are known collectively as Maxwell's equations.. The first of these laws is Gauss' law, named after Carl Friedrich Gauss which relates the electric fieldto the electric charge densityof a volume

(2)

whereis the permittivity of free space. This expression shows that the amount of total electric flux through a given closed surface is proportional to the amount of electric charge in the volume contained by that surface. This also implies that a particle containing a given electric charge has an electric field associated with it.

The second of Maxwell's equations tells of the continuity of magnetic flux through a surface. Sometimes referred to as Gauss' law for magnetic fields, this expression states that the magnetic field *B * is divergenceless, and is given by

(3)

This law is similar to Gauss' law for electric fields, since it tells us about the amount of total magnetic flux through a given closed surface, which is zero. 'This states that all magnetic field lines which enter a particular closed surface must eventually leave the surface; thus there are no magnetic monopoles or sources of 'magnetic charge'.

Michael Faraday discovered the law of electromagnetic induction, which describes how a magnetic field that changes in time can also act as a source for the electric field. Faraday's law is given by

(4)

Andre-Marie Ampere discovered that current was a source of the magnetic field, thus the magnetic field is related to the current density *j * (in A m ^{-2 }) by

(5)

whereis the permeability of free space and is related toand the velocity of lightby the relation

(6)

Ampere's law implies that electric charge is conserved since, if we take the divergence of both sides of ( **5 **), we get

(7)

However, Maxwell noticed thatis only valid for steady state situations and that the complete relation for the continuity of electric charge also includes the variation of the electric charge densitywith time, which is given by

(8)

With this knowledge, Maxwell modified Ampere's law to relate the magnetic field to time-varying electric fields, as well as to the current density, obtaining

(9)

The second term of Ampere's law is called the Maxwell displacement current. Maxwell showed that it was needed in order to combine self-consistently the laws of electromagnetism. It was for this insight that equationsandwhich explain the theory of electric and magnetic fields became known as Maxwell's equations.