Electromagnetic Waves

Abstract

In this chapter we derive the properties of electromagnetic waves. We start from the differential form of the macroscopic Maxwell’s equations and deduce the wave equation for the electric field in vacuum. For its most frequently used solution to describe light - the plane harmonic wave - we formulate the essential properties of light waves, the relations between the oscillating electromagnetic fields, and the propagation parameters. Then we include the linear polarizability of non-magnetic, dielectric media to arrive at the wave equation in matter. By use of the complex quantities dielectric function, refractive index, and wave vector we elaborate the propagation of light in matter including its damping by extinction. We will also shortly review the properties of meta-materials, in particular the consequences of negative or zero refractive indizes.

Problems

2.1

The intensity of the sunlight falling on the earth is, for normal incidence and before its passage through the atmosphere, about 1.5 kW m$^{-2}$. Calculate the electric-field strength.

2.2

Pulsed high power lasers can be easily focused to a power density I of 10 GW/cm$^{2}$. Calculate the E and B fields. Compare them with the electric field in an H atom at a distance of one Bohr radius, and the magnetic field on the surface of the earth, respectively.

2.3

Show qualitatively the $\varvec{B}$, $\varvec{H}$ and $\varvec{M}$ fields of a homogeneously magnetized, brick-shaped piece of iron and for a hollow sphere with inner radius $R_0$ and outer radius $R_0+\Delta R$, which is radially magnetized. Use especially for the second case symmetry considerations together with (2.1).

2.4

Compare the contribution of the electric conductivity of a typical semiconductor to that of the polarisation in (2.26) or (2.28). For which frequencies does the second one dominate?

2.5

Write down the time and space dependence of a spherical wave. Note that the energy flux density varies usually like the amplitude squared. Is it possible to create a spherical vector wave?

2.6

Inspect (with the help of a textbook or a computer program) the electric field of a static electric dipole and the near and far fields of an oscillating electric dipole. Note that in the near field the electric and magnetic fields are not orthogonal.

2.7

Consider or find in a textbook the pattern of the collective motion of the H$_{2}$O molecules in a surface water wave. Which effects contribute to the restoring force? Are water waves harmonic waves? What happens at a seashore, where the depth of the water decreases gradually? Is there a net transport of matter? Assume that the particles have an electric charge and move relative to a fixed background of opposite charge. Which charge pattern do you expect close to the surface? Compare with Fig. 7.4b.