Electromagnetic Energy, Poynting Theorem, and Conservation

Unit 2: Plane waves and electromagnetic energy

What the unit establishes

Derive propagation from Maxwell equations, identify phase, polarization and intrinsic impedance, and distinguish true wave attenuation from an arbitrary drawing that shrinks with distance.

Reproducible investigation

Select a uniform dielectric and construct E and H fields for a forward plane wave. Verify the orthogonality, amplitude ratio and average energy flux at five positions and two times. Change the dielectric constant while fixing source frequency: determine how wavelength, impedance and speed change, and which properties remain unchanged. Then create orthogonal electric components with an explicit relative phase to show linear, elliptical and circular trajectories. Finally add a good-conductor half-space only as a separate approximation, and explain why its exponentially decaying field cannot be represented by the same lossless sinusoid. Derive the formula for skin depth, note the required σ≫ωε regime, and compare field-amplitude decay with power-related decay. The final report should distinguish physical E and H units from normalized chart axes.

Four connected lessons

1. Deriving the electromagnetic wave equation

The starting point is the physical situation considered in the [full lesson](../lessons/U02-L01-deriving-the-electromagnetic-wave-equation.html). Its key mathematical relationship is the following, stated with the particular geometrical or material assumptions of this topic:

From ∇×E=−μ∂H/∂t and ∇×H=ε∂E/∂t, take curl of the first equation and use ∇×(∇×E)=∇(∇·E)−∇²E. With ρ=0 and constant ε, ∇·E=0, hence ∇²E−με∂²E/∂t²=0. A forward solution is E=E0 cos(ωt−βz), with β=ωsqrt(με) and phase speed v_p=ω/β.

Work through its numerical example on paper before using the model. Compare the predicted result with the situation described in the lesson's engineering case; check which assumption restricts the validity of the formula. The goal is not memorization of symbols but identifying whether a simplified scalar parameter is sufficient for this geometry and frequency.

2. Intrinsic impedance and Poynting energy flux

The starting point is the physical situation considered in the [full lesson](../lessons/U02-L02-intrinsic-impedance-and-poynting-energy-flux.html). Its key mathematical relationship is the following, stated with the particular geometrical or material assumptions of this topic:

For a real lossless η, H0=E0/η and ⟨S⟩=E0²/(2η) for peak-amplitude phasors. If RMS electric field is supplied instead, ⟨S⟩=E_rms²/η without the factor 1/2. Poynting's theorem relates −∂u/∂t−∇·S to J·E, where u=(εE²+μH²)/2 under nondispersive linear assumptions. Use a consistent convention for complex phasors and peak versus RMS.

Work through its numerical example on paper before using the model. Compare the predicted result with the situation described in the lesson's engineering case; check which assumption restricts the validity of the formula. The goal is not memorization of symbols but identifying whether a simplified scalar parameter is sufficient for this geometry and frequency.

3. Linear, circular and elliptical polarization

The starting point is the physical situation considered in the [full lesson](../lessons/U02-L03-linear-circular-and-elliptical-polarization.html). Its key mathematical relationship is the following, stated with the particular geometrical or material assumptions of this topic:

At a fixed z choose Ex=A cos(ωt−βz) and Ey=B cos(ωt−βz+δ). Linear occurs when δ=0 or π or one amplitude is zero. If A=B and δ=±π/2, the electric vector tip traces a circle. For ideal linear receiving polarization unit vectors p_r,p_t, the power coupling factor is |p_r·p_t|²; a 45° mismatch gives 1/2 or approximately −3.01 dB.

Work through its numerical example on paper before using the model. Compare the predicted result with the situation described in the lesson's engineering case; check which assumption restricts the validity of the formula. The goal is not memorization of symbols but identifying whether a simplified scalar parameter is sufficient for this geometry and frequency.

4. Conducting media and electromagnetic skin depth

The starting point is the physical situation considered in the [full lesson](../lessons/U02-L04-conducting-media-and-electromagnetic-skin-depth.html). Its key mathematical relationship is the following, stated with the particular geometrical or material assumptions of this topic:

Under e^{jωt} convention, complex propagation γ=α+jβ=sqrt(jωμ(σ+jωε)). In a good conductor σ≫ωε, α≈β≈sqrt(πfμσ), and skin depth δ=1/α≈sqrt(2/(ωμσ)). The field amplitude falls approximately as e^{−z/δ}; field power density therefore scales approximately with e^{−2z/δ}. Surface resistance Rs≈1/(σδ) for a good smooth conductor.

Work through its numerical example on paper before using the model. Compare the predicted result with the situation described in the lesson's engineering case; check which assumption restricts the validity of the formula. The goal is not memorization of symbols but identifying whether a simplified scalar parameter is sufficient for this geometry and frequency.

Unit test of understanding

Use the [eight solved questions](../assessments/unit-02.html) as a closed-book audit. Then complete the [unit investigation](../projects/unit-02.html) and compare its computational trend with the [unit interactive model](../simulations/model-02.html). Write down any mismatch between the ideal analytical limit and the plotted values; if an output appears to violate conservation or dimensional consistency, diagnose the implementation before adjusting a physical parameter to force agreement.

Evidence standard

A completed study record contains the source equations, declared time and orientation convention, parameter units, at least two numerical cases, a model-limitation statement and a reproducible test. Nothing in this unit requires live mains or high-voltage apparatus; synthetic numerical work is adequate for the computational project. A credible claim about real equipment would additionally require calibration and qualified verification appropriate to that application.

U02-L01: Deriving the electromagnetic wave equation

Book 22 · Unit 2 — Plane waves and electromagnetic energy Learning objective: Reconstruct the physical model, calculate one representative case and explain where its assumptions cease to apply.

Physical principles

In a source-free homogeneous isotropic medium, time-varying electric and magnetic fields sustain coupled propagation. Curling Faraday's law and inserting the Ampere-Maxwell law yields a second-order spatial equation. When material parameters are constant, the wave speed is 1/sqrt(με), showing why electromagnetic velocity depends on the medium. A plane-wave solution travels without a preferred amplitude scale in the ideal infinite, lossless environment; boundaries or sources set its actual amplitude. For engineering intuition, a change in time at one point cannot instantaneously establish a spatially separated field value. A single lumped capacitor cannot describe a cable once the signal wavelength becomes comparable to its length.

Derivation and governing equations

From ∇×E=−μ∂H/∂t and ∇×H=ε∂E/∂t, take curl of the first equation and use ∇×(∇×E)=∇(∇·E)−∇²E. With ρ=0 and constant ε, ∇·E=0, hence ∇²E−με∂²E/∂t²=0. A forward solution is E=E0 cos(ωt−βz), with β=ωsqrt(με) and phase speed v_p=ω/β.

Fully specified worked example

For ε_r=4 and μ_r≈1, v_p≈c/2=1.499e8 m/s. At 100 MHz the wavelength is λ=v/f≈1.50 m. At 1 GHz it becomes approximately 0.150 m. Changing frequency, not merely cable length, determines whether a distributed-wave representation becomes essential.

Independent numerical verification. Reconstruct the arithmetic in the preceding illustration using SI base units, then change one input by a factor of two and predict the consequence from the actual governing expression. Specify which inputs remain fixed. When the result involves a signed voltage or complex amplitude, report the chosen polarity or phase reference; a magnitude alone cannot settle the sign. Compare the result against an analytically known limit rather than blindly trusting significant figures from a calculator.

Engineering decision scenario

A design team treats a 10 cm cable as electrically short at a low control frequency. At 1 GHz in a dielectric of ε_r=4, it spans two-thirds of a wavelength and can introduce large phase shifts and reflections. Document the dielectric model and frequency when applying an electrical-length rule of thumb; a fixed geometry is not always lumped or always distributed.

Scope, failure modes and falsification

The derivation omits conductivity, frequency dependence and boundaries. It also presumes local linear material behavior. Strong dispersion requires β(ω), and a resonant or nonuniform structure can alter phase and energy propagation in ways this simple plane-wave equation does not capture.

What evidence would change this conclusion? Propose one measurement or full-wave simulation that would expose the limitation described above, name the raw observable, and determine whether it distinguishes an omitted physical effect from a code implementation error. Numerical convergence and physical validation are different claims. No idealized wave or electromagnetic enclosure model in this chapter constitutes a hardware safety approval.

Three thinking problems

1. Compute and check: Recalculate the central numerical result above with a clearly stated changed parameter. Show unit conversion and the primary scaling law, not just a final decimal.

2. Derive and compare: Starting with the main field law above, reconstruct the next relation using the assumptions explicitly stated in the lesson. Say which boundary condition or sign convention is indispensable.

3. Design critique: Apply the limitation above to the engineering scenario. Specify the experiment, frequency range, geometry or calibration that would need to be checked before an engineer could accept the simple calculation.

Continue: [Unit 2 overview](../units/unit-02.html) · [Assessment and answer key](../assessments/unit-02.html) · [Interactive model](../simulations/model-02.html).

U02-L02: Intrinsic impedance and Poynting energy flux

Book 22 · Unit 2 — Plane waves and electromagnetic energy Learning objective: Reconstruct the physical model, calculate one representative case and explain where its assumptions cease to apply.

Physical principles

In a lossless plane wave, E, H and propagation direction form an orthogonal right-handed triad. The ratio of field magnitudes is intrinsic impedance η=sqrt(μ/ε), distinct from the impedance of an arbitrary antenna terminal. The Poynting vector E×H carries electromagnetic energy-flow information; for an isolated forward sinusoidal wave in a lossless medium its average is positive along propagation. Energy density alternates between electric and magnetic contributions, but the two terms are equal for the ideal traveling plane wave. Reflections produce standing waves where local instantaneous power flow may alternate signs, so peak field alone is not sufficient to infer net delivered power.

Derivation and governing equations

For a real lossless η, H0=E0/η and ⟨S⟩=E0²/(2η) for peak-amplitude phasors. If RMS electric field is supplied instead, ⟨S⟩=E_rms²/η without the factor 1/2. Poynting's theorem relates −∂u/∂t−∇·S to J·E, where u=(εE²+μH²)/2 under nondispersive linear assumptions. Use a consistent convention for complex phasors and peak versus RMS.

Fully specified worked example

In vacuum η0≈376.73 Ω. An ideal forward wave with E_peak=10 V/m has H_peak≈26.5 mA/m and average power flux≈100/(2×376.73)=0.133 W/m². Using E_rms=10 V/m instead would double that numerical flux. The difference is a definition of the supplied amplitude, not a different physical law.

Independent numerical verification. Reconstruct the arithmetic in the preceding illustration using SI base units, then change one input by a factor of two and predict the consequence from the actual governing expression. Specify which inputs remain fixed. When the result involves a signed voltage or complex amplitude, report the chosen polarity or phase reference; a magnitude alone cannot settle the sign. Compare the result against an analytically known limit rather than blindly trusting significant figures from a calculator.

Engineering decision scenario

A radiated-field survey may report RMS E while a numerical package plots peak E. Before comparing an exposure level, instrument reading and theoretical signal, normalize the averaging time, polarization and field amplitude convention. A plane-wave power conversion is unsuitable very close to a small antenna, where reactive near fields are significant.

Scope, failure modes and falsification

In lossy or strongly dispersive media, energy density and impedance require care, and E and H can differ in phase. Near fields, interference and evanescent modes can store substantial energy without delivering corresponding net outward power. Do not interpret a static arrow plot of S as a universal statement about time-averaged radiation.

What evidence would change this conclusion? Propose one measurement or full-wave simulation that would expose the limitation described above, name the raw observable, and determine whether it distinguishes an omitted physical effect from a code implementation error. Numerical convergence and physical validation are different claims. No idealized wave or electromagnetic enclosure model in this chapter constitutes a hardware safety approval.

Three thinking problems

1. Compute and check: Recalculate the central numerical result above with a clearly stated changed parameter. Show unit conversion and the primary scaling law, not just a final decimal.

2. Derive and compare: Starting with the main field law above, reconstruct the next relation using the assumptions explicitly stated in the lesson. Say which boundary condition or sign convention is indispensable.

3. Design critique: Apply the limitation above to the engineering scenario. Specify the experiment, frequency range, geometry or calibration that would need to be checked before an engineer could accept the simple calculation.

Continue: [Unit 2 overview](../units/unit-02.html) · [Assessment and answer key](../assessments/unit-02.html) · [Interactive model](../simulations/model-02.html).

U02-L03: Linear, circular and elliptical polarization

Book 22 · Unit 2 — Plane waves and electromagnetic energy Learning objective: Reconstruct the physical model, calculate one representative case and explain where its assumptions cease to apply.

Physical principles

Polarization specifies the evolution of the electric-field tip at a fixed spatial position as time passes. A single harmonic component along one transverse axis gives linear polarization. Equal orthogonal components in quadrature give circular polarization, with handedness determined by the viewing convention. General unequal amplitudes and phase offsets produce an ellipse. Matching the intended polarization matters for receiving antennas, polarizing optics and communication links: an ideal orthogonally polarized receiver can reject a desired linear signal despite adequate total field intensity. Polarization is not identical to the ray's direction of travel, and phase differences should always be reported with the sign convention used.

Derivation and governing equations

At a fixed z choose Ex=A cos(ωt−βz) and Ey=B cos(ωt−βz+δ). Linear occurs when δ=0 or π or one amplitude is zero. If A=B and δ=±π/2, the electric vector tip traces a circle. For ideal linear receiving polarization unit vectors p_r,p_t, the power coupling factor is |p_r·p_t|²; a 45° mismatch gives 1/2 or approximately −3.01 dB.

Fully specified worked example

For a transmitter linearly polarized at 0° and a receiving dipole rotated 30°, ideal polarization power transfer is cos²30°=0.75, a −1.25 dB penalty. At 90° the ideal transfer is zero, although real antennas commonly have finite cross-polar response. Circular-to-linear ideal coupling gives a −3 dB power factor under matched propagation assumptions.

Independent numerical verification. Reconstruct the arithmetic in the preceding illustration using SI base units, then change one input by a factor of two and predict the consequence from the actual governing expression. Specify which inputs remain fixed. When the result involves a signed voltage or complex amplitude, report the chosen polarity or phase reference; a magnitude alone cannot settle the sign. Compare the result against an analytically known limit rather than blindly trusting significant figures from a calculator.

Engineering decision scenario

Consider an indoor telemetry link with a device that may rotate unpredictably. The field strength at the receiver can remain almost unchanged while an ideal single-axis antenna experiences a polarization loss. A diversity receiver or carefully selected circular polarization may reduce orientation sensitivity, but component cost, antenna pattern and multipath response still matter.

Scope, failure modes and falsification

Definitions of clockwise/counterclockwise circular polarization differ across disciplines depending on viewing direction. Lossy anisotropic media can alter polarization along a path. Reflections and multipath may create elliptical states even when the transmitted field is nominally linear; a simplistic cosine mismatch assumes a uniform plane wave.

What evidence would change this conclusion? Propose one measurement or full-wave simulation that would expose the limitation described above, name the raw observable, and determine whether it distinguishes an omitted physical effect from a code implementation error. Numerical convergence and physical validation are different claims. No idealized wave or electromagnetic enclosure model in this chapter constitutes a hardware safety approval.

Three thinking problems

1. Compute and check: Recalculate the central numerical result above with a clearly stated changed parameter. Show unit conversion and the primary scaling law, not just a final decimal.

2. Derive and compare: Starting with the main field law above, reconstruct the next relation using the assumptions explicitly stated in the lesson. Say which boundary condition or sign convention is indispensable.

3. Design critique: Apply the limitation above to the engineering scenario. Specify the experiment, frequency range, geometry or calibration that would need to be checked before an engineer could accept the simple calculation.

Continue: [Unit 2 overview](../units/unit-02.html) · [Assessment and answer key](../assessments/unit-02.html) · [Interactive model](../simulations/model-02.html).

U02-L04: Conducting media and electromagnetic skin depth

Book 22 · Unit 2 — Plane waves and electromagnetic energy Learning objective: Reconstruct the physical model, calculate one representative case and explain where its assumptions cease to apply.

Physical principles

Electric fields in conductors drive current density, and Ohmic loss converts electromagnetic energy into heat. A harmonic wave entering a good conductor decays with depth while its phase changes. The skin effect concentrates alternating current near conductor surfaces at sufficiently high frequencies. This alters resistance, current distribution and electromagnetic shielding, but the familiar skin-depth formula depends on good-conductor and homogeneous-medium approximations. At low frequency or in poor conductors, the full complex propagation constant matters. Skin depth is an amplitude decay distance, not a fixed physical boundary beyond which the field becomes exactly zero.

Derivation and governing equations

Under e^{jωt} convention, complex propagation γ=α+jβ=sqrt(jωμ(σ+jωε)). In a good conductor σ≫ωε, α≈β≈sqrt(πfμσ), and skin depth δ=1/α≈sqrt(2/(ωμσ)). The field amplitude falls approximately as e^{−z/δ}; field power density therefore scales approximately with e^{−2z/δ}. Surface resistance Rs≈1/(σδ) for a good smooth conductor.

Fully specified worked example

Take copper-like σ=5.8×10^7 S/m, μ≈μ0 and f=1 MHz. δ≈66 μm. One skin depth leaves field amplitude at approximately 36.8%, but power proportional to its squared amplitude falls to 13.5%. This is not an exact reflection or shielding result, because surface boundary matching also determines the amplitude at the interface.

Independent numerical verification. Reconstruct the arithmetic in the preceding illustration using SI base units, then change one input by a factor of two and predict the consequence from the actual governing expression. Specify which inputs remain fixed. When the result involves a signed voltage or complex amplitude, report the chosen polarity or phase reference; a magnitude alone cannot settle the sign. Compare the result against an analytically known limit rather than blindly trusting significant figures from a calculator.

Engineering decision scenario

A PCB trace may conduct nearly uniformly at DC and have surface-concentrated current at radio frequencies. A designer checking only DC resistance underestimates RF losses. Thickness, proximity to neighboring traces and plating roughness affect real losses; changing a conductor material may alter both electrical and thermal behavior.

Scope, failure modes and falsification

The formula assumes an infinite smooth homogeneous conductor and a sinusoidal steady-state excitation. It omits edge crowding, proximity effects, finite thickness, anisotropy and nonlinear magnetic permeability. Skin-depth arguments alone cannot quantify shielding effectiveness of an enclosure with seams or apertures.

What evidence would change this conclusion? Propose one measurement or full-wave simulation that would expose the limitation described above, name the raw observable, and determine whether it distinguishes an omitted physical effect from a code implementation error. Numerical convergence and physical validation are different claims. No idealized wave or electromagnetic enclosure model in this chapter constitutes a hardware safety approval.

Three thinking problems

1. Compute and check: Recalculate the central numerical result above with a clearly stated changed parameter. Show unit conversion and the primary scaling law, not just a final decimal.

2. Derive and compare: Starting with the main field law above, reconstruct the next relation using the assumptions explicitly stated in the lesson. Say which boundary condition or sign convention is indispensable.

3. Design critique: Apply the limitation above to the engineering scenario. Specify the experiment, frequency range, geometry or calibration that would need to be checked before an engineer could accept the simple calculation.

Continue: [Unit 2 overview](../units/unit-02.html) · [Assessment and answer key](../assessments/unit-02.html) · [Interactive model](../simulations/model-02.html).

Unit 2 engineering project: Plane-wave propagation and polarization lab

Type: Computational desk investigation using synthetic data only. No mains wiring or high-voltage hardware work.

The problem and experimental model

Build a forward wave in two dielectric materials at five frequencies. Compute λ, η, peak H, average S and a phase-distance chart. Then combine orthogonal field components to display linear and circular polarization with declared viewing convention. Verify E/H consistency and phase velocity from displacement of the plotted wave.

Required evidence

Deliver two independent checks for each medium, a labeled phasor convention, and a paragraph explaining why these plots are not measured waveguide fields.

Independent benchmark and failure test

Before generating graphs, derive a baseline from the four unit lessons. Establish a numerical prediction from known dimensions, wave speed or material law. Run a second numerical calculation that uses a different formulation or a limiting case. Inspect whether the model conserves charge or power where the assumptions require it; use the proper amplitude or power reference. Compare input perturbations one at a time so the direction of the observed effect can be explained physically.

Analysis requirements

Every plot must label input units, dependent quantities, coordinate reference and whether the curve is synthetic. A statement such as 'the signal became better' is incomplete without specifying the chosen metric and the control variables. Include a short error budget separating numerical resolution, material uncertainty, geometrical approximation and measurement effects that would matter in a real experiment. An unexplained mismatch must remain visible in the report rather than being edited out.

Deliverables and grading

A concise technical report should include: purpose and assumptions (15%), derivation and units (20%), numerical verification (25%), parameter sensitivity and omitted physics (25%), and evidence-based engineering recommendation (15%). Name one specific datum whose collection would change the recommendation. Do not claim that a synthetic dataset confirms real equipment performance.

References inside the book: [Unit guide](../units/unit-02.html) · [Analytical model](../simulations/model-02.html) · [Assessment](../assessments/unit-02.html).

Unit 2 assessment: Plane waves and electromagnetic energy

Eight subject-specific questions, with explanatory answers. Attempt all questions before reading the key. These are educational examples, not external exam past papers.

Question 1

If εr doubles in a μr=1 lossless medium, does wave speed halve?

Answer and rationale: No, speed scales as 1/sqrt(εr), so it decreases by 1/sqrt(2).

Question 2

A wave at 200 MHz travels at 2e8 m/s. Determine wavelength.

Answer and rationale: λ=v/f=1 m.

Question 3

Why is η not the same as an antenna input impedance?

Answer and rationale: η is the local E/H wave ratio in a homogeneous traveling wave; antenna input impedance relates feed voltage and current with stored energy, radiation and loss.

Question 4

A 20 V/m peak plane wave in vacuum has roughly what average power density?

Answer and rationale: 20²/(2×376.73)≈0.531 W/m².

Question 5

What polarization occurs for equal orthogonal sinusoidal E components with a 90° phase offset?

Answer and rationale: Circular polarization in the ideal plane-wave model; the sense depends on time/propagation conventions.

Question 6

How much ideal power penalty is caused by a 60° linear polarization mismatch?

Answer and rationale: cos²60°=1/4, corresponding to about −6.02 dB.

Question 7

In a good conductor, why does power typically decrease faster with depth than field amplitude?

Answer and rationale: Field amplitude decays as exp(−z/δ), while quadratic power-related terms scale as exp(−2z/δ).

Question 8

What limitation makes a skin-depth estimate inadequate for a slotted shield?

Answer and rationale: Aperture leakage and boundary matching can dominate; bulk conductor attenuation alone does not give enclosure performance.

Beyond the answer key

For at least two numerical problems, write a second route to the result using a limiting case or power/charge conservation argument. For conceptual questions, identify a counterexample to an overgeneralized version of the answer. A problem is not fully understood until the applicable physical assumptions are stated alongside the formula.