Related Concepts: 1.04 Media Properties & Conductor-Insulator Behaviour

5.02 Plane Waves in Lossless vs. Lossy Media & Skin Depth Calculations

Core Idea

Whether a material behaves as a conductor or as a dielectric is not a fixed property of the material — it depends on the frequency. One dimensionless number, the loss tangent $\sigma /\omega ϵ$, decides it. Get that number first and every other quantity ($\alpha$, $\beta$, $\eta$, $u_p$, $\delta$) follows from the appropriate approximation.

The single most important habit in this chapter

Before you write anything else in a plane-wave numerical, compute $\frac{\sigma }{\omega ϵ}$. If it is $\gg 1$ you are in a good conductor; if $\ll 1$ you are in a low-loss dielectric. Using the wrong formula set is the most expensive mistake available in Chapter 5, and it is entirely avoidable.

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1. Core Terminology

Uniform Plane Wave

A wave whose $\vec{E}$ and $\vec{H}$ vectors lie entirely in planes perpendicular to the direction of propagation, and which have the same magnitude, direction and phase everywhere on any such plane. Because $\vec{E}$ and $\vec{H}$ are both transverse, it is also called a TEM (Transverse Electromagnetic) wave.

Complex Permittivity ( ${ϵ}_c$) [PYQ: 2019, 2023]

A single complex quantity that folds a medium’s energy storage and energy loss into one number: ${ϵ}_c={ϵ}^{\prime }-j{ϵ}^{\prime \prime }=ϵ-j\frac{\sigma }{\omega }$ The real part ${ϵ}^{\prime }$ represents the capacitive (reversible) polarisation energy stored in the dielectric. The imaginary part ${ϵ}^{\prime \prime }$ represents the ohmic (irreversible) energy dissipated as heat. It lets us reuse every lossless formula by simply replacing $ϵ\to {ϵ}_c$.

Complex Permeability ( ${\mu }_c$) Represents the magnetic counterpart of complex permittivity under high-frequency time-varying fields, taking domain-wall damping, eddy currents, and hysteresis losses into account: ${\mu }_c={\mu }^{\prime }-j{\mu }^{\prime \prime }$

• Real Part (${\mu }^{\prime }$): Measures the magnetic energy storage capacity (ideal permeability, representing domain alignment).
• Imaginary Part (${\mu }^{\prime \prime }$): Measures the magnetic power loss (representing irreversible dissipation due to hysteresis and damping). It lets us model magnetic core losses in waves propagating through ferrite or steel media.

Loss Tangent ( $\tan {\delta }_c$) [PYQ: 2015, 2019]

The ratio of the conduction current density to the displacement current density in the medium: $\tan {\delta }_c=\frac{{ϵ}^{\prime \prime }}{{ϵ}^{\prime }}=\frac{|\vec{J}|}{|{\vec{J}}_d|}=\frac{|\sigma \vec{E}|}{|j\omega ϵ\vec{E}|}=\frac{\sigma }{\omega ϵ}$ It measures how “lossy” a material is at a given frequency. The angle ${\delta }_c$ is the loss angle, the phase by which the total current lags the ideal 90° displacement current.

Intrinsic Impedance ( $\eta$) [PYQ: 2022, 2023]

The ratio of the electric field amplitude to the magnetic field amplitude of a uniform plane wave in that medium: $\eta =\frac{E}{H}=\sqrt{\frac{\mu }{ϵ}}[\Omega ],{\eta }_0=\sqrt{\frac{{\mu }_0}{{ϵ}_0}}=120\pi \approx 377\Omega \text{ (free space)}$ It is the electromagnetic analogue of characteristic impedance on a transmission line: it tells you how much magnetic field a given electric field drags along with it, and mismatches in $\eta$ at an interface are what cause reflection (see 5.03 Poynting Vector, Power Flow & Normal Incidence Reflection (Gamma, Tau)).

[FIGURE: Phasor diagram in the complex plane showing displacement current density jωεE on the imaginary axis, conduction current density σE on the real axis, their resultant, and the loss angle δ_c between the resultant and the imaginary axis. — source: Sadiku 7th Ed., Fig. 10.5 / lecture slide L 13.pdf]

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2. Classifying a Medium

[PYQ: 2015, 2017, 2020, 2023]

Property	Low-Loss Dielectric (Good Insulator)	Good Conductor
Mathematical condition	$\sigma \ll \omega ϵ\Rightarrow \tan {\delta }_c\ll 1$	$\sigma \gg \omega ϵ\Rightarrow \tan {\delta }_c\gg 1$
Dominant current	Displacement current (${\vec{J}}_d\gg \vec{J}$)	Conduction current ($\vec{J}\gg {\vec{J}}_d$)
Energy behaviour	Mostly stored and returned (reactive)	Mostly dissipated as ohmic heat
Attenuation	Very low — wave travels far	Extremely high — confined to a thin skin
Field penetration	Deep	Skin depth $\delta$ only
$\eta$ is	Real (E and H in phase)	Complex at $45°$ (H lags E by $45°$)
Examples	Glass, PTFE, dry air, polyethylene	Copper, silver, seawater below ~10 MHz

"The same medium can act as a good conductor or a good insulator" [PYQ: 2020 — 10 Marks]

This is a frequency question, not a material question. Because $\tan {\delta }_c=\sigma /\omega ϵ$ has $\omega$ in the denominator:

• At low frequency, $\omega ϵ$ is small, so $\sigma /\omega ϵ$ is large → the medium behaves as a good conductor.
• At high frequency, $\omega ϵ$ becomes large, so $\sigma /\omega ϵ$ falls → the same medium behaves as a good insulator/dielectric.

Worked example: seawater ($\sigma =4$ S/m, ${ϵ}_r=72$). At 50 kHz, $\sigma /\omega ϵ\approx 200\gg 1$ — a good conductor, which is why submarine radio uses very low frequencies. At 10 GHz the same seawater has $\sigma /\omega ϵ\approx 10^{-1}$ — it now behaves as a lossy dielectric.

The boundary condition separating the two regimes is $\sigma =\omega ϵ$, i.e. the transition frequency $f_t=\frac{\sigma }{2\pi ϵ}$.

Characteristics of a lossy dielectric [PYQ: 2015]

A lossy dielectric is a partially conducting medium ($\sigma \ne 0$ but small) in which a wave loses amplitude as it travels. Its defining features: $\gamma$ is complex so the wave attenuates as $e^{-\alpha z}$; $\eta$ is complex so $\vec{H}$ lags $\vec{E}$ by a small angle; the permittivity is complex, ${ϵ}_c=ϵ-j\sigma /\omega$; and power is continuously drained from the wave as ohmic heat at the rate $\frac{1}{2}\sigma E_0^2$ per unit volume.

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3. The Complex Propagation Constant $\gamma =\alpha +j\beta$

In a source-free lossy medium, the phasor Helmholtz equation becomes:

${\nabla }^2{\vec{E}}_s-{\gamma }^2{\vec{E}}_s=0,\gamma =j\omega \sqrt{\mu {ϵ}_c}=j\omega \sqrt{\mu ϵ\left(1-j\frac{\sigma }{\omega ϵ}\right)}=\alpha +j\beta$

A wave travelling in $+z$ therefore behaves as:

$\vec{E}(z,t)={\hat{a}}_x{E}_0\underset{\text{decay}}{\underbrace{e^{-\alpha z}}}\underset{\text{oscillation}}{\underbrace{\cos (\omega t-\beta z)}}$

The two constants

• $\alpha$ — Attenuation constant $[\text{Np/m}]$: the rate of exponential amplitude decay per metre.
• $\beta$ — Phase constant $[\text{rad/m}]$: the phase shift per metre travelled.

3.1 Exact general expressions

Square $\gamma$ and equate real and imaginary parts of $(\alpha +j\beta {)}^2=-{\omega }^2\mu ϵ+j\omega \mu \sigma$:

$\alpha =\omega \sqrt{\frac{\mu ϵ}{2}\left[\sqrt{1+{\left(\frac{\sigma }{\omega ϵ}\right)}^2}-1\right]}[\text{Np/m}]$

$\beta =\omega \sqrt{\frac{\mu ϵ}{2}\left[\sqrt{1+{\left(\frac{\sigma }{\omega ϵ}\right)}^2}+1\right]}[\text{rad/m}]$

These exact forms are almost never needed in the exam

Every PYQ numerical falls cleanly into one of the two limiting cases below. Quote the exact expressions to show you know where the approximations come from, then use the approximation.

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4. Case A — Low-Loss Dielectric ($\sigma \ll \omega ϵ$)

[PYQ: 2019, 2025]

Derivation by binomial expansion. With $x=j\sigma /\omega ϵ$ small, use $(1-x{)}^{1/2}\approx 1-\frac{1}{2}x+\frac{1}{8}{x}^2$… :

$\gamma =j\omega \sqrt{\mu ϵ}{\left(1-j\frac{\sigma }{\omega ϵ}\right)}^{1/2}\approx j\omega \sqrt{\mu ϵ}\left[1-j\frac{\sigma }{2\omega ϵ}+\frac{1}{8}{\left(\frac{\sigma }{\omega ϵ}\right)}^2\right]$

$\gamma \approx \underset{\alpha }{\underbrace{\frac{\sigma }{2}\sqrt{\frac{\mu }{ϵ}}}}+j\underset{\beta }{\underbrace{\omega \sqrt{\mu ϵ}\left[1+\frac{1}{8}{\left(\frac{\sigma }{\omega ϵ}\right)}^2\right]}}$

Low-Loss Dielectric Formula Set

$\alpha \approx \frac{\sigma }{2}\sqrt{\frac{\mu }{ϵ}}[\text{Np/m}]\beta \approx \omega \sqrt{\mu ϵ}\left[1+\frac{1}{8}{\left(\frac{\sigma }{\omega ϵ}\right)}^2\right][\text{rad/m}]$ ${\eta }_c\approx \sqrt{\frac{\mu }{ϵ}}\left(1+j\frac{\sigma }{2\omega ϵ}\right)[\Omega ]u_p=\frac{\omega }{\beta }\approx \frac{1}{\sqrt{\mu ϵ}}\left[1-\frac{1}{8}{\left(\frac{\sigma }{\omega ϵ}\right)}^2\right][\text{m/s}]$

Practical shortcut for numericals

Since $\sigma =\omega ϵ\tan \delta$, the attenuation constant can be written directly in terms of the given loss tangent: $\alpha \approx \frac{\omega \tan \delta }{2}\sqrt{\mu ϵ}=\frac{\pi f\tan \delta \sqrt{{ϵ}_r}}{c}$ This saves you from having to compute $\sigma$ at all when the question gives you $\tan \delta$ directly.

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5. Case B — Good Conductor ($\sigma \gg \omega ϵ$)

[PYQ: 2019, 2025]

Since $\sigma /\omega ϵ\gg 1$, drop the $1$ inside the bracket:

$\gamma =j\omega \sqrt{\mu \left(-j\frac{\sigma }{\omega }\right)}=\sqrt{j\omega \mu \sigma }$

Use $\sqrt{j}=e^{j\pi /4}=\frac{1+j}{\sqrt{2}}$:

$\gamma =(1+j)\sqrt{\frac{\omega \mu \sigma }{2}}=(1+j)\sqrt{\pi f\mu \sigma }$

Good Conductor Formula Set

$\boxed{\alpha =\beta =\sqrt{\frac{\omega \mu \sigma }{2}}=\sqrt{\pi f\mu \sigma }}$ ${\eta }_c=(1+j)\sqrt{\frac{\omega \mu }{2\sigma }}=\sqrt{\frac{\omega \mu }{\sigma }}\angle 45°[\Omega ]u_p=\frac{\omega }{\beta }=\sqrt{\frac{2\omega }{\mu \sigma }}[\text{m/s}]$

Key Exam Checkpoint — the 45° phase lag

In a good conductor ${\eta }_c$ has a phase angle of exactly $+45°$, which means the magnetic field lags the electric field by 45°. When you convert $\vec{H}$ back to the time domain you must subtract $\pi /4$ from the cosine argument: $\vec{H}(z,t)={\hat{a}}_y\frac{E_0}{|{\eta }_c|}{e}^{-\alpha z}\cos \left(\omega t-\beta z-\frac{\pi }{4}\right)$ Omitting this is one of the most commonly penalised errors in the seawater numerical.

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6. Master Comparison Table

[PYQ: 2019, 2025 — “Determine (i) attenuation constant (ii) phase constant (iii) intrinsic impedance and (iv) phase velocity for both low-loss dielectrics and good conductor”]

Parameter	Lossless Dielectric ($\sigma =0$)	Low-Loss Dielectric ($\sigma \ll \omega ϵ$)	Good Conductor ($\sigma \gg \omega ϵ$)
Attenuation $\alpha$	$0$	$\frac{\sigma }{2}\sqrt{\frac{\mu }{ϵ}}$	$\sqrt{\pi f\mu \sigma }$
Phase constant $\beta$	$\omega \sqrt{\mu ϵ}$	$\omega \sqrt{\mu ϵ}\left[1+\frac{1}{8}{\left(\frac{\sigma }{\omega ϵ}\right)}^2\right]$	$\sqrt{\pi f\mu \sigma }$
Intrinsic impedance $\eta$	$\sqrt{\frac{\mu }{ϵ}}$ (real)	$\sqrt{\frac{\mu }{ϵ}}\left(1+j\frac{\sigma }{2\omega ϵ}\right)$	$\sqrt{\frac{\omega \mu }{\sigma }}\angle 45°$
Phase velocity $u_p=\omega /\beta$	$\frac{1}{\sqrt{\mu ϵ}}$	$\approx \frac{1}{\sqrt{\mu ϵ}}$	$\sqrt{\frac{2\omega }{\mu \sigma }}$
Wavelength $\lambda =2\pi /\beta$	$\frac{2\pi }{\omega \sqrt{\mu ϵ}}$	$\approx \frac{2\pi }{\omega \sqrt{\mu ϵ}}$	$2\pi \delta$
Skin depth $\delta =1/\alpha$	$\infty$	$\frac{2}{\sigma }\sqrt{\frac{ϵ}{\mu }}$	$\frac{1}{\sqrt{\pi f\mu \sigma }}$
$\vec{H}$ vs $\vec{E}$ phase	In phase	Lags slightly	Lags by $45°$

Sanity checks you can apply instantly

• In a good conductor $\alpha =\beta$ always. If your two numbers differ, you made an arithmetic slip.
• $\lambda =2\pi \delta$ in a good conductor — the wave dies within one wavelength.
• $u_p$ in a conductor depends on $\sqrt{\omega }$, so a conductor is inherently dispersive (see 5.05 Dispersion, Phase & Group Velocity, Doppler Effect & Brewster’s Angle).

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7. Skin Depth ($\delta$)

[PYQ: 2015, 2016, 2021, 2025] for the concept; [PYQ: 2018, 2020, 2024] for the numericals.

Definition

Skin depth ($\delta$), also called depth of penetration, is the distance a wave must travel into a conducting medium for its field amplitude to fall to $1/e$ (≈ 36.8%) of its value at the surface.

Derivation. The field decays as $E(z)=E_0{e}^{-\alpha z}$. Set $E(\delta )=E_0{e}^{-1}$:

$e^{-\alpha \delta }=e^{-1}⟹\alpha \delta =1⟹\delta =\frac{1}{\alpha }$

For a good conductor, substitute $\alpha =\sqrt{\pi f\mu \sigma }$:

Key Result — Skin Depth

$\boxed{\delta =\frac{1}{\alpha }=\frac{1}{\sqrt{\pi f\mu \sigma }}=\sqrt{\frac{2}{\omega \mu \sigma }}}[\text{m}]$

[GRAPH: Field amplitude E(z) = E₀e^(−αz) versus depth z into a conductor, showing the curve crossing 0.368E₀ at exactly z = δ. Governing equation: E(z)/E₀ = e^(−z/δ). — source: L 13.pdf, Slide 8 / Sadiku Fig. 10.6]

Physical consequences

• $\delta \propto 1/\sqrt{f}$ — the higher the frequency, the thinner the conducting skin. At 50 Hz, copper’s skin depth is ~9 mm; at 1 GHz it is ~2 µm.
• The skin effect — at high frequency, AC current abandons the interior of a conductor and crowds into a thin surface layer, so the effective cross-section shrinks and the AC resistance rises far above the DC value.
• Engineering consequences: hollow/tubular conductors work as well as solid ones at RF; Litz wire subdivides a conductor into many insulated strands to fight the skin effect; RF shielding cans need only be a few skin depths thick; submarine communication must use ELF because seawater’s skin depth at HF is centimetres.

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8. Average Power Dissipated in a Lossy Medium

[PYQ: 2017, 2022, 2023, 2024] — supports the recurring numerical.

For a sinusoidal field of amplitude $E_0$ in a medium of conductivity $\sigma$, the time-average ohmic power dissipated per unit volume is:

$\boxed{P_{av}=\frac{1}{2}\sigma E_0^2}[{\text{W/m}}^3]$

with $\sigma$ obtained from the loss tangent when it is not given directly:

$\sigma =\omega ϵ\tan \delta =2\pi f{ϵ}_r{ϵ}_0\tan \delta$

The factor of $\frac{1}{2}$

It comes from $⟨{\cos }^2\omega t⟩=\frac{1}{2}$. Only omit it if the question gives an RMS field rather than an amplitude. Full worked solution in 5.07 Solved PYQ Numerical Bank - Waves & Propagation.

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9. Common Mistakes That Cost Marks

Avoid these

1. Not computing $\sigma /\omega ϵ$ first. Everything downstream depends on which regime you are in.
2. Using $ϵ={ϵ}_r$ instead of $ϵ={ϵ}_r{ϵ}_0$. Multiply by $8.854\times 10^{-12}$ every time.
3. Forgetting the $45°$ lag when writing $\vec{H}$ in a good conductor.
4. Confusing Np/m with dB/m. $1$ Np $=8.686$ dB. Attenuation constants are in nepers unless stated otherwise.
5. Writing $\delta =1/\beta$. Skin depth is the reciprocal of the attenuation constant $\alpha$. (They happen to be equal in a good conductor, but the definition is $1/\alpha$.)
6. Dropping unit vectors on $\vec{E}$ and $\vec{H}$ in the answer expressions.

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10. PYQ Bank — Verbatim Questions & Answer Plans

Q1 — Full parameter set for both media [PYQ: 2019 — 12 Marks]

“Determine (i) attenuation constant (ii) phase constant (iii) intrinsic impedance and (iv) phase velocity for both low-loss dielectrics and good conductor.”

Answer plan: Start from $\gamma =j\omega \sqrt{\mu {ϵ}_c}$ (§3), then do the binomial expansion for the low-loss case (§4) and the drop-the-1 simplification for the conductor (§5). Present all eight results in the §6 comparison table. The derivations, not just the results, carry the marks.

(Tagging note: the 2025 version of this question asks only for parts (i) and (ii) — see Q2. Your corrupted pyq.md correctly flags that the earlier categorisation truncated the 2019 question to force a merge.)

Q2 — Attenuation and phase constants only [PYQ: 2025 — 13 Marks]

“Determine: (i) attenuation constant and (ii) phase constant for both low-loss dielectrics and good conductor.”

Answer plan: As Q1 but restricted to $\alpha$ and $\beta$. With 13 marks for only two quantities, the expected depth is the full binomial derivation for the dielectric and the full $\sqrt{j}$ manipulation for the conductor.

Q3 — Explain low-loss dielectrics and good conductors [PYQ: 2017, 2023 — 09/12 Marks]

“Explain low loss dielectrics and good conductors with your knowledge of electromagnetics.”

Answer plan: §1 (loss tangent as the current ratio) → §2 classification table → the physical picture of what dominates in each → the §6 parameter comparison. Conclude with the frequency-dependence point from §2.

Q4 — Behaviour under external fields [PYQ: 2023 — 09 Marks] (flagged as completely missing in corrupted pyq.md)

“Explain low loss dielectrics and good conductors. Explain their behavior under the impact of external fields.”

Answer plan: Q3 content, then add the behaviour half: in a low-loss dielectric the external field induces bound-charge polarisation, energy is stored and largely returned, and the wave penetrates deeply with slight attenuation. In a good conductor free carriers respond almost instantly, set up a conduction current that opposes and cancels the interior field, dissipate the energy as heat, and confine the field to one skin depth.

Q5 — Loss tangent + three characteristics [PYQ: 2015 — 12 Marks]

“What do you mean by loss tangent? Discuss the characteristics of i) good conductor, ii) lossy dielectric and iii) skin depth.”

Answer plan: Loss tangent definition from §1 (with the current-ratio derivation) → good conductor characteristics from §2/§5 → lossy dielectric callout in §2 → skin depth definition, formula, $1/\sqrt{f}$ behaviour and the skin effect from §7.

Q6 — Skin depth as a short-note term [PYQ: 2016, 2021, 2025 — part of 09/12 Marks]

“Briefly explain the following terms: i) Vector magnetic potential, ii) Magnetic Susceptibility iii) Skin depth.” (2025) / “Explain the following terms: (i) Doppler effect, (ii) Skin depth and (iii) Plasma frequency.” (2016, 2021)

Answer plan: For the skin depth part: definition ($1/e$ decay), formula $\delta =1/\sqrt{\pi f\mu \sigma }$, the $\delta \propto 1/\sqrt{f}$ consequence, and one engineering example. 4–5 lines. The other parts route to 3.03 Ampere’s Circuital Law & Vector Magnetic Potential (A), 3.04 Magnetization, Magnetic Materials, Boundary Conditions & Hall Effect, 5.05 Dispersion, Phase & Group Velocity, Doppler Effect & Brewster’s Angle and 5.06 Radio Wave Propagation Modes & Ionospheric Effects.

Q7 — Conductor/insulator conditions [PYQ: 2020 — 10 Marks]

“Write short description on the conditions when a same medium can be act as a good conductor or a good insulator.”

Answer plan: The §2 info callout in full — the loss tangent criterion, the frequency dependence, the transition frequency $f_t=\sigma /2\pi ϵ$, and the seawater worked example.

Q8 — Short-note terms: loss tangent, complex permittivity, intrinsic impedance [PYQ: 2019, 2022, 2023]

“Write short notes on (i) loss tangent (ii) homogenous medium (iii) complex permittivity.” (2019) / “Briefly discuss the following terms: i) Intrinsic impedance, ii) Complex permittivity, iii) Displacement current density.” (2023) / “Briefly explain the following terms: i) Intrinsic impedance ii) Virtual height.” (2022)

Answer plan: Use the §1 abstract callouts verbatim for loss tangent, complex permittivity and intrinsic impedance. “Homogeneous medium” → 1.04 Media Properties & Conductor-Insulator Behaviour; “displacement current density” → 4.02 Faraday’s Law of Induction & Maxwell’s Displacement Current; “virtual height” → 5.06 Radio Wave Propagation Modes & Ionospheric Effects.

Q9 — Numericals using this note

• Seawater parameter set (α, β, η, u_p, λ, δ, and 1% decay distance) [PYQ: 2018, 2024 — 14/15 Marks]
• Narrow-band signal in a lossy dielectric (α, β, u_p, u_g, dispersion) [PYQ: 2017, 2019, 2021 — 10/11/12 Marks]
• Average power dissipated per cubic metre [PYQ: 2017, 2022, 2023, 2024 — 08/09/10 Marks]
• Graphite skin depth → conductivity → 30 dB distance [PYQ: 2020 — 10 Marks]

All four are worked out step by step in 5.07 Solved PYQ Numerical Bank - Waves & Propagation.

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11. Self-Check Before Moving On

• Can you derive $\tan {\delta }_c=\sigma /\omega ϵ$ from the ratio of conduction to displacement current?
• Can you explain why the same medium is a conductor at low $f$ and a dielectric at high $f$?
• Can you get the low-loss $\alpha$ and $\beta$ by binomial expansion, showing the expansion step?
• Can you get $\alpha =\beta =\sqrt{\pi f\mu \sigma }$ using $\sqrt{j}=(1+j)/\sqrt{2}$?
• Can you write down $\eta$ for both cases and state the $45°$ lag for the conductor?
• Can you define skin depth, derive $\delta =1/\alpha$, and give the good-conductor formula?
• Can you state $P_{av}=\frac{1}{2}\sigma E_0^2$ and get $\sigma$ from a given loss tangent?

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Source: 05 electromagnetic_waves_master_notes.md §2 (master dump), ECE 2105 Syllabus Week 12, PYQ bank 2015–2025, Cheng Ch. 8, Sadiku Ch. 10, L 13.pdf, L 14.pdf