Week 12

Basic Facts

The Ampere-Maxwell Law: if there are time-varying electric fields present then Ampere's Law must be modified by the addition of a second term to account for their effect. In integral form the final version is
where S is any open surface bounded by the curve Γ and I is the total current flowing through that surface.

In differential form this becomes
where J is the current density.

By combining the Ampere-Maxwell law with Faraday's Law James Clerk Maxwell deduced that the electric field E should obey a wave equation of the form
.

This equation predicts the existance of electro-magnetic waves that propagate through empty space at the fixed speed
which is found to be the speed of light.

Plane Waves: one of the most common and useful solutions of the wave equation is the plane
wave. In this situation the wave fronts are parallel planes in space and the wave vector k is a
constant that points in the direction of motion of the wave, perpendicular to the wave fronts.
The simplest form of plane wave has an electric field that points entirely in a single
direction. Such a solution is called a Linearly Polarized Plane Wave and has the form
.

In free space and in non-conducting media the electromagnetic wave is strictly transverse so
that and the electric component is accompanied by a magnetic component of
the save phase but at right angles to both k and E. The magnetic field is given by
.

The general solution of the wave equation is a sum of travelling waves of the form
where k is a vector which points in the direction of propagaion wave
and has magnitude . k is called the Wave Vector.

In a non-magnetic, linear, isotropic medium with dielectric constant e the wave equation is unchanged in form
but the velocity is reduced to
where n = √ε is called the Refractive Index of the material.