5.03 Poynting Vector, Power Flow & Normal Incidence Reflection (Gamma, Tau)

Related Concepts: 5.02 Plane Waves in Lossless vs. Lossy Media & Skin Depth Calculations | 4.04 Dynamic Boundary Conditions for Electromagnetic Fields | 4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem | 5.05 Dispersion, Phase & Group Velocity, Doppler Effect & Brewster’s Angle | 5.07 Solved PYQ Numerical Bank - Waves & Propagation

5.03 Poynting Vector, Power Flow & Normal Incidence Reflection ($\Gamma$, $\tau$)

Core Idea

When a plane wave meets a boundary between two media, the two media generally demand different $E/H$ ratios — different intrinsic impedances. A single forward-travelling wave cannot satisfy both boundary conditions at once, so part of the wave must bounce back. The reflection coefficient $\Gamma$ and transmission coefficient $\tau$ fall straight out of the tangential-field boundary conditions from 4.04 Dynamic Boundary Conditions for Electromagnetic Fields.

The most-asked single result in the entire course

“Show that $1+\Gamma =\tau$” appears in 2015, 2016, 2017, 2019, 2021, 2022, 2023 and 2024 — eight of the eleven papers, for 10–15 marks. Your previous note tagged it as only four years; the correct list is above.

---

1. Power Flow Recap — The Poynting Vector

The full derivation of Poynting’s theorem lives in 4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem. The results you need here:

Poynting vector

$\vec{\mathcal{P}}=\vec{E}\times \vec{H}[{\text{W/m}}^2]{\vec{\mathcal{P}}}_{av}=\frac{1}{2}\text{Re}\left\{{\vec{E}}_s\times {\vec{H}}_s^{\ast }\right\}$

For a uniform plane wave in a lossless medium with $|\vec{E}|=E_0$ and $\eta$ real:

${\mathcal{P}}_{av}=\frac{E_0^2}{2\eta }=\frac{1}{2}\eta H_0^2[{\text{W/m}}^2]$

Power fractions at a boundary

Because power scales as $E^2$, the fraction of incident power reflected and transmitted is: $\frac{P_r}{P_i}=|\Gamma {|}^2\frac{P_t}{P_i}=\frac{{\eta }_1}{{\eta }_2}|\tau {|}^2|\Gamma {|}^2+\frac{{\eta }_1}{{\eta }_2}|\tau {|}^2=1$ Note that $|\Gamma {|}^2+|\tau {|}^2\ne 1$ in general — power conservation requires the impedance ratio. This trips up a lot of students.

---

2. Master Derivation: Normal Incidence at a Plane Dielectric Boundary

[PYQ: 2015, 2016, 2017, 2019, 2021, 2022, 2023, 2024] — ⭐⭐⭐⭐⭐

[FIGURE: Interface at z = 0 separating Medium 1 (η₁) for z < 0 and Medium 2 (η₂) for z > 0. Show three waves: incident (E_i, H_i) travelling +z; reflected (E_r, H_r) travelling −z in Medium 1; transmitted (E_t, H_t) travelling +z in Medium 2. Mark E along â_x and H along ±â_y for each. — source: Cheng 2nd Ed., Fig. 8-6 / L 15_updated.pdf, Slide 10]

2.1 Set up the three waves

Incident wave ($+z$ in Medium 1):

${\vec{E}}_i(z)={\hat{a}}_x{E}_{i0}{e}^{-{\gamma }_{1}z},{\vec{H}}_i(z)={\hat{a}}_y\frac{E_{i0}}{{\eta }_1}{e}^{-{\gamma }_{1}z}$

Reflected wave ($-z$ in Medium 1) — note the sign reversal on $\vec{H}$:

${\vec{E}}_r(z)={\hat{a}}_x{E}_{r0}{e}^{+{\gamma }_{1}z},{\vec{H}}_r(z)=-{\hat{a}}_y\frac{E_{r0}}{{\eta }_1}{e}^{+{\gamma }_{1}z}$

Transmitted wave ($+z$ in Medium 2):

${\vec{E}}_t(z)={\hat{a}}_x{E}_{t0}{e}^{-{\gamma }_{2}z},{\vec{H}}_t(z)={\hat{a}}_y\frac{E_{t0}}{{\eta }_2}{e}^{-{\gamma }_{2}z}$

Key Exam Checkpoint — why ${\vec{H}}_r$ carries a minus sign

The Poynting vector $\vec{E}\times \vec{H}$ must point in the direction of propagation. For the reflected wave, propagation is $-{\hat{a}}_z$. With ${\vec{E}}_r$ still along $+{\hat{a}}_x$, we need ${\hat{a}}_x\times (\text{something})=-{\hat{a}}_z$, and since ${\hat{a}}_x\times (-{\hat{a}}_y)=-{\hat{a}}_z$, the magnetic field must reverse. Explain this sentence in the exam — examiners award a mark for it and most students just assert the sign.

2.2 Apply the boundary conditions at $z=0$

At the interface, the tangential components of $\vec{E}$ and $\vec{H}$ must be continuous (from 4.04 Dynamic Boundary Conditions for Electromagnetic Fields — with no surface current since neither medium is a perfect conductor):

Tangential $\vec{E}$: E_{i0} + E_{r0} = E_{t0} \tag{1}

Tangential $\vec{H}$: \frac{E_{i0}}{\eta_1} - \frac{E_{r0}}{\eta_1} = \frac{E_{t0}}{\eta_2} \tag{2}

2.3 Solve the pair

From (1), $E_{t0}=E_{i0}+E_{r0}$. Substitute into (2):

$\frac{E_{i0}-E_{r0}}{{\eta }_1}=\frac{E_{i0}+E_{r0}}{{\eta }_2}$

${\eta }_2(E_{i0}-E_{r0})={\eta }_1(E_{i0}+E_{r0})⟹E_{i0}({\eta }_2-{\eta }_1)=E_{r0}({\eta }_1+{\eta }_2)$

Reflection Coefficient

$\boxed{\Gamma =\frac{E_{r0}}{E_{i0}}=\frac{{\eta }_2-{\eta }_1}{{\eta }_2+{\eta }_1}}$

Substituting back into (1):

Transmission Coefficient

$\boxed{\tau =\frac{E_{t0}}{E_{i0}}=\frac{2{\eta }_2}{{\eta }_2+{\eta }_1}}$

2.4 The fundamental identity $1+\Gamma =\tau$

Method 1 — directly from the boundary condition (the elegant proof). Divide equation (1) throughout by $E_{i0}$:

$\frac{E_{i0}}{E_{i0}}+\frac{E_{r0}}{E_{i0}}=\frac{E_{t0}}{E_{i0}}⟹\boxed{1+\Gamma =\tau }$

Method 2 — algebraic verification:

$1+\Gamma =1+\frac{{\eta }_2-{\eta }_1}{{\eta }_2+{\eta }_1}=\frac{({\eta }_2+{\eta }_1)+({\eta }_2-{\eta }_1)}{{\eta }_2+{\eta }_1}=\frac{2{\eta }_2}{{\eta }_2+{\eta }_1}=\tau ✓$

Physical meaning of $1+\Gamma =\tau$

It is nothing more than continuity of the tangential electric field restated. The total field on the left of the boundary is incident + reflected; the total field on the right is transmitted. They must match at $z=0$. The identity holds for any pair of media, lossy or lossless.

Answer both methods

Method 1 shows you understand the physics; Method 2 shows the algebra is consistent. A full-mark answer derives $\Gamma$ and $\tau$ from the boundary conditions first (§2.1–2.3), then presents Method 1 as the one-line proof and Method 2 as verification.

graph TD
    A["Incident wave in Medium 1<br/>Ei, Hi = Ei/η₁"] --> B["Interface at z = 0<br/>η₁ ≠ η₂"]
    B --> C["Tangential E continuous<br/>Ei0 + Er0 = Et0"]
    B --> D["Tangential H continuous<br/>Ei0/η₁ − Er0/η₁ = Et0/η₂"]
    C --> E["Γ = (η₂−η₁)/(η₂+η₁)"]
    D --> E
    C --> F["τ = 2η₂/(η₂+η₁)"]
    D --> F
    E --> G["1 + Γ = τ"]
    F --> G

---

3. Special Cases

Situation	${\eta }_2$	$\Gamma$	$\tau$	Physical outcome
Matched media	${\eta }_2={\eta }_1$	$0$	$1$	No reflection; all power transmitted. SWR $=1$.
Perfect conductor	${\eta }_2=0$	$-1$	$0$	Total reflection with $180°$ phase reversal; pure standing wave.
Denser dielectric (${ϵ}_{r2}>{ϵ}_{r1}$)	${\eta }_2<{\eta }_1$	negative	$0<\tau <1$	Reflected $\vec{E}$ flips sign; transmitted wave weaker.
Rarer dielectric (${ϵ}_{r2}<{ϵ}_{r1}$)	${\eta }_2>{\eta }_1$	positive	$\tau >1$	Reflected $\vec{E}$ in phase; transmitted $E$ larger than incident.

" $\tau >1$ — isn't that more energy out than in?"

No. $\tau$ compares field amplitudes, not power. When ${\eta }_2>{\eta }_1$ the transmitted medium needs a larger $E$ to carry the same power because $\mathcal{P}=E^2/2\eta$. Power is conserved via $|\Gamma {|}^2+\frac{{\eta }_1}{{\eta }_2}|\tau {|}^2=1$. Stating this correctly is a discriminator between a good and an average script.

---

4. Standing Waves at a Conducting Boundary

[PYQ: 2018, 2020 — 08/10 Marks] — “Why standing wave is created when a plane electromagnetic wave incident normally on a plane conducting boundary? Explain it with necessary equation.”

4.1 The physical reason

A perfect conductor has $\sigma \to \infty$, so its intrinsic impedance is:

${\eta }_2=\sqrt{\frac{j\omega \mu }{\sigma +j\omega ϵ}}\stackrel{\sigma \to \infty }{\to }0$

Therefore:

$\Gamma =\frac{0-{\eta }_1}{0+{\eta }_1}=-1,\tau =1+\Gamma =0$

The wave is totally reflected with a $180°$ phase reversal, and nothing is transmitted (consistent with $\vec{E}=0$ inside a perfect conductor). Medium 1 now contains two waves of equal amplitude travelling in opposite directions. Their superposition does not travel — it oscillates in place with fixed nodes (permanent zeros) and antinodes (permanent maxima). That is a standing wave.

4.2 Mathematical proof

Let Medium 1 be lossless, so ${\gamma }_1=j{\beta }_1$. The total field in Medium 1:

${\vec{E}}_{1s}(z)={\hat{a}}_x\left(E_{i0}{e}^{-j{\beta }_{1}z}+E_{r0}{e}^{+j{\beta }_{1}z}\right)$

Substitute $E_{r0}=\Gamma E_{i0}=-E_{i0}$:

${\vec{E}}_{1s}(z)={\hat{a}}_x{E}_{i0}\left(e^{-j{\beta }_{1}z}-e^{+j{\beta }_{1}z}\right)$

Apply Euler’s identity $e^{j\theta }-e^{-j\theta }=2j\sin \theta$:

${\vec{E}}_{1s}(z)=-{\hat{a}}_{x}j2E_{i0}\sin ({\beta }_{1}z)$

Convert to the time domain, ${\vec{E}}_1(z,t)=\text{Re}\{{\vec{E}}_{1s}{e}^{j\omega t}\}$, using $\text{Re}\{-j{e}^{j\omega t}\}=\sin \omega t$:

Key Result — Standing Wave

$\boxed{{\vec{E}}_1(z,t)={\hat{a}}_{x}2E_{i0}\sin ({\beta }_{1}z)\sin (\omega t)}$

Why this is a standing wave and not a travelling wave

A travelling wave has the form $\cos (\omega t-\beta z)$ — space and time are locked together in a single argument, so the pattern moves. Here the spatial factor $\sin ({\beta }_{1}z)$ and the temporal factor $\sin (\omega t)$ are completely separated. The shape in space never changes; only its overall amplitude pulses up and down. The zeros stay put.

Node locations (where ${\vec{E}}_1=0$ permanently):

$\sin ({\beta }_{1}z)=0⟹{\beta }_{1}z=-n\pi ⟹z=-\frac{n\lambda }{2},n=0,1,2,\dots$

So nodes occur at the conductor surface and at every half-wavelength back from it. Antinodes sit midway between, at odd multiples of $\lambda /4$.

[FIGURE: Standing wave pattern in front of a conducting plane at z = 0. Plot |E₁| envelope versus z showing zeros at z = 0, −λ/2, −λ, … and maxima at z = −λ/4, −3λ/4, …; overlay |H₁| showing the complementary pattern (maximum at the conductor surface). — source: Cheng 2nd Ed., Fig. 8-8 / L 15_updated.pdf]

---

5. Standing Wave Ratio (SWR)

Tagging note — this is foundational, not a PYQ topic

Your previous note tagged SWR as [PYQ: 2019, 2023]. No PYQ in the 2015–2025 bank asks for SWR directly. It is retained here because it is genuinely useful for interpreting reflection problems and may be needed as a sub-step, but do not budget exam time for it as a standalone question.

Interference between the incident and reflected waves produces maxima and minima in Medium 1:

$|E{|}_{max}=|E_{i0}|(1+|\Gamma |),|E{|}_{min}=|E_{i0}|(1-|\Gamma |)$

$\text{SWR}=\frac{|E{|}_{max}}{|E{|}_{min}}=\frac{1+|\Gamma |}{1-|\Gamma |}$

Case	$\Gamma$	SWR
Perfect match (${\eta }_1={\eta }_2$)	$0$	$1$ (no standing wave)
Partial reflection	$0<\Vert \Gamma \Vert <1$	$1<\text{SWR}<\infty$
Total reflection (conductor)	$-1$	$\infty$ (pure standing wave)

---

6. Common Mistakes That Cost Marks

Avoid these

1. Forgetting the minus sign on ${\vec{H}}_r$. Without it you get the wrong $\Gamma$ entirely.
2. Writing $\Gamma =({\eta }_1-{\eta }_2)/({\eta }_1+{\eta }_2)$. The convention is ${\eta }_2$ first: $\Gamma =({\eta }_2-{\eta }_1)/({\eta }_2+{\eta }_1)$. Sign errors here propagate into every subsequent part.
3. Claiming $|\Gamma {|}^2+|\tau {|}^2=1$. It is not — the impedance ratio ${\eta }_1/{\eta }_2$ is required.
4. Not stating which boundary conditions you are using. “Tangential $\vec{E}$ and tangential $\vec{H}$ are continuous” must appear explicitly.
5. In the standing-wave question, only stating $\Gamma =-1$ without the algebra. The question says “explain it with necessary equation” — the Euler-identity step to $\sin (\beta z)\sin (\omega t)$ is where the marks are.
6. Forgetting that $\vec{H}$ has a maximum where $\vec{E}$ has a node. At a conductor surface $E$ is zero but $H$ is maximum — that is what sustains the surface current $J_s$.

---

7. PYQ Bank — Verbatim Questions & Answer Plans

Q1 — Prove $1+\Gamma =\tau$ [PYQ: 2015, 2016, 2017, 2019, 2021, 2022, 2023, 2024 — 10 to 15 Marks] ⭐⭐⭐⭐⭐

“If an EM wave follows normal incidence at a plain dielectric boundary, show that the reflection coefficient, $\Gamma$ and the transmission coefficient, $\tau$ are related by $1+\Gamma =\tau$.”

Answer plan (this is a full-page answer at 15 marks):

1. Draw the interface figure with all three waves labelled.
2. Write the six field expressions (§2.1), explaining the sign reversal on ${\vec{H}}_r$.
3. State the two boundary conditions and write equations (1) and (2) (§2.2).
4. Solve for $\Gamma$ and $\tau$ (§2.3) — box both.
5. Prove $1+\Gamma =\tau$ by Method 1, verify by Method 2 (§2.4).
6. Close with the physical interpretation: it is tangential-$E$ continuity restated.

Q2 — Polarization + $1+\Gamma =\tau$ combined [PYQ: 2015 — 11 Marks] (flagged as completely missing in corrupted pyq.md)

“What is meant by polarization of a wave? Show that the reflection and transmission coefficients are related with the following expression $1+\Gamma =\tau$, where the symbols have their usual meaning.”

Answer plan: Part 1 → the polarization definition from 5.04 Wave Polarization & Ionospheric Sky-Wave Radio Propagation. Part 2 → the Q1 derivation. Budget roughly 3 marks / 8 marks.

Q3 — Standing wave at a conducting boundary [PYQ: 2018, 2020 — 08/10 Marks]

“Why standing wave is created when a plane electromagnetic wave incident normally on a plane conducting boundary? Explain it with necessary equation.”

Answer plan: §4.1 (show ${\eta }_2\to 0\Rightarrow \Gamma =-1$, $\tau =0$, total reflection with phase reversal) → §4.2 (full algebra to $2E_{i0}\sin {\beta }_{1}z\sin \omega t$) → the separation-of-variables explanation → node locations at $z=-n\lambda /2$.

Note: the 2018 paper pairs this with “Define plasma frequency” in the same question — that part routes to 5.06 Radio Wave Propagation Modes & Ionospheric Effects.

Q4 — Numericals using this note

• y-polarized wave normally incident on a perfect conductor — phasor and instantaneous expressions for incident, reflected and total waves, plus nearest node location [PYQ: 2015, 2016 — 08/19 Marks] (The 2016 version asks only for parts (i) and (ii) — incident and reflected — per corrupted pyq.md.)

Worked in full in 5.07 Solved PYQ Numerical Bank - Waves & Propagation.

---

8. Self-Check Before Moving On

• Can you write the six field expressions for incident, reflected and transmitted waves with correct signs?
• Can you justify the minus sign on ${\vec{H}}_r$ using the Poynting direction?
• Can you name the two boundary conditions and derive $\Gamma$ and $\tau$ from them?
• Can you prove $1+\Gamma =\tau$ in one line from the tangential-$E$ condition?
• Can you explain why $\tau$ can exceed 1 without violating energy conservation?
• Can you show $\Gamma =-1$ for a perfect conductor and derive the standing-wave expression?
• Can you say where the nodes and antinodes are, and why $H$ is maximum where $E$ is zero?

---

Source: 05 electromagnetic_waves_master_notes.md §5 (master dump), ECE 2105 Syllabus Weeks 11–13, PYQ bank 2015–2025 + corrupted pyq.md, Cheng Ch. 8-6, L 15_updated.pdf