Lesson 6

Lesson 6 Applications of Electrodynamics

Introduction

You have now built a solid foundation in the basic laws of electromagnetism and learned to express them using the language of tensors. In this lesson we turn to the rich and beautiful applications of these ideas. We begin by exploring the different types of waves that can exist in electromagnetic fields, then see how these waves behave when they encounter matter—in dielectrics, conductors, and magnetic materials.

Transverse Waves

One of the most important types of waves you will encounter in electromagnetism—and indeed in all of physics—is the transverse wave. In a transverse wave the disturbance (the oscillation) is perpendicular to the direction of the wave propagation.

We start with the wave equation in one dimension

The general solution describes waves traveling at speed c. We can write the phase velocity in terms of the frequency, ω, and wave number, k,

The wave number is related to the wavelength, λ, by

Exercise 6.1: A transverse wave on a string is described by y(x,t)=A cos⁡(k x−ω t).
    1) Find the phase velocity.
    2) Find the velocity of a particle on the string at x=0.
    3) What is the maximum particle velocity in terms of A, ω, and k?

Exercise 6.2: Derive (6.3) from (6.2).

The velocity of a particle is then

Exercise 6.3: For the wave y(x,t)=A sin(k x−ω t).
    1) Find and .
    2) Show that .
    3) What does this relationship tell you physically about energy and momentum transport in the wave?

The displacement of the particle is in terms of the amplitude

If we are modeling waves as transmitting along a string, with a given tension, T, we can define an impedance

where ρ is the linear mass density of the string.

Exercise 6.4: Derive the expression for the impedance Z of a wave on a string. Start from the definitions of force and velocity and derive (6.6).

We can also define relationships about various impedances, where is the impedance for the incident region of the string and is the impedance for the reflected region of the string,

These are collectively known as the reflection and transmission coefficients.

Exercise 6.5: A wave travels from a string with impedance to one with . Calculate:
    1) The amplitude reflection and transmission coefficients.
    2) The energy reflection and transmission coefficients.
    3) Verify that energy is conserved.

Given and , we can match these impedances by inserting an intermediate impedance , where .

Exercise 6.6: Explain why inserting an intermediate impedance eliminates reflection. Derive the reflection coefficient when this matching layer is used and show it is zero.

We can separate variables to get a time-independent wave equation

Standing waves on a string of length L have wavelengths

Exercise 6.7: Derive (6.12) and an expression for the frequency of the nth harmonic.

The displacement of the nth harmonic is

The energy of the nth harmonic, assuming a string mass of m is

Exercise 6.8: A string of mass m=0.05 kg and length L=1 m has standing waves with amplitudes m for the first three harmonics. Calculate the total energy in each harmonic and the total energy in the first three harmonics combined.

In a medium that disperses waves, the wave velocity (group velocity) changes with frequence (or wave number) and a dispersion relation appears

where the dispersion parameter is,

Exercise 6.9: Explain the difference between phase velocity and group velocity. In a dispersive medium, which one carries the energy and information of the wave? Why is this distinction physically important?

An emitted wave of frequency ν and velocity c experiences a frequency shift as the source moves with velocity v

this is the famous Doppler shift.

Exercise 6.10: A source emitting waves of frequency ν=500 Hz moves toward a stationary observer at v=20 m/s. The speed of the waves is c=340 m/s. Calculate the observed frequency.

Transverse waves are fundamental to electromagnetism because the electric and magnetic fields in a propagating electromagnetic wave oscillate perpendicular to the direction of propagation. Understanding transverse waves gives you deep insight into the nature of light and all electromagnetic radiation.

Longitudinal Waves

You have already explored transverse waves, where the disturbance is perpendicular to the direction of propagation. Longitudinal waves are similar, but with an important difference: the disturbance (the oscillation) is parallel to the direction in which the wave propagates. In a longitudinal wave the particles move back and forth along the same line as the wave’s travel.

The wave velocity in matter is given by

where the adiabatic index is

where is the specific heat at constant pressure, and is the specific heat at constant volume. For an ideal gas, γ tells us how the gas behaves when compressed or expanded adiabatically (without heat exchange with the surroundings). For air at room temperature, γ≈1.4 (for a diatomic gas). This is why the speed of sound in air uses the adiabatic bulk modulus (γ P) rather than the isothermal one. Sound waves involve rapid compressions and rarefactions, so the process is essentially adiabatic.

The acoustic impendence is

This is for waves traveling in the positive direction. For waves traveling in the negative direction the pressure and particle velocity are in antiphase, so

Some of the reflection and transmission coefficients take on new significance in the longitudinal case

These relationships are particularly useful when studying sound waves in fluids and gases, where the compression and rarefaction of the medium create the longitudinal disturbance. Understanding longitudinal waves gives you insight into sound, seismic waves, and many other important physical phenomena. It also prepares you to appreciate the purely transverse nature of electromagnetic waves in vacuum.

The Wave Equation

You have already seen how waves propagate in vacuum. Now we consider what happens when waves travel through real materials—conductors and dielectrics—where the medium itself affects the propagation.

For waves in a conductor having permeability μ and permittivity ε we have the wave equation

Or, in terms of magnetization

The impedance is

The energy density is

this is equal to the mean energy flow, or the intensity,

In a conductor we add the diffusion equation to our wave equation as there are surface loss effects

This can be solved

where we have k=ω μ σ/2.

Exercise 6.11: For a good conductor with conductivity σ, permeability μ, and permittivity ε, the wave equation includes a diffusion term. Show that the solution has the form and derive the expression for k=ω μ σ/2.

We also have the skin depth, this is the depth where about a third of the current collects near the surface of the conductor,

Exercise 6.12: Derive (6.31). Explain physically why most of the current flows within a distance δ of the surface in a good conductor.

We also have the following ratio

and this determines whether we have a dielectric or a conductor. If this ratio is much greater than 1, then we have a good conductor (the conduction current dominates). If the ratio is much less than 1 then the displacement current dominates and the material is a good dielectric. If the ratio is of order 1, then the material is in an intermediate state having poor conductivity or a lossy dielectric.

Exercise 6.13: For a material with conductivity σ and permittivity ε, at what frequency does the conduction current equal the displacement current? What does this frequency tell you about whether the material behaves as a conductor or a dielectric at different frequencies?

The impedance for a conductor is

We have a reflection coefficient

Exercise 6.14: A plane wave is incident from vacuum onto a good conductor. The impedance of vacuum is and the conductor impedance is given by (6.34). Calculate the reflection coefficient (6.35).

and the transmission coefficient

We have the Fresnel equations for a dielectric

Exercise 6.15: Derive the Fresnel equations for a dielectric.

These results show how electromagnetic waves behave when they encounter real materials. Understanding these effects is essential for applications ranging from optics to radio engineering and beyond.

Exercise 6.16: Reflect on the differences between wave propagation in vacuum, dielectrics, and conductors. Why does the wave equation gain a diffusion term in conductors? How does this affect practical applications such as radio wave propagation or shielding?

DC Circuits

AC Circuits

Energy and Momentum in Electromagnetic Waves

Energy and Momentum in Electromagnetic Waves in Tensor Notation

Polarization of Electromagnetic Waves

Reflection and Refraction of Electromagnetic Waves

Electromagnetic Waves in Dielectrics

Conductors and Electromagnetic Waves

Magnetic Materials

Doing This Stuff in Mathematica

I begin with deriving the wave equations from Maxwell’s equations.

I will use Cartesian coordinates and I clear all of the variables being used.