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

(6.1)

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,

(6.2)

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

(6.3)

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

(6.4)

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

(6.5)

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

(6.6)

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,

(6.7)

(6.8)

(6.9)

(6.10)

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

(6.11)

Standing waves on a string of length L have wavelengths

(6.12)

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

The displacement of the nth harmonic is

(6.13)

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

(6.14)

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

(6.15)

where the dispersion parameter is,

(6.16)

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

(6.17)

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

(6.18)

where the adiabatic index is

(6.19)

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

(6.20)

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

(6.21)

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

(6.22)

(6.23)

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

(6.24)

Or, in terms of magnetization

(6.25)

The impedance is

(6.26)

The energy density is

(6.27)

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

(6.28)

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

(6.29)

This can be solved

(6.30)

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,

(6.31)

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.

so

(6.32)

We also have the following ratio

(6.33)

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

(6.34)

We have a reflection coefficient

(6.35)

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

(6.36)

We have the Fresnel equations for a dielectric

(6.37)

(6.38)

(6.39)

(6.40)

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

You turn on a battery and current flows steadily through a circuit. This simple situation—constant currents and voltages—is the realm of direct current (DC) circuits, one of the most practical applications of electromagnetic theory.

In a DC circuit the currents and voltages are constant in time. The fundamental tools for analyzing such circuits are Kirchhoff’s two laws:

Kirchhoff’s Current Law (KCL): The algebraic sum of currents entering a node is zero. Charge is conserved.

Kirchhoff’s Voltage Law (KVL): The algebraic sum of voltages around any closed loop is zero. Energy is conserved.

These two simple statements, combined with Ohm’s law

(6.41)

for resistors, allow you to solve for currents and voltages in even quite complicated networks.

For capacitors in DC circuits the current is zero once the capacitor is fully charged, and the voltage across it is

(6.42)

where C is the capacitance.

. For inductors in steady state the voltage is zero and the inductor acts like a short circuit

(6.43)

A particularly useful technique is the concept of equivalent resistance. Resistors in series add directly

(6.44)

Resistors in parallel combine as

(6.45)

Similar rules exist for capacitors (series and parallel) and for more complex networks using Thevenin’s or Norton’s theorems.

Thevenin’s theorem is one of the most useful tools in circuit analysis. It tells us that any linear network of voltage sources, current sources, and resistors—as seen from two terminals—can be replaced by a single equivalent circuit consisting of one voltage source in series with one resistor.

To use this, first remove any load connected to the terminals and calculate the voltage that appears across them. This is . Then deactivate all independent sources (replace voltage sources with short circuits and current sources with open circuits). Then calculate the equivalent resistance seen from the terminals. This is .

The resulting simple circuit in series with ) produces exactly the same voltage and current at the terminals as the original complicated network.

Norton’s theorem is the dual of Thevenin’s theorem and is equally powerful. It tells us that any linear network of voltage sources, current sources, and resistors—as seen from two terminals—can be replaced by a single equivalent circuit consisting of one current source in parallel with one resistor.

To find the Norton equivalent short the two terminals together and calculate the current that flows through the short. This is . Deactivate all independent sources (replace voltage sources with short circuits and current sources with open circuits). Then calculate the equivalent resistance seen from the terminals. This is (and it is the same as from Thevenin’s theorem).

The Thevenin and Norton equivalents are related by

(6.46)

You can freely convert between them depending on which form is more convenient for the problem at hand.

The power dissipated in a resistor is

(6.47)

reminding us that real circuits convert electrical energy into heat.

DC circuit analysis is not only practically important for power supplies, batteries, and simple electronics—it also serves as a gentle introduction to the broader ideas of conservation laws and network theory that appear throughout physics and prepare you for the richer behavior of alternating current circuits.

AC Circuits

You turn on a household outlet and the voltage varies sinusoidally with time. This is the realm of alternating current (AC) circuits—the type of electricity that powers most of our modern world.

In AC circuits the voltages and currents vary sinusoidally with time. The most powerful tool for analyzing them is the use of phasors. We represent a sinusoidal quantity as a complex number whose magnitude is the amplitude and whose angle is the phase. For example, a voltage

(6.48)

is represented by the phasor

(6.49)

The great advantage of phasors is that differentiation with respect to time becomes multiplication by i ω. This turns differential equations into algebraic ones.

The basic circuit elements in phasor notation have the following impedances

(6.50)

(6.51)

(6.52)

The total impedance of elements in series is the sum of their individual impedances, and for parallel elements we add the reciprocals.

Kirchhoff’s laws still hold in phasor form, so you can solve AC circuits using the same techniques as DC circuits, but now with complex numbers.

The power in an AC circuit has both real and reactive components. The average power dissipated is

(6.53)

Resonance occurs when the inductive and capacitive reactances cancel, making the impedance purely resistive. At resonance the current is maximum and the circuit behaves in a particularly interesting way.

AC circuit analysis is essential for understanding power systems, radio, audio equipment, and virtually all modern electronics. It shows how the simple sinusoidal variation of voltage and current leads to a rich variety of behaviors that are beautifully described by complex numbers and phasors. This also prepares you for more advanced topics such as filters, transformers, and electromagnetic waves.

Energy and Momentum in Electromagnetic Waves

You have seen how electromagnetic waves propagate through space carrying oscillating electric and magnetic fields. These waves also carry energy and momentum. Understanding how energy and momentum flow in an electromagnetic wave is essential for understanding radiation, light pressure, and many practical applications.

The Poynting Vector

The instantaneous power flow per unit area in an electromagnetic wave is given by the Poynting vector

(6.54)

The direction of is the direction of energy flow—the direction in which the wave is propagating. For a plane wave the magnitude of the Poynting vector is

(6.55)

The time-averaged Poynting vector for a sinusoidal wave gives the average intensity (average power per unit area)

(6.56)

Energy Density

An electromagnetic wave stores energy in both its electric and magnetic fields. The energy density in the electric field is

(6.57)

and the energy density in the magnetic field is

(6.58)

For a plane electromagnetic wave in vacuum these two contributions are equal at every instant. The total instantaneous energy density is therefore

(6.59)

The average energy density for a sinusoidal wave is half of the peak value.

Exercise 6.17: A plane electromagnetic wave has electric field amplitude V/m. Calculate the average intensity ⟨S⟩ and the average energy density ⟨u⟩.

Momentum Density and Radiation Pressure

Electromagnetic waves also carry momentum. The momentum density (momentum per unit volume) is

(6.60)

When an electromagnetic wave strikes a surface it exerts a force. This is radiation pressure. For a perfectly absorbing surface the radiation pressure is

(6.61)

For a perfectly reflecting surface it is twice as large

(6.62)

Radiation pressure is the reason why solar sails can propel spacecraft and why comet tails point away from the Sun.

Exercise 6.18: A laser beam with intensity W/m² strikes a perfectly absorbing surface. Calculate the radiation pressure. If the surface has area 1 , what force does the beam exert?

Energy Flow and the Poynting Theorem

The Poynting theorem expresses local conservation of energy

(6.63)

The term represents the work done on charges (ohmic heating or mechanical work). In vacuum, with no charges or currents, the energy simply flows from one region to another at the speed of light.

Exercise 6.19: Starting from the Poynting theorem, show that in vacuum (no charges or currents) the energy conservation equation reduces to . Interpret this equation physically.

The Plane Wave Example

For a monochromatic plane wave traveling in the +z direction with

(6.64)

and

(6.65)

the instantaneous Poynting vector points in the +z direction and its magnitude oscillates at twice the frequency of the wave. The time-averaged intensity is constant and equal to the values given above.

These ideas show that electromagnetic waves are not just oscillating fields—they are carriers of energy and momentum that can do work and exert force on matter. This is the foundation for understanding light pressure, laser cooling, solar sails, and the energy transport in all electromagnetic radiation. It also prepares you for the deeper relativistic treatment of electromagnetic fields in later lessons.

Exercise 6.20: For a plane wave in vacuum show that the electric and magnetic energy densities are equal at every instant. What does this equality tell you about the relationship between E and B in an electromagnetic wave?

Exercise 6.21: For a monochromatic plane wave traveling in vacuum, calculate the time-averaged Poynting vector and show that its magnitude equals c times the average energy density.

Exercise 6.22: Reflect on the fact that electromagnetic waves carry both energy and momentum. How does this explain phenomena such as radiation pressure and solar sails? Why is it significant that light can exert force on matter?

Energy and Momentum in Electromagnetic Waves in Tensor Notation

You have seen how electromagnetic waves carry energy and momentum using vector notation. Now we can express these quantities more elegantly using the language of tensors. This formulation makes the relationships coordinate-independent and prepares us for the full electromagnetic field tensor.

The instantaneous power flow per unit area is given by the Poynting vector in component form

(6.66)

The energy density stored in the electric field is

(6.67)

and the energy density in the magnetic field is

(6.68)

For a plane electromagnetic wave in vacuum these two contributions are equal. The total instantaneous energy density is therefore

(6.69)

Electromagnetic waves also carry momentum. The momentum density in tensor notation is

(6.70)

The local conservation of energy is expressed by the Poynting theorem

(6.71)

These ideas show that electromagnetic waves are not just oscillating fields — they are carriers of energy and momentum that can do work and exert force on matter. Expressing them in tensor notation makes their geometric nature clear and prepares you for the full electromagnetic field tensor and more advanced topics.

Polarization of Electromagnetic Waves

You have seen that electromagnetic waves are transverse  the electric and magnetic fields oscillate perpendicular to the direction of propagation. The particular way that the electric field vector oscillates is called the polarization of the wave. Understanding polarization is essential for optics, antennas, and many applications of electromagnetism.

Linear Polarization

The simplest case is linear polarization. Here the electric field vector oscillates back and forth along a fixed straight line. For a wave propagating in the +z direction we can write

(6.72)

The electric field always lies in the x-direction. Linearly polarized light is what you get from most simple lasers and from light reflected at Brewster’s angle.

Exercise 6.23: A linearly polarized electromagnetic wave propagating in the +z direction has electric field (6.72). Write the corresponding magnetic field B(t,z). What is the direction of the Poynting vector?

Circular Polarization

When the electric field vector rotates in a circle at a constant rate we have circular polarization. There are two senses, right-handed and left-handed. For a wave propagating in the +z direction a right-circularly polarized wave can be written as

(6.73)

The tip of the electric field vector traces out a circle. The magnetic field rotates in phase with the electric field, always perpendicular to it.

Exercise 6.24:  Write the electric field for a right-circularly polarized wave propagating in the +z direction. Show that the tip of the vector traces a circle and that the magnetic field rotates in phase with it.

Elliptical Polarization

The general case is elliptical polarization, where the tip of the electric field vector traces out an ellipse. Linear and circular polarization are special cases of elliptical polarization (with eccentricity 0 and 1, respectively). Any elliptically polarized wave can be decomposed into two orthogonal linear polarizations with a phase difference.

Exercise 6.25: Show that any elliptically polarized wave can be written as the sum of two orthogonal linearly polarized waves with a phase difference. Derive the condition for the polarization to be circular.

Description with Tensors

Polarization is naturally described using tensors. The polarization state can be characterized by the polarization tensor or by the Stokes parameters, which form a convenient set of four real numbers. In tensor language the electric field of a monochromatic wave can be written as a complex vector whose components transform under rotations.

The general form is

(6.74)

where the relative phases determine whether the polarization is linear, circular, or elliptical. The intensity is proportional to .

Exercise 6.26: For a monochromatic wave the electric field is described by the complex vector . Show that the intensity is proportional to . Explain why this is a scalar invariant.

This tensorial description makes it straightforward to handle changes of coordinate systems and the effect of polarizing filters or media on the wave.

Exercise 6.27:  A wave has electric field components , , . Determine the type of polarization and the handedness.

Polarization is one of the most useful properties of electromagnetic waves. It is exploited in 3D movies, liquid crystal displays, radio antennas, and many optical instruments. Understanding polarization in tensor language gives you a powerful tool for working with light and other electromagnetic radiation.

Exercise 6.28: Reflect on why polarization is such a useful property of electromagnetic waves. Give two practical applications (one from optics, one from communications or astronomy) where controlling polarization is important. How does the tensor description make these applications easier to analyze?

Reflection and Refraction of Electromagnetic Waves

When an electromagnetic wave encounters a boundary between two different media, part of the wave is reflected and part is transmitted (refracted). Understanding these phenomena is essential for optics, antennas, and many practical applications of electromagnetism.

Boundary Conditions

At the interface between two media the electromagnetic fields must satisfy certain continuity conditions derived from Maxwell’s equations. In 3D tensor notation these conditions become particularly clear.

Assume the boundary is thex-y-plane, with the normal along z direction. The tangential components of are continuous

(6.75)

The normal component of is also continuous

(6.76)

The tangential components of and normal involve the material properties (permittivity and permeability) of the two media. These continuity conditions, expressed in terms of the field components, determine the amplitudes and directions of the reflected and transmitted waves.

Exercise 6.29:  An electromagnetic wave is incident normally on a boundary between two dielectrics. The tangential component of must be continuous. If the incident electric field amplitude is V/m and the reflected amplitude is V/m, calculate the transmitted amplitude .

Fresnel Equations

The relationships between the amplitudes of the incident, reflected, and transmitted waves are given by the Fresnel equations. For the electric field we distinguish two cases, one polarization parallel to the plane of incidence (p-polarization or parallel) and one polarization perpendicular to the plane of incidence (s-polarization or perpendicular).

For p-polarization we have (6.37) and (6.38)

For s-polarization we have (6.39) and (6.40)

Here φ is the angle of incidence, and θ is the angle of refraction. These equations tell us how much of the wave is reflected and how much is transmitted at different angles.

Exercise 6.30: Using the Fresnel equations for s-polarization, calculate the reflection coefficient at normal incidence (φ=0). Show that it reduces to the familiar form and explain physically why there is a phase change upon reflection from a denser medium.

Brewster’s Angle

There is a special angle of incidence, called Brewster’s angle, where the reflection coefficient for p-polarized light is zero. At this angle the reflected wave is completely s-polarized. Brewster’s angle is given by

(6.77)

where and   are the refractive indices of the two media. This phenomenon is widely used in optics to produce polarized light and in laser systems to reduce unwanted reflections.

Exercise 6.31: Light is incident from air () onto glass (). Calculate Brewster’s angle. At this angle, what is the polarization of the reflected light?

Exercise 6.32: Derive the expression for Brewster’s angle starting from the condition that the reflection coefficient for p-polarization is zero. Show that .

Physical Interpretation and Applications

When light passes from air to glass, for example, the speed decreases and the wave bends toward the normal (refraction). Part of the energy is reflected back into the first medium. The Fresnel equations quantify exactly how much is reflected and transmitted at each angle. These effects are responsible for the shimmering of water surfaces, the operation of polarizing sunglasses, and the design of anti-reflection coatings on lenses and solar cells.

In tensor language these phenomena can be described using the boundary conditions on the field tensor components, making the treatment coordinate-independent and ready for more advanced calculations.

Reflection and refraction are among the most visible and useful properties of electromagnetic waves. Mastering them gives you deep insight into the behavior of light and the design of optical devices.

The 3D tensor description makes these phenomena coordinate-independent within a given frame and greatly simplifies calculations when changing bases or dealing with more complicated geometries.

Reflection and refraction are among the most visible and useful properties of electromagnetic waves. Mastering them gives you deep insight into the behavior of light and the design of optical devices.

Exercise 6.33: A wave goes from air to water. Explain qualitatively why the reflection coefficient is different for p- and s-polarization at oblique incidence. How does this explain the glare on water surfaces and the usefulness of polarizing sunglasses?

Exercise 6.34: Reflect on how the boundary conditions and Fresnel equations arise from the continuity of the field components. Why is the tensor (or component) approach powerful here? Give an example of a practical device (e.g., anti-reflection coating, polarizer, or optical fiber) that relies on these principles.

Electromagnetic Waves in Dielectrics

You have seen how electromagnetic waves propagate in vacuum at the speed of light. When these waves enter a material medium the situation becomes richer. The electric field of the wave polarizes the atoms or molecules in the material, and the resulting polarization affects the wave itself. This is the realm of electromagnetic waves in dielectrics.

Dielectrics

A dielectric is a material that can be polarized by an applied electric field but does not conduct electricity well. When an electromagnetic wave passes through a dielectric, the oscillating electric field causes the charges in the material to oscillate, producing a polarization field . The total electric field inside the material is related to the displacement field by

(6.78)

where is the permittivity of the material and is the relative permittivity (dielectric constant). We can write this in tensor notation

(6.79)

where is the permittivity tensor. For isotropic dielectrics this simplifies to , but in anisotropic materials (such as crystals) the tensor nature becomes important and the refractive index depends on direction.

Exercise 6.35: A dielectric material has relative permittivity . Calculate the speed of an electromagnetic wave in this material and the refractive index n. How does this compare to the speed in vacuum?

Dispersion

In many dielectrics the response of the material depends on the frequency of the wave. This frequency dependence is called dispersion. The permittivity ε(ω) becomes a function of frequency, and therefore the speed of the wave inside the material also depends on frequency. This is why a prism can separate white light into colors—different frequencies travel at slightly different speeds.

Exercise 6.36: Explain in your own words why the permittivity ε(ω) is frequency-dependent in most dielectrics. What physical process in the material causes dispersion?

Refractive Index

The refractive index n of a medium is defined as the ratio of the speed of light in vacuum to the speed of light in the medium

(6.80)

For non-magnetic dielectrics )) this simplifies to . The refractive index determines how much a wave bends when it enters the material (refraction) and how much is reflected at the interface.

Exercise 6.37:  Light passes from air () into a dielectric with . Using Snell’s law, calculate the angle of refraction if the angle of incidence is 30°. What is the speed of the wave inside the dielectric?

Propagation in Linear Dielectrics

In a linear dielectric the wave equation becomes

(6.81)

The solutions are still plane waves, but they travel at reduced speed v=c/n and have a modified wave number k=n ω/c. The electric and magnetic fields remain perpendicular to each other and to the direction of propagation, but their amplitudes are related by B=n E/c.

In tensor notation, this becomes

(6.82)

For monochromatic plane waves we assume a solution of the form . Substituting this into the wave equation yields the dispersion relation

(6.83)

and for isotropic cases

(6.84)

The phase velocity inside the medium is v=c/n, where the refractive index for non-magnetic dielectrics.

The magnetic field is related to the electric field by

(6.85)

This shows that , , and the propagation direction remain mutually perpendicular, but the magnitudes are scaled by the properties of the medium.

In a lossless dielectric the wave propagates without attenuation. In real materials there is some absorption, and the refractive index becomes complex, leading to exponential damping of the wave amplitude.

Exercise 6.36: Starting from the wave equation in a dielectric, derive the dispersion relation (6.84) for an isotropic medium. Show that the phase velocity is v=c/n.

In tensor language these relations make it straightforward to handle changes of coordinate systems and anisotropic materials. The permittivity tensor encodes how the material responds to the electric field in different directions, and the wave equation becomes a compact statement about how the field propagates through that response.

Exercise 6.37:  In an anisotropic dielectric the permittivity is a tensor . Explain why the wave velocity can depend on the direction of propagation. Give a simple example of how the refractive index becomes direction-dependent.

Electromagnetic waves in dielectrics are responsible for many familiar optical phenomena—the bending of light in lenses, the colors of the rainbow, and the operation of optical fibers. The tensor formulation gives you a powerful, coordinate-independent tool for analyzing these effects and for designing devices that control light.

Exercise 6.39: Reflect on the differences between wave propagation in vacuum and in dielectrics. Why does the introduction of a permittivity tensor make the description more powerful? Give two practical applications (e.g., lenses, optical fibers, or liquid crystal displays) that rely on the behavior of electromagnetic waves in dielectrics.

Conductors and Electromagnetic Waves

In the earlier parts of this lesson you explored how electromagnetic waves travel cleanly through vacuum and through insulating dielectrics, with the electric and magnetic fields oscillating in phase, perpendicular to each other and to the direction of propagation, carrying energy at the speed of light. Those waves arise from the coupled, source-free Maxwell equations and satisfy a simple wave equation whose solutions are plane waves with real wave number.

When the medium contains free charges that can drift under the influence of an electric field, everything changes. Currents appear, the displacement current is joined (or even dominated) by a conduction current, and the wave equation acquires damping terms. The elegant propagating solutions become exponentially decaying disturbances that penetrate only a finite distance into the material. This is not a mathematical inconvenience; it is nature’s practical way of confining electromagnetic energy and momentum where mobile charges can respond most effectively.

Understanding conductors and the waves that travel through them therefore bridges the ideal vacuum case you already know with the real materials that surround us — metals, sea water, the ionosphere, and even biological tissue. It explains why radio signals fade quickly underwater, why high-frequency currents hug the surface of a wire, why metal enclosures shield sensitive electronics, and why the mathematics of complex wave numbers and tensor formulations ultimately describe one unified physical reality.

Conductors

Begin with the physical idea. In an ordinary conductor (a metal, an electrolyte, a plasma), some charges are not bound to specific atoms or molecules. They are free to move throughout the volume when an electric field is applied. Their collective drift constitutes an electric current. The simplest description of this response, valid for many materials over a wide range of conditions, is Ohm’s law in local form

(6.86)

Here is the current density, is the electric field, and σ is the electrical conductivity (measured is Siemens per meter). The conductivity is a constant that measures how readily the material allows charge to flow; good conductors such as copper or silver have enormous σ, while sea water or the human body have modest but still significant values.

You already know from electrostatics that, in a perfect conductor (σ->∞)), any internal electric field is immediately canceled by the rearrangement of free charges. The surface becomes an equipotential, tangential vanishes just inside the material, and magnetic fields (if steady) can penetrate according to other rules. For time-varying fields the story is richer: the free charges still try to cancel , but the finite conductivity and the inertia of the charges (or, equivalently, the inductive effects) prevent perfect cancellation everywhere at once. Fields and currents therefore decay with depth inside the conductor rather than vanishing abruptly.

Definition 6.1 Linear Isotropic Conductor: A linear isotropic conductor is characterized by a scalar conductivity σ\sigma\sigma
that relates current density to electric field via . . (Anisotropic or nonlinear conductors require tensors or more complicated constitutive relations.)

At power-line frequencies (50–60 Hz) the skin depth in copper is several millimeters; most of the current still flows through the bulk of a thick cable. At microwave frequencies the same copper confines current to a layer thinner than a human hair. The mathematics you are about to derive captures both regimes with a single formula.

Skin Effect

The core physical idea is simple and profound where an alternating or propagating electromagnetic field cannot instantly rearrange charges throughout the entire volume of a good conductor. The induced currents themselves generate opposing magnetic fields (Lenz’s law) that oppose further penetration. Consequently the fields and the associated currents are forced to concentrate near the surface. This concentration is called the skin effect.

To see it quantitatively, start from Maxwell’s equations inside the conductor (no free charge accumulation on the scale of interest, or already incorporated). Take the curl of Faraday’s law and substitute Ampère’s law with the conduction current

(6.87)

For good conductors at frequencies where the conduction current dominates the displacement current (σ≫ω ε), the second term on the right is negligible and you obtain the diffusion equation for the electric field

(6.88)

Assume a plane-wave-like solution propagating in the +z direction into a semi-infinite conductor occupying z>0: with complex wave number. Substituting yields

(6.89)

The physically decaying solution (fields must remain finite as z-> +∞) is the one with positive imaginary part in the exponent, giving an exponential decay factor  . The characteristic distance over which the amplitude drops by a factor of 1/e is the skin depth

(6.90)

The same factor appears in the current density . Thus both the fields and the currents decay exponentially away from the surface, with an accompanying phase shift of 45° between and (or between and ) deep inside the good-conductor limit.

Figure 6.1: The skin effect.

Principle 6.1: In a good conductor the electromagnetic disturbance diffuses inward rather than propagating as a wave; the diffusion length is set by the competition between inductive opposition (μ) and dissipative flow (σ) at the driving frequency ω.

The simple skin-depth formula assumes linear response, isotropic material, frequencies low enough that displacement current is negligible, and a geometry where the radius of curvature is much larger than δ. At optical frequencies in metals the plasma frequency intervenes and the good-conductor approximation fails; one must use the full complex dielectric function.

Exercise 6.40: Copper has conductivity S/m and . Calculate the skin depth δ at
    1) 60 Hz (power-line frequency).
    2) 2.4 GHz (Wi-Fi frequency).
Express your answers in millimeters and micrometers, respectively. What does the dramatic change tell you about why high-frequency cables often use stranded or tubular conductors?

Attenuation

The exponential factor is an attenuation of the wave amplitude. Define the attenuation constant α as the real number such that the field behaves as (for propagation in the +z direction). In the good-conductor limit,

(6.91)

The power (time-averaged Poynting vector magnitude) therefore decays as , twice as fast. This rapid attenuation is why a thin sheet of aluminum foil stops microwave oven leakage, why submarine communication at very low frequencies still requires enormous transmitter power, and why the interior of a good conductor is effectively shielded from external high-frequency fields.

Attenuation is nature’s bookkeeping where the energy that disappears from the wave is continuously dissipated as Joule heat () inside the conductor. The process is irreversible on macroscopic scales, converting ordered electromagnetic energy into microscopic thermal motion of the lattice ions and electrons.

Exercise 6.41: Sea water has σ≈4 S/m and relative permittivity . For a 10 kHz signal, estimate the skin depth and the distance at which the time-averaged power density drops by a factor of . Discuss why extremely-low-frequency (ELF) waves are used for submarine communication while higher frequencies are useless beyond a few meters of depth.

Propagation in a Conducting Medium

Now drop the “good-conductor” approximation and keep the full displacement-current term. The wave equation inside a linear isotropic conductor becomes

(6.92)

For a monochromatic plane wave , substitution yields the dispersion relation

(6.93)

The complex wave number has real part β (phase progression) and imaginary part α (attenuation). Solving the square root gives explicit but somewhat lengthy expressions for α(ω) and β(ω). In the two limiting regimes you recover the results already derived:

Good conductor (σ/ ε ω≫1): , phase velocity , impedance .

Poor conductor or dielectric (σ/ ε ω≪1): α small, , , recovering ordinary wave propagation with weak damping.

Inside a good conductor, and are no longer in phase. The magnetic field lags the electric field by up to 45°. The time-averaged Poynting vector still points in the direction of propagation (or into the surface for a reflected wave), but its magnitude is reduced by the factor involving the skin depth. The wave is said to be inhomogeneous; surfaces of constant amplitude and constant phase are not the same.

Radio waves at 3 MHz incident on sea water (σ≈4 S/m) have skin depth of only a few meters. A submarine at 100 m depth is essentially invisible to ordinary radio communication; extremely low-frequency (ELF) transmissions with skin depths of kilometers are required.

Exercise 6.42: A monochromatic plane wave of frequency ω is incident normally from vacuum onto a good conductor (σ≫ω ε). Using the good-conductor approximation for the intrinsic impedance inside the material, derive the amplitude reflection coefficient for the electric field. Show that it approaches −1 in the perfect-conductor limit and interpret the small deviation physically.

Exercise 6.43: Keep both conduction and displacement currents. For a material with given σ, ε r, and μ r=1, write a short Wolfram Language snippet (or solve analytically in the high- and low-frequency limits) that computes the attenuation constant α and phase constant β as functions of frequency. Evaluate numerically at a frequency where σ/(ε ω)≈1 and comment on how the phase difference between and behaves in that transition region.

Translation of These Results into Tensor Notation

You have now seen how conductivity turns the ordinary wave equation into one with damping. The next natural step is to express the same physics using index notation (Cartesian tensors). This makes the isotropy (or possible anisotropy) of the material explicit, prepares you for crystals or plasmas later, and keeps everything firmly within the three-dimensional vector calculus you already know — no four-dimensional spacetime is required yet.

The core physical idea remains the same: free charges respond to the local electric field by producing a current. When the material may be anisotropic (conductivity depends on direction, as in a crystal or a magnetized plasma), the simple scalar relation generalizes. The current in one direction can depend on electric-field components in all three directions.

Definition 6.2: The conductivity tensor. This is a second-rank Cartesian tensor, , that relates the current-density vector to the electric field via

(6.94)

For an isotropic conductor, and you recover the familiar Ohm’s law.

For time-harmonic fields (), it is convenient to combine conduction and displacement currents into a single effective term. Ampère’s law with Maxwell’s correction becomes

(6.95)

where the effective displacement field absorbs the conductivity

(6.96)

We define the complex permittivity tensor

(6.97)

All the results you derived earlier (skin depth, attenuation constant, phase lag) follow immediately once you replace the scalar ε by this tensor in the wave equation. For an isotropic medium the tensor is diagonal and the earlier scalar formulas are recovered exactly.

Starting from the curl equations and eliminating , the electric field inside the linear medium satisfies

(6.98)

where is the Levi-Civita symbol. For plane-wave solutions , this becomes an algebraic eigenvalue problem for the wave vector .

(6.99)

The dispersion relation and the complex you met in the scalar case are the eigenvalues of this tensor equation. When is diagonal and isotropic, it collapses to the same quadratic you solved before.

Principle 6.1: Tensor notation does not change the physics—it simply makes the directional dependence transparent. Attenuation, skin depth, and the 45° phase shift all emerge from the imaginary part of . In isotropic metals this imaginary part dominates at radio and microwave frequencies, driving the strong skin effect you already calculated.

For sea water treated as isotropic, . The complex permittivity becomes , exactly the factor that appeared inside the square root of the earlier dispersion relation. The skin-depth formula follows at once.

This index/tensor language is the natural bridge between the vector equations of your earlier sections and the fully relativistic formulation you will meet in a future lesson. It lets you keep using familiar three-dimensional pictures while gaining the generality needed for real materials.

Exercise 6.44: Write the complex permittivity tensor for
    1) an isotropic conductor.
    2) a uniaxial material in which conductivity is along one crystal axis and perpendicular to it.
Show that the dispersion relation you derived earlier is recovered in the isotropic limit. What new phenomenon (birefringence or dichroism) might appear in the anisotropic case?

Exercise 6.45: Throughout this section you saw how a seemingly simple addition of the term turns perfect propagating waves into exponentially damped disturbances, with energy steadily converted into heat. Step back and reflect; in what sense is the skin effect nature’s way of letting mobile charges “protect” the interior of the conductor? How does this picture connect to the electrostatic result that inside a perfect conductor? Finally, imagine you are designing a shield for sensitive electronics—how would the frequency dependence of δ guide your choice of material thickness and conductivity? What broader wonder about the universe does this evoke for you?

Principle 6.2: In good conductors, fields and currents concentrate near surfaces (skin effect) because induced currents oppose deeper penetration.  

Principle 6.3: Electromagnetic energy is continuously converted to Joule heat inside conductors, producing attenuation.

Magnetic Materials

In the previous section you explored how free charges in conductors respond to electric fields, turning clean propagating waves into damped disturbances confined near surfaces. Nature is symmetric in many ways, where materials also respond to magnetic fields by developing microscopic magnetic moments that can align, oppose, or amplify the applied field. These responses give us permanent magnets, transformers, data storage, and the shielding or focusing of magnetic fields in countless devices.  

You will now see the same elegance you met with conductivity tensors—only now the response is magnetic. We use tensor notation from the start so the directional character of real materials (crystals, magnetized iron, ferrites) is built-in naturally rather than added later.

Magnetization

When you place a material in an external magnetic field or , the atoms, molecules, or domains inside develop net magnetic moments. These moments arise from the orbital motion and spin of electrons. The collective effect per unit volume is a magnetization vector field , with units of amperes per meter (magnetic moment per unit volume).

In tensor language we allow the response to be anisotropic. The magnetization produced by an applied field is written

(6.100)

this is the linear response, but we will build up to that definition carefully. For the moment, treat as the fundamental quantity that appears in the macroscopic Maxwell equations in the next section.

The auxiliary field is defined so that Ampère’s law in materials reads cleanly

(6.101)

while the fundamental field satisfies

and

Definition 6.3 Magnetization : This vector  is the magnetic moment per unit volume. In component form the contribution to the macroscopic is .

For example, A bar magnet has roughly uniform inside and zero outside; the “bound” surface currents produce the external field you feel when you pick it up.

Permeability

Many materials respond linearly to a weak applied . The total is then proportional to , but the constant of proportionality can be a tensor when the material is anisotropic (e.g., a crystal or a material with a preferred magnetization direction).

Definition 6.4 The permeability tensor  : This tensor can be defined by

(6.102)

For isotropic materials this reduces to the familiar scalar , where is the relative permeability. In vacuum or non-magnetic media, .

From the relation you immediately obtain the connection to magnetization

(6.103)

where is the magnetic susceptibility tensor (introduced next). Diamagnetic materials have (tiny negative χ), paramagnetic have (tiny positive χ), and ferromagnetic materials can have that is both large and history-dependent.

Principle 6.4: The permeability tensor encodes how the material’s internal moments amplify or reduce the applied field. Because it is a tensor, need not be parallel to in anisotropic crystals—exactly analogous to the dielectric tensor you may have met earlier.

For example,  In transformer steel, can reach thousands, concentrating magnetic flux and making efficient energy transfer possible. In ferrites used at microwave frequencies the tensor character (especially when magnetized) allows non-reciprocal devices such as isolators and circulators.

Magnetic Susceptibility

It is often convenient to think in terms of the material’s response rather than the total . The magnetic susceptibility tensor directly relates the induced magnetization to the applied .

Definition 6.5 Magnetic Susceptibility: Beginning with (6.100)

For linear isotropic media this is a single number . Diamagnetism gives χ<0 (weak repulsion from fields), paramagnetism χ>0 (weak attraction), and the values are usually small (|χ|≪1) except in ferro- and ferrimagnetic materials.

Because is a tensor, it can have off-diagonal elements in crystals with lower symmetry, leading to interesting phenomena such as the Faraday effect (rotation of polarization in a longitudinal magnetic field) when combined with time-dependent fields.

Each tiny current loop or spin inside the material acts like a little compass. In diamagnetic materials the induced loops oppose the applied field (Lenz’s law at the atomic scale). In paramagnetic materials existing moments partially align with the field. The tensor simply keeps track of how easily alignment occurs along each direction.

Exercise 6.46: A linear isotropic material has . Calculate the magnetic susceptibility χ. If an applied field H=200 A/m produces a magnetization , find both and the resulting . What fraction of comes from the material’s own moments?

Exercise 6.47: A uniaxial crystal has principal permeabilities . Write the permeability tensor in matrix form. An applied field is present. Find the components of and show that is not parallel to unless or .

Ferromagnetism

In certain materials (iron, nickel, cobalt, many alloys and oxides) the magnetic moments interact so strongly with each other that they spontaneously align over macroscopic regions called domains, even in the absence of an external field. This cooperative behavior is ferromagnetism (or ferrimagnetism in oxides).

The macroscopic magnetization can be large and remanent—it remains after the external field is removed. The relation between and is no longer linear or single-valued since it follows a hysteresis loop. In tensor language the permeability or susceptibility becomes field- and history-dependent where (the history of ).

There are a number of key features that you should recognize:

Spontaneous magnetization below the Curie temperature. Where the Curie temperature is the critical temperature at which a ferromagnetic (or ferrimagnetic) material undergoes a second-order phase transition from the ordered ferromagnetic state to the disordered paramagnetic state. We will learn more about this beginning with Lessons 9, 24, 25, and 39.

Domain walls that move or rotate when an external field is applied. Where a  domain wall is a transition region between two magnetic domains in which the magnetization vector rotates or reverses its direction. In tensor language the local magnetization direction varies continuously across the wall, so the susceptibility or permeability tensors are position-dependent inside the wall.

Saturation where once essentially all domains are aligned, cannot increase further.

Hysteresis loss where energy dissipated as heat each time the material traverses a loop (important for transformer efficiency).

For example, a permanent magnet is a ferromagnet whose domains have been aligned and “frozen” by the manufacturing process. A transformer core is a soft ferromagnet with narrow hysteresis loop so that little energy is wasted as heat during each AC cycle.

Exercise 6.48: Soft iron used in transformer cores has easily mobile domain walls and a narrow hysteresis loop where a hard permanent-magnet alloy has strongly pinned walls and a wide loop. Explain, in terms of domain-wall motion and energy dissipation, why the soft material is preferred for AC power devices while the hard material is preferred for permanent magnets. What role does wall width play in these differences?

Principle 6.6: Ferromagnetism shows that collective, cooperative behavior of microscopic moments can produce macroscopic effects far larger than the sum of isolated atoms. The tensor description remains useful where even in polycrystals an effective anisotropic appears when the material is under stress or partially magnetized.

In wave problems the frequency dependence of the ferromagnetic response leads to gyromagnetic effects and tensor permeability of the form

(6.104)

(in a coordinate system with a static bias field along z). This off-diagonal structure enables microwave devices that let waves pass in one direction but not the other.

Principle 6.7: Materials amplify or screen magnetic fields through microscopic moment alignment; the tensorial character captures anisotropy.  

Principle 6.8: Energy stored in a linear magnetic material is ; hysteresis adds dissipative losses.

Principle 6.9: Cooperative interactions in ferromagnets produce effects orders of magnitude stronger than simple paramagnetism.

Exercise 6.49: A long solenoid of n turns per meter carries free current I and is filled with a linear isotropic material of relative permeability . Write expressions for (inside) and (inside). At the solenoid–air interface, which field component is continuous and which may jump? Sketch the field lines and explain how the material concentrates flux.

Exercise 6.50: Domain walls, the Curie temperature, and the separation of from all arise from the same microscopic competition where quantum exchange that wants moments to align versus thermal energy and magnetostatic energy that prefer disorder or flux closure. Reflect on how this competition produces macroscopic phenomena—permanent magnets, transformers, and the very existence of magnetic domains—yet remains invisible in vacuum. How does the tensor description help you see the directional and history-dependent aspects of the same competition? What larger sense of wonder about the emergence of order from microscopic interactions does this section leave you with?

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.

I then define the fields as functions of space and time.

We take the curl of both sides of Faraday’s law.

We take the curl of both sides again.

Further Reading