Chapter 4 - Time-Varying Fields & Maxwell Equations

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📄 Section: 04 Chapter Map - Time-Varying Fields & Maxwell Equations

04 Chapter Map — Time-Varying Fields & Maxwell’s Equations

Chapter Overview

This is the chapter where electrostatics and magnetostatics stop being two separate subjects. Once fields are allowed to vary with time, a changing $\vec{B}$ creates a circulating $\vec{E}$ (Faraday) and a changing $\vec{E}$ creates a circulating $\vec{H}$ (Maxwell) — a feedback loop that produces self-sustaining electromagnetic waves. The chapter covers charge conservation and the continuity equation, Faraday’s law and the three forms of induced EMF, displacement current, the four complete Maxwell’s equations, dynamic boundary conditions, and Poynting’s theorem for electromagnetic power flow.

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📚 Study Sequence

Study these in order — each note depends on the one before it.

#	Note	Covers	Weight
1	4.01 Charge Conservation & Continuity Equation Mechanics	Conservation of charge, continuity equation derivation, physical significance, KCL reduction	⭐⭐⭐⭐⭐
2	4.02 Faraday’s Law of Induction & Maxwell’s Displacement Current	Faraday’s law, transformer/motional/combined EMF, displacement current, AC capacitor proof	⭐⭐⭐⭐
3	4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem	The four equations, physical significance, deduction from static postulates, Poynting’s theorem, DC wire verification	⭐⭐⭐⭐⭐
4	4.04 Dynamic Boundary Conditions for Electromagnetic Fields	Why dynamic = static BCs, the four general conditions, lossless–lossless and dielectric–PEC cases	⭐⭐⭐⭐⭐
5	00 Chapter 4 Active-Recall Diagnostic Quiz	Self-test before moving to Chapter 5	—

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🔗 Dependency Graph

graph TD
    C1["Ch.1 Vector Calculus<br/>Divergence & Stokes's Theorems"] --> N1
    C2["Ch.2 Electrostatics<br/>∇·D = ρᵥ, ∇×E = 0"] --> N3
    C3["Ch.3 Magnetostatics<br/>∇·B = 0, ∇×H = J"] --> N3
    N1["4.01 Continuity Equation<br/>∇·J = −∂ρᵥ/∂t"] --> N2["4.02 Faraday's Law<br/>& Displacement Current"]
    N2 --> N3["4.03 Maxwell's Equations<br/>& Poynting's Theorem"]
    N3 --> N4["4.04 Dynamic<br/>Boundary Conditions"]
    N3 --> C5["Ch.5 Wave Equations<br/>& EM Wave Propagation"]
    N4 --> C5

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🎯 PYQ Weight Map (2015–2025)

Topic	Note	Years asked	Frequency
Conservation of charge → continuity equation → significance	4.01	2015, 2018, 2019, 2021, 2022, 2024, 2025	⭐⭐⭐⭐⭐ (7)
Maxwell’s equations: differential + integral forms	4.03	2015, 2016, 2017, 2021, 2023, 2024, 2025	⭐⭐⭐⭐⭐ (7)
Physical significance of Maxwell’s equations	4.03	2015, 2018, 2021, 2023, 2024, 2025	⭐⭐⭐⭐⭐ (6)
Dynamic boundary conditions (lossless + PEC)	4.04	2018, 2019, 2021, 2022, 2023, 2024	⭐⭐⭐⭐⭐ (6)
Deduce Maxwell’s from static postulates	4.03	2018, 2019, 2023	⭐⭐⭐ (3)
Total power flowing through a closed surface (Poynting derivation)	4.03	2016, 2022, 2024	⭐⭐⭐ (3)
State & explain Poynting’s theorem	4.03	2015, 2023	⭐⭐ (2)
Displacement current: define + capacitor derivation	4.02	2016, 2017	⭐⭐ (2)
Identify each Maxwell equation with its experimental law	4.03	2016, 2017	⭐⭐ (2)
Why dynamic BCs equal static BCs	4.04	2020	⭐ (1)
Poynting vector on a DC conducting wire + verification	4.03	2017	⭐ (1)
Displacement current density (short-note term)	4.02	2023	⭐ (1)

Effort-vs-reward verdict

Must-master: the continuity equation derivation, the Maxwell’s equations master table with significance, and the dynamic boundary conditions. Those three items alone have appeared in every paper in the range and typically account for 25–30 marks. Safe-pass: Poynting’s theorem derivation, displacement current in a capacitor. Lower ROI (but cheap): the DC-wire Poynting verification — one year only, but it is short and memorable.

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🧮 Chapter Formula Quick Reference

Quantity	Formula	Note
Continuity equation	$\nabla \cdot \vec{J}=-\frac{\partial {\rho }_v}{\partial t}$	4.01
Steady-current / KCL	$\nabla \cdot \vec{J}=0\Rightarrow {\sum }_k{I}_k=0$	4.01
Faraday’s law	${\oint }_C\vec{E}\cdot d\vec{l}=-\frac{d\Phi }{dt}$	4.02
Transformer EMF	$e=-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}$	4.02
Motional EMF	$e={\oint }_C(\vec{u}\times \vec{B})\cdot d\vec{l}$	4.02
Displacement current density	${\vec{J}}_d=\frac{\partial \vec{D}}{\partial t}=ϵ\frac{\partial \vec{E}}{\partial t}$	4.02
Maxwell 1	$\nabla \cdot \vec{D}={\rho }_v$	4.03
Maxwell 2	$\nabla \cdot \vec{B}=0$	4.03
Maxwell 3	$\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$	4.03
Maxwell 4	$\nabla \times \vec{H}=\vec{J}+\frac{\partial \vec{D}}{\partial t}$	4.03
Instantaneous Poynting vector	$\vec{\mathcal{P}}=\vec{E}\times \vec{H}$ [W/m²]	4.03
Time-average Poynting vector	${\vec{\mathcal{P}}}_{av}=\frac{1}{2}\text{Re}\{{\vec{E}}_s\times {\vec{H}}_s^{\ast }\}$	4.03
Boundary: tangential $\vec{E}$	$E_{1t}=E_{2t}$	4.04
Boundary: normal $\vec{D}$	$D_{1n}-D_{2n}={\rho }_s$	4.04
Boundary: tangential $\vec{H}$	$H_{1t}-H_{2t}=J_s$	4.04
Boundary: normal $\vec{B}$	$B_{1n}=B_{2n}$	4.04

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🔗 Related Resources

• Course Teaching Plan: ECE 2105 - Electromagnetic Fields and Waves (Weeks 8–9, 11, 13)
• Master Course Portal: 00 ECE 2105 Complete Course Master Map
• Glossary & Formula Sheet: 00 Master Glossary & Formula Sheet for ECE 2105
• Next chapter: 05 Chapter Map - Electromagnetic Waves & Radio Propagation
• Raw source dump (immutable): electrostatics/04 time_varying_fields_and_maxwells_equations.md

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📄 Section: 4.01 Charge Conservation & Continuity Equation Mechanics

Related Concepts: 3.01 Fundamental Postulates of Magnetostatics & Lorentz Force Equation | 4.02 Faraday’s Law of Induction & Maxwell’s Displacement Current | 4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem

4.01 Charge Conservation & Continuity Equation Mechanics

Core Idea

Electric charge can never be created or destroyed — it can only be moved. Writing that single sentence in the language of vector calculus produces the continuity equation, $\nabla \cdot \vec{J}=-\partial {\rho }_v/\partial t$. This one equation is the hinge on which the whole of Chapter 4 turns: it is what later forces Maxwell to invent displacement current, and it is what Kirchhoff’s Current Law reduces to when nothing changes with time.

Highest-Value Item in Chapter 4

This derivation has appeared in seven of the last eleven papers (2015, 2018, 2019, 2021, 2022, 2024, 2025), always for 8–10 marks, and the question always has the same three parts: state the principle → derive → give the physical significance. Losing marks here is losing free marks.

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0. Why This Chapter Exists

Everything up to now has been static. Electrostatics gave us fields from charges that sit still; magnetostatics gave us fields from currents that never change. The two worlds never spoke to each other — an $\vec{E}$ field and a $\vec{B}$ field were completely independent objects.

From this chapter on, fields are allowed to change with time, and the moment they do, the two worlds become one:

graph LR
    A["Time-varying B field"] -->|"Faraday's Law"| B["induces E field"]
    B -->|"Maxwell's Displacement Current"| A
    A --> C["Self-sustaining<br/>Electromagnetic Wave"]
    B --> C

This mutual regeneration is what allows a wave to detach from its source and travel through empty space. But before any of it works, the bookkeeping of charge has to be airtight — which is exactly what the continuity equation provides.

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1. The Principle of Conservation of Charge

Statement (write this verbatim in the exam)

The Principle of Conservation of Charge states that electric charge can neither be created nor destroyed; it can only be transported from one region of space to another. Consequently, the net electric charge of an isolated system remains constant for all time.

The engineering consequence, stated for a closed region:

For any arbitrary closed volume $V$ bounded by a closed surface $S$, if charge flows outward through $S$ (constituting a net outgoing current $I_{out}$), then the total charge $Q$ remaining inside $V$ must be decreasing at exactly the same rate. No charge quietly disappears at the boundary and none is spontaneously manufactured inside.

Why this is a postulate, not a theorem

Conservation of charge is not derived from Maxwell’s equations — it is an independent experimental law of nature, on the same footing as conservation of energy. Maxwell’s equations were later made consistent with it. That direction of logic matters: examiners ask you to derive the continuity equation from the conservation principle, never the other way round.

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2. Master Derivation: The Continuity Equation

[PYQ: 2015, 2018, 2019, 2021, 2022, 2024, 2025] — Heavily Tested

[FIGURE: An arbitrary closed volume V bounded by closed surface S. Current density vector J piercing outward through a differential surface patch ds, with volume charge density ρ_v distributed inside. — source: Cheng, "Field and Wave Electromagnetics" 2nd Ed., Fig. 5-3 / lecture slide "03 solution to em eqns.pdf"]

Step 1 — Express the outward current as a surface integral

The total current leaving the closed volume $V$ through its bounding surface $S$ is the flux of the current density vector $\vec{J}$ through that surface:

$I_{out}={\oint }_S\vec{J}\cdot d\vec{s}$

Step 2 — Impose charge conservation

Conservation demands that this outgoing current equals the rate of decrease of the enclosed charge $Q$ (hence the minus sign):

$I_{out}=-\frac{dQ}{dt}⟹{\oint }_S\vec{J}\cdot d\vec{s}=-\frac{dQ}{dt}$

Step 3 — Write $Q$ in terms of volume charge density

$Q={\int }_V{\rho }_{v}dv⟹{\oint }_S\vec{J}\cdot d\vec{s}=-\frac{d}{dt}{\int }_V{\rho }_{v}dv$

Step 4 — Move the time derivative inside the integral

The volume $V$ is stationary and rigid, so its limits of integration do not depend on time. The derivative may therefore be taken inside. Because ${\rho }_v={\rho }_v(x,y,z,t)$ is a function of both space and time, the total derivative becomes a partial derivative:

${\oint }_S\vec{J}\cdot d\vec{s}=-{\int }_V\frac{\partial {\rho }_v}{\partial t}dv$

Key Exam Checkpoint

State explicitly that $V$ is stationary and rigid, and justify the change $\frac{d}{dt}\to \frac{\partial }{\partial t}$. Examiners routinely award a separate mark for this single line, and most students skip it.

Step 5 — Apply the Divergence Theorem

Convert the closed surface integral on the left into a volume integral:

${\oint }_S\vec{J}\cdot d\vec{s}={\int }_V(\nabla \cdot \vec{J})dv$

Equating the two volume integrals:

${\int }_V(\nabla \cdot \vec{J})dv=-{\int }_V\frac{\partial {\rho }_v}{\partial t}dv$

${\int }_V\left(\nabla \cdot \vec{J}+\frac{\partial {\rho }_v}{\partial t}\right)dv=0$

Step 6 — Shrink the volume to a point

This result must hold for any arbitrary volume $V$, no matter how small or where it is placed. The only way an integral can vanish for every possible domain is if the integrand itself is identically zero everywhere:

Key Result — Continuity Equation (Differential / Point Form)

$\boxed{\nabla \cdot \vec{J}=-\frac{\partial {\rho }_v}{\partial t}}[{\text{A/m}}^3]$

Integral form (occasionally asked as a follow-up):

${\oint }_S\vec{J}\cdot d\vec{s}=-\frac{d}{dt}{\int }_V{\rho }_{v}dv$

Symbol definitions — write these out, they carry marks

• $\vec{J}$ = volume current density $[{\text{A/m}}^2]$
• ${\rho }_v$ = volume charge density $[{\text{C/m}}^3]$
• $\nabla \cdot$ = divergence operator $[{\text{m}}^{-1}]$
• $t$ = time $[\text{s}]$
• $d\vec{s}$ = outward-directed differential surface element $[{\text{m}}^2]$

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3. Physical Significance & Interpretation

[PYQ: 2015, 2018, 2019, 2021, 2022, 2024, 2025] — every one of those papers asked for this explicitly as a separate marked part

The continuity equation is a local, point-by-point statement of charge bookkeeping:

Mathematical condition	Physical meaning at that point
$\nabla \cdot \vec{J}>0$	Current diverges — more current leaves than enters, so charge density there is draining ($\partial {\rho }_v/\partial t<0$). The point acts as a source.
$\nabla \cdot \vec{J}<0$	Current converges — more current arrives than departs, so charge is piling up ($\partial {\rho }_v/\partial t>0$). The point acts as a sink.
$\nabla \cdot \vec{J}=0$	Whatever flows in flows straight back out; charge density is constant in time.

In one sentence for the answer script:

The divergence of the current density at any point equals the time rate of decrease of the volume charge density at that same point. Charge does not vanish — if it leaves a region as current, the charge density in that region must fall by precisely the corresponding amount.

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4. Steady-Current Reduction & Kirchhoff’s Current Law

[PYQ: 2015, 2018, 2019, 2021, 2022, 2024, 2025] — this is the standard “significance” follow-up

For steady (direct) currents, charge neither accumulates nor depletes anywhere, so $\frac{\partial {\rho }_v}{\partial t}=0$ and the continuity equation collapses to:

$\nabla \cdot \vec{J}=0$

Two immediate consequences:

1. Steady current is solenoidal. A divergence-free field has no sources or sinks, so steady current must flow in closed loops. This is precisely why a DC circuit must be a complete loop for current to flow at all.

2. This is Kirchhoff’s Current Law. Integrate over a closed surface drawn around a circuit junction and apply the divergence theorem in reverse:

${\oint }_S\vec{J}\cdot d\vec{s}=0⟹{\sum }_k{I}_k=0$

graph TD
    A["Conservation of Charge<br/>(experimental postulate)"] --> B["∮ J·ds = −dQ/dt"]
    B -->|"Divergence Theorem"| C["∇·J = −∂ρᵥ/∂t<br/>CONTINUITY EQUATION"]
    C -->|"set ∂ρᵥ/∂t = 0<br/>(steady currents)"| D["∇·J = 0<br/>J is solenoidal"]
    D -->|"integrate over a junction"| E["Σ Iₖ = 0<br/>KIRCHHOFF'S CURRENT LAW"]
    C -->|"feeds into Ampère's law"| F["Displacement Current Jd<br/>→ see 4.02"]

The one-line takeaway examiners love

KCL is not a separate law — it is the continuity equation evaluated under steady-state conditions. Circuit theory is the low-frequency shadow of field theory.

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5. Where This Equation Goes Next

The continuity equation is not an endpoint. In 4.02 Faraday’s Law of Induction & Maxwell’s Displacement Current you will see that Ampère’s static law $\nabla \times \vec{H}=\vec{J}$ directly contradicts it for time-varying fields, and that resolving the contradiction is exactly what produces displacement current — and therefore electromagnetic waves.

Regime	Continuity equation reads	Consequence
Electrostatics / Magnetostatics	$\nabla \cdot \vec{J}=0$	Ampère’s law $\nabla \times \vec{H}=\vec{J}$ is self-consistent
Time-varying (electrodynamics)	$\nabla \cdot \vec{J}=-\partial {\rho }_v/\partial t\ne 0$	Ampère’s law breaks; must add ${\vec{J}}_d=\partial \vec{D}/\partial t$

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6. Common Mistakes That Cost Marks

Avoid these

1. Dropping the minus sign in $I_{out}=-dQ/dt$. The sign is the physics — it encodes “outflow depletes the interior”.
2. Writing $d/dt$ instead of $\partial /\partial t$ after moving the derivative inside the volume integral, with no justification.
3. Forgetting to state “for arbitrary $V$” in the final step. Without that sentence, going from the integral to the point form is unjustified.
4. Using an open surface integral ${\int }_S$ instead of the closed ${\oint }_S$. Charge enclosure only makes sense for a closed surface.
5. Stopping at the equation. Every single PYQ on this topic asks for the physical significance as a marked sub-part. Always write the KCL reduction.

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7. PYQ Bank — Verbatim Questions & Answer Plans

Q1 — Conservation → Continuity → Significance [PYQ: 2015, 2018, 2022 — 09/10 Marks]

“State the principle of conservation of charge. Based on this principle, derive the continuity equation, $\nabla \cdot \vec{J}=-\partial {\rho }_v/\partial t$, where the symbols have their usual meanings. Also write down the physical significance of this equation.”

Answer plan (3 marked parts):

1. Statement — reproduce Section 1 verbatim.
2. Derivation — all six steps of Section 2, explicitly naming the Divergence Theorem and justifying $d/dt\to \partial /\partial t$.
3. Significance — the interpretation table of Section 3, then the steady-current reduction to KCL from Section 4.

Q2 — "Deduce" phrasing [PYQ: 2019 — 10 Marks]

“State the principle of conservation of charge. Through this principle, deduce the continuity equation, $\nabla \cdot \vec{J}=-\frac{\partial {\rho }_v}{\partial t}$, where the symbols have their usual meanings. Also, write down the physical significance of this equation.”

Answer plan: Identical content to Q1. “Deduce” and “derive” are interchangeable here — do not shorten the derivation.

Q3 — "Extend to" phrasing [PYQ: 2021 — 10 Marks]

“State the principle of conservation of charge. Explain how this principle can extend to continuity equation and the significance of the equation.”

Answer plan: Same derivation, but the word “explain” means you should narrate the physical reasoning between steps rather than presenting bare algebra. Spend an extra line on why the volume must be stationary and why the result must hold for arbitrary $V$.

Q4 — "Law of conservation" phrasing [PYQ: 2025 — 10 Marks]

“Explain the law of conservation of charge. Starting from this law, derive the continuity equation. $\nabla \cdot J=-\frac{\partial {\rho }_v}{\partial t}$. Where the symbols have their usual meanings. Briefly discuss its physical interpretation.”

Answer plan: Same content. Note the question says “where the symbols have their usual meanings” — you must still define every symbol (Section 2 tip box). The word “briefly” on the interpretation means 3–4 lines is enough; do not sacrifice derivation detail for it.

Q5 — Short form [PYQ: 2024 — 08 Marks]

“Derive and explain the continuity equation along with its significance.”

Answer plan: At 8 marks with no explicit “state the principle” prompt, still open with a one-line statement of conservation of charge (it is the starting premise and carries a mark), then the full derivation, then significance + KCL.

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8. Self-Check Before Moving On

• Can you state the conservation of charge principle in one clean sentence, from memory?
• Can you reproduce all six derivation steps without looking, naming the Divergence Theorem at the correct step?
• Can you explain why the derivative becomes partial when it moves inside the integral?
• Can you explain why the integrand must vanish (the “arbitrary volume” argument)?
• Can you show, in three lines, that KCL is the steady-state case of this equation?
• Can you state what goes wrong with Ampère’s law if $\nabla \cdot \vec{J}\ne 0$? → if not, go to 4.02 Faraday’s Law of Induction & Maxwell’s Displacement Current

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Source: 04 time_varying_fields_and_maxwells_equations.md (master dump), ECE 2105 Syllabus Week 8, PYQ bank 2015–2025, 03 solution to em eqns.pdf, Masuk sir-2309008.pdf

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📄 Section: 4.02 Faraday_s Law of Induction & Maxwell_s Displacement Current

Related Concepts: 4.01 Charge Conservation & Continuity Equation Mechanics | 4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem | 3.01 Fundamental Postulates of Magnetostatics & Lorentz Force Equation

4.02 Faraday’s Law of Induction & Maxwell’s Displacement Current

Core Idea

Two symmetric statements turn static field theory into electrodynamics:

• Faraday (experiment, 1831): a time-varying magnetic field creates a circulating electric field.
• Maxwell (pure mathematics, 1865): a time-varying electric field must create a circulating magnetic field — otherwise Ampère’s law contradicts conservation of charge.

Faraday’s half was discovered in a laboratory. Maxwell’s half was discovered on paper, purely to repair a mathematical inconsistency — and it predicted electromagnetic waves before anyone had ever produced one.

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1. Faraday’s Law of Electromagnetic Induction

1.1 Statement and Equation

Faraday's Law

The induced electromotive force (EMF) around any closed circuit equals the negative time rate of change of the total magnetic flux $\Phi$ linking that circuit.

$e={\oint }_C\vec{E}\cdot d\vec{l}=-\frac{d\Phi }{dt}=-\frac{d}{dt}{\int }_S\vec{B}\cdot d\vec{s}$

Terminology & Concept Breakdown

• Electromotive force ($e$): despite the name, not a force — it is the work done per unit charge in carrying that charge once around the closed loop $C$. Units: volts.
• Magnetic flux ($\Phi$): the total amount of $\vec{B}$ threading through the open surface $S$ that the loop $C$ bounds. Units: webers (Wb).
• Lenz’s Law (the minus sign): the induced current flows in whatever direction produces a magnetic flux that opposes the change that caused it. Nature resists change in flux. Without this minus sign you would have a machine that amplifies its own output — a violation of energy conservation.

The single most important structural consequence

In electrostatics, ${\oint }_C\vec{E}\cdot d\vec{l}=0$ — the electric field was conservative. Faraday’s law says that once $\vec{B}$ changes with time, this line integral is no longer zero. The dynamic electric field is non-conservative: it circulates. This is precisely why $\nabla \times \vec{E}=0$ (static) must be upgraded to $\nabla \times \vec{E}=-\partial \vec{B}/\partial t$ (dynamic).

1.2 Differential (Point) Form

Apply Stokes’s Theorem to the left-hand side and assume a stationary loop so the derivative may be taken inside as a partial derivative:

${\int }_S(\nabla \times \vec{E})\cdot d\vec{s}=-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}$

Since this holds for any arbitrary surface $S$, the integrands must be equal:

Faraday's Law — Differential Form (Maxwell's 3rd Equation)

$\boxed{\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}}$

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2. The Three Operational Cases of Induced EMF

Flux $\Phi ={\int }_S\vec{B}\cdot d\vec{s}$ can change for two independent reasons: $\vec{B}$ itself changes, or the loop moves/deforms. This gives three cases.

[FIGURE: Three side-by-side panels — (a) stationary loop with a time-varying B field through it (transformer EMF); (b) rigid loop sliding with velocity u through a uniform static B field (motional EMF); (c) moving loop in a time-varying B field (combined). — source: Cheng 2nd Ed., Figs. 7-2 to 7-4 / lecture slide L 9.pdf]

Case 1 — Transformer EMF (stationary loop, time-varying $\vec{B}$)

$e_{trans}=-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}⟺\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$

The loop sits still; only the field changes. Physical example: an ordinary transformer — the secondary winding never moves, but the alternating primary current makes $\vec{B}$ in the core oscillate.

Case 2 — Motional EMF (moving loop, static $\vec{B}$)

$e_{motion}={\oint }_C(\vec{u}\times \vec{B})\cdot d\vec{l}$

The field is frozen in time; the conductor cuts through it at velocity $\vec{u}$. The force on the carriers comes straight from the magnetic part of the Lorentz force, $\vec{F}=q(\vec{u}\times \vec{B})$. Physical example: a DC generator, or a conducting bar sliding along rails.

Case 3 — Combined EMF (moving loop, time-varying $\vec{B}$)

$e_{total}=\underset{\text{transformer part}}{\underbrace{-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}}}+\underset{\text{motional part}}{\underbrace{{\oint }_C(\vec{u}\times \vec{B})\cdot d\vec{l}}}$

The most general case — both mechanisms contribute and simply add.

Case	Loop	$\vec{B}$ field	EMF expression	Everyday device
Transformer	Stationary	Time-varying	$-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}$	Transformer, induction cooktop
Motional	Moving ($\vec{u}$)	Static	${\oint }_C(\vec{u}\times \vec{B})\cdot d\vec{l}$	DC generator, dynamo
Combined	Moving ($\vec{u}$)	Time-varying	Sum of both	AC machine with a rotating rotor

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3. Master Derivation: Maxwell’s Displacement Current

[PYQ: 2016, 2017] for the derivation; the definition alone also appeared [PYQ: 2023] as a short-note term.

PYQ tagging note

Displacement current is asked as a standalone question in 2016 and 2017 only (“Define displacement current. Determine the displacement current in between two parallel plates of a capacitor energized by an alternating current source.”). It also appears as a required step inside the “Deduce Maxwell’s equations” question [PYQ: 2018, 2019, 2023] — see 4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem. So its real exam weight is higher than the two standalone years suggest.

3.1 The Fatal Flaw in Ampère’s Static Law

In magnetostatics, Ampère’s circuital law in differential form is:

$\nabla \times \vec{H}=\vec{J}$

Take the divergence of both sides:

$\nabla \cdot (\nabla \times \vec{H})=\nabla \cdot \vec{J}$

The left side is identically zero by the null identity (the divergence of the curl of any vector field is always zero, $\nabla \cdot \nabla \times \vec{A}\equiv 0$). Therefore Ampère’s static law forces:

$0=\nabla \cdot \vec{J}$

But from 4.01 Charge Conservation & Continuity Equation Mechanics, the continuity equation says:

$\nabla \cdot \vec{J}=-\frac{\partial {\rho }_v}{\partial t}\ne 0\text{whenever charge density changes with time}$

The contradiction

Ampère’s static law demands $\nabla \cdot \vec{J}=0$ everywhere and always. Conservation of charge says $\nabla \cdot \vec{J}\ne 0$ whenever ${\rho }_v$ varies. Both cannot be true. Since conservation of charge is an experimental law of nature that cannot be abandoned, Ampère’s law must be the thing that is incomplete.

3.2 Maxwell’s Repair

Maxwell postulated a missing term — an unknown displacement current density ${\vec{J}}_d$ — added to the right-hand side:

$\nabla \times \vec{H}=\vec{J}+{\vec{J}}_d$

Step 1. Take the divergence of both sides. The left side is still identically zero:

$0=\nabla \cdot \vec{J}+\nabla \cdot {\vec{J}}_d⟹\nabla \cdot {\vec{J}}_d=-\nabla \cdot \vec{J}$

Step 2. Substitute the continuity equation $\nabla \cdot \vec{J}=-\partial {\rho }_v/\partial t$:

$\nabla \cdot {\vec{J}}_d=\frac{\partial {\rho }_v}{\partial t}$

Step 3. Substitute Gauss’s postulate ${\rho }_v=\nabla \cdot \vec{D}$:

$\nabla \cdot {\vec{J}}_d=\frac{\partial }{\partial t}\left(\nabla \cdot \vec{D}\right)$

Step 4. Spatial and temporal derivatives are independent, so their order may be exchanged:

$\nabla \cdot {\vec{J}}_d=\nabla \cdot \left(\frac{\partial \vec{D}}{\partial t}\right)$

Step 5. Strip the divergence operator from both sides:

Key Result — Displacement Current Density

$\boxed{{\vec{J}}_d=\frac{\partial \vec{D}}{\partial t}=ϵ\frac{\partial \vec{E}}{\partial t}}[{\text{A/m}}^2]$

and Ampère’s law becomes the Ampère–Maxwell Law: $\nabla \times \vec{H}=\vec{J}+\frac{\partial \vec{D}}{\partial t}$

Definition to memorise

Displacement current is a fictitious current, proportional to the time rate of change of the electric flux density $\vec{D}$, introduced by Maxwell to make Ampère’s circuital law consistent with the conservation of charge in time-varying fields. It involves no motion of physical charge, yet it produces a magnetic field exactly as a real conduction current does.

graph TD
    A["Ampère's static law<br/>∇×H = J"] -->|"take divergence"| B["∇·J = 0<br/>(forced)"]
    C["Conservation of charge<br/>∇·J = −∂ρᵥ/∂t ≠ 0"] --> D{"CONTRADICTION"}
    B --> D
    D -->|"Maxwell adds Jd"| E["∇×H = J + Jd"]
    E -->|"divergence + continuity<br/>+ Gauss ρᵥ = ∇·D"| F["Jd = ∂D/∂t<br/>DISPLACEMENT CURRENT"]
    F --> G["∇×H = J + ∂D/∂t<br/>AMPÈRE–MAXWELL LAW"]
    G --> H["Time-varying E creates H<br/>→ EM waves possible"]

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4. High-Yield Derivation: Displacement Current in a Capacitor

[PYQ: 2016, 2017 — 07/09 Marks] — this is the standard second half of the displacement-current question.

The physical puzzle: current flows in the wires feeding a capacitor, yet the gap between the plates is an insulator where no charge can cross. If you draw an Ampèrean loop around the wire and stretch its surface so it passes between the plates instead of cutting the wire, the enclosed conduction current is zero — so where did the magnetic field go? Displacement current is the answer.

[FIGURE: Parallel-plate capacitor of plate area A and separation d energised by an AC source v(t) = V₀ sin ωt. Show conduction current i_c in the connecting wires, E-field lines between the plates, and two alternative Ampèrean surfaces bounded by the same loop — one cutting the wire, one passing through the gap. — source: Cheng 2nd Ed., Fig. 7-1 / "01 Static Electric Field.pdf"]

Step 1. Energise a parallel-plate capacitor (plate area $A$, separation $d$, dielectric $ϵ$) with a time-harmonic source:

$v(t)=V_0\sin \omega t$

Step 2. Neglecting fringing, the field between the plates is uniform:

$\vec{E}(t)={\hat{a}}_n\frac{v(t)}{d}={\hat{a}}_n\frac{V_0}{d}\sin \omega t$

Step 3. The electric flux density in the dielectric:

$\vec{D}(t)=ϵ\vec{E}(t)={\hat{a}}_n\frac{ϵV_0}{d}\sin \omega t$

Step 4. Differentiate to get the displacement current density:

${\vec{J}}_d=\frac{\partial \vec{D}}{\partial t}={\hat{a}}_n\frac{ϵV_0\omega }{d}\cos \omega t$

Step 5. Integrate over the plate area $A$ to get the total displacement current:

$i_d={\int }_A{\vec{J}}_d\cdot d\vec{s}=J_{d}A=\frac{ϵA}{d}{V}_0\omega \cos \omega t$

Step 6. Recognise the parallel-plate capacitance $C=\frac{ϵA}{d}$:

$i_d=C{V}_0\omega \cos \omega t=C\frac{dv(t)}{dt}$

Step 7. From elementary circuit theory, the conduction current in the external wires is:

$i_c(t)=C\frac{dv(t)}{dt}$

Conclusion

$\boxed{i_d(t)=i_c(t)}$ The displacement current between the plates is exactly equal to the conduction current in the wires. Total current is therefore continuous around the whole circuit loop, and Ampère’s law gives the same magnetic field regardless of which surface you stretch across the loop.

The sentence that earns the last mark

“Displacement current is not a flow of physical charge — it is a time-varying electric field that behaves, magnetically, exactly as a real current does. It is what bridges the insulating gap and keeps total current continuous.”

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5. Comparison: Conduction vs. Displacement Current

Property	Conduction Current $\vec{J}=\sigma \vec{E}$	Displacement Current ${\vec{J}}_d=\partial \vec{D}/\partial t$
Physical nature	Actual drift of free charge carriers	No charge motion; changing electric flux
Requires	A conducting medium ($\sigma \ne 0$)	Any medium including vacuum
Governing relation	Ohm’s law, $\vec{J}=\sigma \vec{E}$	${\vec{J}}_d=ϵ\partial \vec{E}/\partial t$
Exists in DC/static?	Yes	No — vanishes when $\partial /\partial t=0$
Energy behaviour	Dissipates as ohmic heat $\vec{J}\cdot \vec{E}$	Stores/returns energy (reactive)
Dominates when	$\sigma \gg \omega ϵ$ (good conductor)	$\sigma \ll \omega ϵ$ (good dielectric)
Produces $\vec{H}$?	Yes	Yes — identically

Forward link

The ratio of these two currents, $\frac{\sigma }{\omega ϵ}$, is the loss tangent, and it is the single number that decides whether a medium behaves as a conductor or a dielectric at a given frequency. See 5.02 Plane Waves in Lossless vs. Lossy Media & Skin Depth Calculations.

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6. Why This Chapter Produces Waves

Put the two halves together and note the feedback loop:

$\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}\text{(changing }\vec{B}\Rightarrow \text{ circulating }\vec{E})$ $\nabla \times \vec{H}=\vec{J}+\frac{\partial \vec{D}}{\partial t}\text{(changing }\vec{E}\Rightarrow \text{ circulating }\vec{H})$

Each field regenerates the other. In a source-free region ($\vec{J}=0$) this becomes self-sustaining — the disturbance no longer needs its source and propagates away as an electromagnetic wave. That is the entire content of 5.01 Wave Equations & Helmholtz Equations in Source-Free Media.

Without displacement current, there are no radio waves

If Maxwell had not added $\partial \vec{D}/\partial t$, the feedback loop would be broken in one direction and electromagnetic waves would be mathematically impossible. Every wireless technology traces back to this one term.

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7. Common Mistakes That Cost Marks

Avoid these

1. Omitting the minus sign in Faraday’s law — it is Lenz’s law, and it is worth a mark on its own.
2. Calling displacement current a “real current”. It is explicitly fictitious — no charge moves. Say so.
3. Skipping the contradiction step. The whole point of the derivation is that $\nabla \cdot (\nabla \times \vec{H})\equiv 0$ clashes with continuity. If you jump straight to ${\vec{J}}_d=\partial \vec{D}/\partial t$ you lose most of the marks.
4. Forgetting to prove $i_d=i_c$ in the capacitor problem. The equality is the answer; the algebra before it is only setup.
5. Confusing transformer EMF with motional EMF. Transformer = loop still, field changes. Motional = field still, loop moves.

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8. PYQ Bank — Verbatim Questions & Answer Plans

Q1 — Displacement current, definition + capacitor [PYQ: 2016, 2017 — 07/09 Marks]

“Define displacement current. Determine the displacement current in between two parallel plates of a capacitor energized by an alternating current source.”

Answer plan (2 marked parts):

1. Definition — the boxed definition in Section 3.2. Mention explicitly that it is fictitious and that it exists to reconcile Ampère’s law with continuity.
2. Derivation — all seven steps of Section 4, ending with $i_d=i_c$ and the closing interpretation sentence. Bonus marks: briefly showing the $\nabla \cdot (\nabla \times \vec{H})=0$ contradiction (Section 3.1) demonstrates you know why ${\vec{J}}_d$ exists, not just what it equals.

Q2 — Displacement current density as a short-note term [PYQ: 2023 — part of 09 Marks]

“Briefly discuss the following terms: i) Intrinsic impedance, ii) Complex permittivity, iii) Displacement current density.”

Answer plan: For part (iii) only — 3–4 lines: the definition, the formula ${\vec{J}}_d=\partial \vec{D}/\partial t=ϵ\partial \vec{E}/\partial t$, units A/m², and the one-line reason it was introduced. Parts (i) and (ii) belong to 5.02 Plane Waves in Lossless vs. Lossy Media & Skin Depth Calculations and 1.04 Media Properties & Conductor-Insulator Behaviour.

Q3 — Faraday's law inside the Maxwell deduction [PYQ: 2018, 2019, 2023 — 08/10/13 Marks]

“Derive Maxwell’s equation from fundamental electrostatic and magnetostatic expressions by incorporating Faraday’s law of electromagnetic induction and continuity equation.” (2019 wording)

Answer plan: Faraday’s law (Section 1) and the displacement current derivation (Section 3) are both required as sub-steps. The full four-equation deduction is set out in 4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem §3.

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9. Self-Check Before Moving On

• Can you state Faraday’s law in both integral and differential form, and explain the minus sign?
• Can you name and write the EMF expression for all three induction cases, with a device example for each?
• Can you show, in four lines, exactly why Ampère’s static law contradicts conservation of charge?
• Can you derive ${\vec{J}}_d=\partial \vec{D}/\partial t$ from that contradiction without looking?
• Can you prove $i_d=i_c$ for an AC-driven capacitor?
• Can you explain in one sentence why electromagnetic waves would not exist without the ${\vec{J}}_d$ term?

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Source: 04 time_varying_fields_and_maxwells_equations.md (master dump), ECE 2105 Syllabus Week 9, PYQ bank 2015–2025, L 9.pdf, 01 Static Electric Field.pdf

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📄 Section: 4.03 Maxwell_s Equations (Differential & Integral Forms) & Poynting Theorem

Related Concepts: 4.02 Faraday’s Law of Induction & Maxwell’s Displacement Current | 4.04 Dynamic Boundary Conditions for Electromagnetic Fields | 5.01 Wave Equations & Helmholtz Equations in Source-Free Media | 5.03 Poynting Vector, Power Flow & Normal Incidence Reflection (Gamma, Tau)

4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem

Core Idea

Four equations describe every classical electromagnetic phenomenon that exists — from a capacitor charging to starlight crossing a galaxy. Two of them are unchanged from statics; two of them had to be repaired for time-varying fields. Bolt on Poynting’s theorem and you also know exactly where the energy goes.

This is the highest-frequency topic in the entire course

Maxwell’s equations in some form appear in every single paper from 2015 to 2025, usually worth 8–13 marks. There are three distinct question types and they are not interchangeable — see the routing table below.

If the question says…	It wants…	Section	Years
”Write down the differential and integral form … with their physical significance”	The master table, all three columns	§1, §2	2015, 2021, 2023, 2024, 2025
”… and identify each equation with proper experimental law”	The master table, law-attribution column emphasised	§1, §2	2016, 2017
”Deduce / derive Maxwell’s equations from the fundamental governing equations”	A full four-part derivation, not a table	§3	2018, 2019, 2023
”Write down the significance of Maxwell’s equations”	Physical meaning only, no derivation	§2	2018

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1. Master Table: The Four Maxwell’s Equations

[PYQ: 2015, 2016, 2017, 2021, 2023, 2024, 2025] — Heavily Tested

#	Equation Name	Differential (Point) Form	Integral Form	Experimental Law
1	Gauss’s Law for Electricity	$\nabla \cdot \vec{D}={\rho }_v$	${\oint }_S\vec{D}\cdot d\vec{s}=Q_{enc}$	Gauss’s law / Coulomb’s law
2	Gauss’s Law for Magnetism	$\nabla \cdot \vec{B}=0$	${\oint }_S\vec{B}\cdot d\vec{s}=0$	No isolated magnetic monopole
3	Faraday’s Law of Induction	$\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$	${\oint }_C\vec{E}\cdot d\vec{l}=-\frac{d}{dt}{\int }_S\vec{B}\cdot d\vec{s}$	Faraday’s law of electromagnetic induction
4	Ampère–Maxwell Law	$\nabla \times \vec{H}=\vec{J}+\frac{\partial \vec{D}}{\partial t}$	${\oint }_C\vec{H}\cdot d\vec{l}=I_{enc}+{\int }_S\frac{\partial \vec{D}}{\partial t}\cdot d\vec{s}$	Ampère’s circuital law, as modified by Maxwell

Constitutive relations (needed to close the system in a simple medium — write these too):

$\vec{D}=ϵ\vec{E},\vec{B}=\mu \vec{H},\vec{J}=\sigma \vec{E}$

Memory hook — "two divergences, two curls"

• The divergence pair tells you where field lines begin and end: electric lines start on charge; magnetic lines never start at all.
• The curl pair tells you what makes field lines circulate: each field is stirred by the time-rate-of-change of the other.
• Equations 1 and 2 are identical to their static forms. Only 3 and 4 changed. Saying this explicitly earns marks in the “deduce” question.

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2. Physical Significance of Each Equation

[PYQ: 2015, 2018, 2021, 2023, 2024, 2025] — asked as an explicitly marked sub-part in all six years

Equation 1 — $\nabla \cdot \vec{D}={\rho }_v$

The total outward electric flux through any closed surface equals the total free charge enclosed. Electric field lines originate on positive charge and terminate on negative charge. Electric charge is a genuine source of the electric field.

Equation 2 — $\nabla \cdot \vec{B}=0$

The net magnetic flux through any closed surface is always zero. Magnetic monopoles do not exist — an isolated north pole has never been observed. Magnetic flux lines are always closed loops with no beginning and no end; the magnetic field is solenoidal.

Equation 3 — $\nabla \times \vec{E}=-\partial \vec{B}/\partial t$

A magnetic field that changes with time induces a circulating, non-conservative electric field, producing an EMF around any closed path. The negative sign is Lenz’s law: the induced effect always opposes the change that produced it, which is what makes energy conservation hold.

Equation 4 — $\nabla \times \vec{H}=\vec{J}+\partial \vec{D}/\partial t$

A circulating magnetic field is produced by both a conduction current $\vec{J}$ (moving charge) and a displacement current $\partial \vec{D}/\partial t$ (a changing electric field). The second term is Maxwell’s addition and is what allows fields to propagate through empty space.

The overall significance, in one sentence

Together the four equations state that electric and magnetic fields are not independent entities but two aspects of a single electromagnetic field, coupled through time variation — and that this coupling permits self-sustaining waves travelling at $u=1/\sqrt{\mu ϵ}$.

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3. Master Derivation: Deducing Maxwell’s Equations from the Static Postulates

[PYQ: 2018, 2019, 2023 — 08/10/13 Marks] — Heavily Tested

Read the question carefully

This is not the “write down the table” question. Here you must start from the four static governing equations and show, one at a time, which survive unchanged and which must be modified — and why.

The four static starting points

Field theory	Static governing equations
Electrostatics	$\nabla \cdot \vec{D}={\rho }_v$ and $\nabla \times \vec{E}=0$
Magnetostatics	$\nabla \cdot \vec{B}=0$ and $\nabla \times \vec{H}=\vec{J}$

Postulate I — Gauss’s Law for Electricity: unchanged

Time-varying electric fields still originate and terminate on electric charge; nothing in Faraday’s or Maxwell’s discoveries affects the sources of $\vec{D}$. Hence:

$\nabla \cdot \vec{D}={\rho }_v\text{(Maxwell’s 1st Equation)}$

Postulate II — Gauss’s Law for Magnetism: unchanged

No experiment has ever isolated a magnetic monopole, in static or dynamic conditions. Magnetic flux lines remain closed loops. Hence:

$\nabla \cdot \vec{B}=0\text{(Maxwell’s 2nd Equation)}$

Postulate III — Faraday’s Law: modifies $\nabla \times \vec{E}=0$

In electrostatics the field is conservative: $\nabla \times \vec{E}=0$. Faraday’s experiment showed that a changing magnetic flux induces an EMF around a closed loop:

$\text{EMF}={\oint }_C\vec{E}\cdot d\vec{l}=-\frac{d\Phi }{dt}$

Write the flux as a surface integral, $\Phi ={\int }_S\vec{B}\cdot d\vec{s}$:

${\oint }_C\vec{E}\cdot d\vec{l}=-\frac{d}{dt}{\int }_S\vec{B}\cdot d\vec{s}$

For a stationary loop, take the derivative inside as a partial derivative:

${\oint }_C\vec{E}\cdot d\vec{l}=-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}$

Apply Stokes’s Theorem to the left-hand side:

${\int }_S(\nabla \times \vec{E})\cdot d\vec{s}=-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}$

Since this holds for arbitrary $S$, equate the integrands:

$\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}\text{(Maxwell’s 3rd Equation)}$

Postulate IV — Ampère’s Law: modifies $\nabla \times \vec{H}=\vec{J}$

Take the divergence of the static law and use the null identity $\nabla \cdot (\nabla \times \vec{H})\equiv 0$:

$0=\nabla \cdot \vec{J}$

This contradicts the continuity equation $\nabla \cdot \vec{J}=-\partial {\rho }_v/\partial t$ for time-varying fields. Introduce a displacement current density ${\vec{J}}_d$:

$\nabla \times \vec{H}=\vec{J}+{\vec{J}}_d$

Take the divergence again:

$0=\nabla \cdot \vec{J}+\nabla \cdot {\vec{J}}_d=-\frac{\partial {\rho }_v}{\partial t}+\nabla \cdot {\vec{J}}_d⟹\nabla \cdot {\vec{J}}_d=\frac{\partial {\rho }_v}{\partial t}$

Substitute Gauss’s law ${\rho }_v=\nabla \cdot \vec{D}$ and exchange the order of the derivatives:

$\nabla \cdot {\vec{J}}_d=\frac{\partial }{\partial t}(\nabla \cdot \vec{D})=\nabla \cdot \left(\frac{\partial \vec{D}}{\partial t}\right)⟹{\vec{J}}_d=\frac{\partial \vec{D}}{\partial t}$

$\nabla \times \vec{H}=\vec{J}+\frac{\partial \vec{D}}{\partial t}\text{(Maxwell’s 4th Equation)}$

graph TD
    S1["ELECTROSTATICS<br/>∇·D = ρᵥ"] -->|"unchanged"| M1["① ∇·D = ρᵥ"]
    S2["MAGNETOSTATICS<br/>∇·B = 0"] -->|"unchanged"| M2["② ∇·B = 0"]
    S3["ELECTROSTATICS<br/>∇×E = 0"] -->|"+ Faraday's Law<br/>+ Stokes's Theorem"| M3["③ ∇×E = −∂B/∂t"]
    S4["MAGNETOSTATICS<br/>∇×H = J"] -->|"+ Continuity Equation<br/>+ Displacement Current"| M4["④ ∇×H = J + ∂D/∂t"]
    M1 --> W["MAXWELL'S EQUATIONS<br/>complete electrodynamics"]
    M2 --> W
    M3 --> W
    M4 --> W

Scoring structure for this question

Marks are allocated roughly 1 + 1 + 4 + 5: one each for correctly asserting that equations 1 and 2 are unchanged (with justification), four for the Faraday modification, five for the Ampère modification. Students who only derive equation 4 typically score under half.

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4. Poynting’s Theorem & Electromagnetic Power Flow

4.1 Statement

[PYQ: 2015, 2023 — 09/10 Marks]

Poynting's Theorem

Poynting’s theorem is the statement of conservation of energy for the electromagnetic field. It says that the net electromagnetic power flowing out of a closed surface equals the rate of decrease of the electromagnetic energy stored inside that volume, minus the power dissipated as ohmic heat within it.

4.2 Step-by-Step Derivation

[PYQ: 2016, 2022, 2024 — 10/13 Marks] — asked as “Find the equation of the total power flowing in a closed surface due to electromagnetic waves at any instant.”

Start from Maxwell’s two curl equations in a simple medium:

\nabla \times \vec{E} = -\mu\frac{\partial\vec{H}}{\partial t} \tag{1} \nabla \times \vec{H} = \vec{J} + \epsilon\frac{\partial\vec{E}}{\partial t} = \sigma\vec{E} + \epsilon\frac{\partial\vec{E}}{\partial t} \tag{2}

Use the vector identity:

\nabla \cdot (\vec{E} \times \vec{H}) = \vec{H}\cdot(\nabla\times\vec{E}) - \vec{E}\cdot(\nabla\times\vec{H}) \tag{3}

Step 1 — dot (1) with $\vec{H}$:

$\vec{H}\cdot (\nabla \times \vec{E})=-\mu \vec{H}\cdot \frac{\partial \vec{H}}{\partial t}$

Using $\vec{H}\cdot \frac{\partial \vec{H}}{\partial t}=\frac{1}{2}\frac{\partial }{\partial t}(\vec{H}\cdot \vec{H})=\frac{1}{2}\frac{\partial H^2}{\partial t}$:

\vec{H}\cdot(\nabla\times\vec{E}) = -\frac{\partial}{\partial t}\left(\frac{1}{2}\mu H^2\right) \tag{4}

Step 2 — dot (2) with $\vec{E}$:

\vec{E}\cdot(\nabla\times\vec{H}) = \vec{E}\cdot\vec{J} + \epsilon\,\vec{E}\cdot\frac{\partial\vec{E}}{\partial t} = \vec{J}\cdot\vec{E} + \frac{\partial}{\partial t}\left(\frac{1}{2}\epsilon E^2\right) \tag{5}

Step 3 — substitute (4) and (5) into (3):

$\nabla \cdot (\vec{E}\times \vec{H})=-\vec{J}\cdot \vec{E}-\frac{\partial }{\partial t}\left(\frac{1}{2}ϵE^2+\frac{1}{2}\mu H^2\right)$

Step 4 — integrate over a closed volume $V$:

${\int }_V\nabla \cdot (\vec{E}\times \vec{H})dv=-{\int }_V\vec{J}\cdot \vec{E}dv-\frac{\partial }{\partial t}{\int }_V\left(\frac{1}{2}ϵE^2+\frac{1}{2}\mu H^2\right)dv$

Step 5 — apply the Divergence Theorem to the left-hand side:

Poynting's Theorem — Integral Form

$\boxed{{\oint }_S(\vec{E}\times \vec{H})\cdot d\vec{s}=-\frac{\partial }{\partial t}{\int }_V\left(\frac{1}{2}ϵE^2+\frac{1}{2}\mu H^2\right)dv-{\int }_V\sigma E^{2}dv}$

Differential (point) form:

$-\nabla \cdot \vec{\mathcal{P}}=\vec{J}\cdot \vec{E}+\frac{\partial w_e}{\partial t}+\frac{\partial w_m}{\partial t}$

4.3 Physical Interpretation of Each Term

Term	Physical meaning	Units
${\oint }_S(\vec{E}\times \vec{H})\cdot d\vec{s}$	Total electromagnetic power flowing out of the closed volume through $S$	W
$-\frac{\partial }{\partial t}{\int }_V\left(\frac{1}{2}ϵE^2+\frac{1}{2}\mu H^2\right)dv$	Rate of decrease of stored electric + magnetic energy inside $V$	W
${\int }_V\sigma E^{2}dv$	Ohmic power dissipated as heat inside $V$	W

In words: power out = (energy released from storage) − (energy burnt as heat).

4.4 The Poynting Vector

Definitions

Instantaneous Poynting vector: $\vec{\mathcal{P}}=\vec{E}\times \vec{H}[{\text{W/m}}^2]$ It is the electromagnetic power density crossing unit area at a given instant. Its direction is the direction of power flow, which for a uniform plane wave is the direction of propagation.

Time-average Poynting vector (for time-harmonic fields): ${\vec{\mathcal{P}}}_{av}=\frac{1}{2}\text{Re}\left\{{\vec{E}}_s\times {\vec{H}}_s^{\ast }\right\}[{\text{W/m}}^2]$ where ${\vec{H}}_s^{\ast }$ is the complex conjugate of the magnetic field phasor.

Key Exam Checkpoint — three classic sign/factor traps

1. Order matters: $\vec{\mathcal{P}}=\vec{E}\times \vec{H}$, never $\vec{H}\times \vec{E}$. Swapping reverses the direction of power flow.
2. The factor of $\frac{1}{2}$ in the time-average formula comes from $⟨{\cos }^2\omega t⟩=\frac{1}{2}$. Forgetting it is the single most common numerical error.
3. Closed vs. open integral: total power leaving a volume needs ${\oint }_S$; power crossing a flat aperture or cross-section needs ${\int }_S$.

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5. High-Yield Derivation: Poynting Vector on a DC Current-Carrying Wire

[PYQ: 2017 — 08 Marks] — “Find the Poynting vector on the surface of a long straight conducting wire (of radius $b$ and conductivity $\sigma$) that carries a direct current $I$. Also verify Poynting’s theorem.”

This problem is beautiful because it delivers a counter-intuitive result: the energy that heats a wire does not travel along the wire — it flows into it radially from the surrounding space.

[FIGURE: Cylindrical wire of length L and radius b carrying DC current I along +z. Show E directed axially (+â_z) inside the wire, H wrapping circumferentially (â_φ) around the surface, and the resulting Poynting vector P pointing radially INWARD (−â_r) into the wire. — source: Cheng 2nd Ed., Fig. 7-8 / "01 Static Electric Field.pdf"]

Step 1 — uniform current density inside the wire:

$\vec{J}={\hat{a}}_z\frac{I}{\pi b^2}$

Step 2 — electric field from Ohm’s law $\vec{J}=\sigma \vec{E}$:

$\vec{E}={\hat{a}}_z\frac{I}{\sigma \pi b^2}$

Since the wire resistance is $R=\frac{L}{\sigma \pi b^2}$, this can be written compactly as:

$\vec{E}={\hat{a}}_z\frac{IR}{L}$

Step 3 — magnetic field at the surface ($r=b$) from Ampère’s law:

$\vec{H}={\hat{a}}_{\varphi }\frac{I}{2\pi b}$

Step 4 — form the Poynting vector, using ${\hat{a}}_z\times {\hat{a}}_{\varphi }=-{\hat{a}}_r$:

$\vec{\mathcal{P}}=\vec{E}\times \vec{H}=\left({\hat{a}}_z\frac{IR}{L}\right)\times \left({\hat{a}}_{\varphi }\frac{I}{2\pi b}\right)=-{\hat{a}}_r\frac{I^{2}R}{2\pi bL}$

Interpreting the minus sign

$-{\hat{a}}_r$ means the power flows radially inward, from the surrounding electromagnetic field into the wire’s surface. Energy is not conducted along the copper — it is delivered through the space around it and absorbed at the surface. The wire is a sink of electromagnetic energy, not a pipe for it.

Step 5 — integrate over the closed cylindrical surface of length $L$ to get the total power entering:

$P=-{\oint }_S\vec{\mathcal{P}}\cdot d\vec{s}=-{\int }_0^L{\int }_0^{2\pi }\left(-{\hat{a}}_r\frac{I^{2}R}{2\pi bL}\right)\cdot \left({\hat{a}}_{r}bd\varphi dz\right)$

$P=\frac{I^{2}R}{2\pi bL}\cdot (L)(2\pi b)=I^{2}R$

Verification

$\boxed{P=I^{2}R}$ The total electromagnetic power entering the wire through its surface is exactly the ohmic power dissipated as heat inside it. Poynting’s theorem is verified. (Note that $\partial /\partial t=0$ here because the current is steady, so the energy-storage term vanishes and all incoming power goes to heat.)

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6. Common Mistakes That Cost Marks

Avoid these

1. Answering the “deduce” question with the table. §3 and §1 are different questions with different mark schemes. Read the verb.
2. Writing only $\vec{J}$ in Ampère’s law in a time-varying context. The $\partial \vec{D}/\partial t$ term is non-negotiable here.
3. Omitting the constitutive relations $\vec{D}=ϵ\vec{E}$, $\vec{B}=\mu \vec{H}$, $\vec{J}=\sigma \vec{E}$. Maxwell’s four equations alone are underdetermined.
4. Skipping the physical-significance column. It is a separately marked part in six of the last eleven papers.
5. Writing $\vec{H}\times \vec{E}$ or dropping the $\frac{1}{2}$ in the time-average Poynting vector.
6. In the wire problem, forgetting to explain the inward direction. The $-{\hat{a}}_r$ interpretation is where the “verify” marks live.

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7. PYQ Bank — Verbatim Questions & Answer Plans

Q1 — Forms + significance [PYQ: 2015, 2021, 2023, 2024, 2025 — 08/09/10 Marks]

“Write down the differential and integral form of Maxwell’s equations with their physical significance.”

Answer plan: Reproduce the §1 master table with all vector arrows and correct differential elements ($d\vec{s}$, $d\vec{l}$), then write the four significance paragraphs of §2. Add the constitutive relations at the end.

Q2 — Forms + experimental law attribution [PYQ: 2016, 2017 — 07/08 Marks]

“Write down the differential form and integral form of the Maxwell’s equations and identify each equation with proper experimental law.”

Answer plan: Same table, but the fourth column is the answer — name each law explicitly: (1) Gauss’s law, (2) Gauss’s law for magnetism / non-existence of monopoles, (3) Faraday’s law of electromagnetic induction, (4) Ampère’s circuital law as modified by Maxwell.

Q3 — Deduction from static postulates [PYQ: 2018, 2023 — 08/13 Marks]

“Deduce Maxwell’s equations from the four fundamental governing equations of electrostatics and magnetostatics.”

Answer plan: The full §3 derivation. Begin by listing the four static equations, state which two survive unchanged and why, then perform both modifications in full.

Q4 — Deduction, Faraday + continuity wording [PYQ: 2019 — 10 Marks]

“Derive Maxwell’s equation from fundamental electrostatic and magnetostatic expressions by incorporating Faraday’s law of electromagnetic induction and continuity equation.”

Answer plan: Identical to Q3, but the wording explicitly names the two ingredients — so make Faraday’s law (Postulate III) and the continuity equation (Postulate IV) the visible backbone of the answer.

Q5 — Significance only [PYQ: 2018 — 05 Marks]

“Write down the significance of Maxwell’s equations.”

Answer plan: §2 only — four short paragraphs plus the closing “one sentence” summary. No derivation needed at 5 marks.

Q6 — Poynting's theorem, state & explain [PYQ: 2015, 2023 — 09/10 Marks]

“State and explain Poynting’s theorem with necessary equations. Also denotes the pointing vector $\mathcal{P}$.” (2023 wording; 2015 omits the last clause)

Answer plan: Statement (§4.1) → integral-form equation (§4.2 result) → term-by-term interpretation table (§4.3) → define $\vec{\mathcal{P}}=\vec{E}\times \vec{H}$ in W/m² with its directional meaning (§4.4).

Q7 — Total power flowing through a closed surface [PYQ: 2016, 2022, 2024 — 10/13 Marks]

“Find the equation of the total power flowing in a closed surface due to electromagnetic waves at any instant.”

Answer plan: This is asking for the full derivation, not the statement. Write out all five steps of §4.2 with the vector identity named explicitly, then interpret each of the three terms.

Q8 — Poynting vector on a DC wire [PYQ: 2017 — 08 Marks]

“Find the Poynting vector on the surface of a long straight conducting wire (of radius b and conductivity σ) that carries a direct current I as shown in Fig. 8(b). Also verify Poynting’s theorem.”

Answer plan: All five steps of §5. The two mark-critical items are (a) explaining the $-{\hat{a}}_r$ inward direction, and (b) closing with $P=I^{2}R$ and stating explicitly that this equals the ohmic dissipation.

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8. Self-Check Before Moving On

• Can you write all four equations in both forms, from memory, with correct vector notation?
• Can you give the physical significance of each in two sentences?
• Can you say which two are unchanged from statics, and justify why?
• Can you perform the full four-part deduction from the static postulates?
• Can you derive Poynting’s theorem, naming the vector identity and the divergence theorem at the right steps?
• Can you explain why power flows into a resistive wire radially rather than along it?
• Do you know the dynamic boundary conditions? → if not, go to 4.04 Dynamic Boundary Conditions for Electromagnetic Fields

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Source: 04 time_varying_fields_and_maxwells_equations.md (master dump), ECE 2105 Syllabus Week 9 & 13, PYQ bank 2015–2025, Cheng Ch. 7, 01 Static Electric Field.pdf

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📄 Section: 4.04 Dynamic Boundary Conditions for Electromagnetic Fields

Related Concepts: 4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem | 2.04 Electrostatic Boundary Conditions, Refraction Law & Poisson-Laplace Equations | 3.04 Magnetization, Magnetic Materials, Boundary Conditions & Hall Effect | 5.03 Poynting Vector, Power Flow & Normal Incidence Reflection (Gamma, Tau)

4.04 Dynamic Boundary Conditions for Electromagnetic Fields

Core Idea

Maxwell’s equations in differential form only work where the medium is smooth. At an interface between two different materials, the field derivatives blow up, so we must fall back on the integral forms and shrink a loop and a pillbox down onto the surface. The startling result: the boundary conditions for time-varying fields are algebraically identical to the static ones. Every time-derivative term dies in the limit.

Heavily Tested — and it was missing from your notes

Dynamic boundary conditions appear in 2018, 2019, 2020, 2021, 2022, 2023 and 2024, typically for 8–10 marks. The “derive for (i) two lossless media and (ii) dielectric–perfect conductor” phrasing alone accounts for five of those years.

These are also the engine of Chapter 5

The reflection and transmission coefficients $\Gamma$ and $\tau$ in 5.03 Poynting Vector, Power Flow & Normal Incidence Reflection (Gamma, Tau) are obtained by applying exactly these boundary conditions at $z=0$. Learning them here means Chapter 5 costs you almost nothing.

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1. Why Boundary Conditions Are Needed At All

The differential (point) forms of Maxwell’s equations involve spatial derivatives of $\vec{E}$, $\vec{D}$, $\vec{H}$ and $\vec{B}$. At an interface, material properties ($ϵ,\mu ,\sigma$) jump discontinuously, so the fields themselves may jump and their derivatives become undefined.

The fix: apply the integral forms to a small geometry that straddles the interface, then shrink the geometry onto the surface.

• Tangential components ← use a thin rectangular loop (Faraday’s law, Ampère’s law)
• Normal components ← use a thin cylindrical pillbox (both Gauss’s laws)

[FIGURE: Interface between Medium 1 (ε₁, μ₁, σ₁) and Medium 2 (ε₂, μ₂, σ₂). Show (a) a thin rectangular loop abcd of tangential width Δw and normal height Δh straddling the boundary, and (b) a thin cylindrical pillbox of face area ΔA and height Δh straddling the boundary. Mark the unit normal â_n2 pointing from medium 2 into medium 1. — source: Cheng 2nd Ed., Figs. 7-11 and 7-12 / "01 Static Electric Field.pdf"]

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2. Master Proof: Why the Dynamic Conditions Equal the Static Ones

[PYQ: 2020 — 10 Marks] — “Why the boundary conditions for electromagnetic fields are same to the boundary conditions for static electric and static magnetic field.”

This is a purely conceptual question and the answer is a scaling argument.

The argument for Faraday’s law

Take the integral form around the rectangular loop $abcd$:

${\oint }_C\vec{E}\cdot d\vec{l}=-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}$

1. Let the loop have tangential width $\Delta w$ (along the interface) and normal height $\Delta h$ (across it).
2. The area enclosed by the loop is $S=\Delta w\cdot \Delta h$.
3. Now shrink the loop onto the boundary: $\Delta h\to 0$ while $\Delta w$ stays finite.
4. The enclosed area collapses: $S=\Delta w\Delta h\to 0$.
5. In any physically real material, $\vec{B}$ and $\partial \vec{B}/\partial t$ are finite. A finite integrand over a vanishing area gives a vanishing integral:

${\lim }_{\Delta h\to 0}\left(-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}\right)=0$

6. The dynamic term therefore disappears and Faraday’s law degenerates to the electrostatic conservative form:

${\oint }_C\vec{E}\cdot d\vec{l}=0⟹E_{1t}=E_{2t}$

The identical argument for Ampère’s law

${\lim }_{\Delta h\to 0}\left({\int }_S\frac{\partial \vec{D}}{\partial t}\cdot d\vec{s}\right)=0$

so Ampère–Maxwell reduces to the magnetostatic form ${\oint }_C\vec{H}\cdot d\vec{l}=I_{enc}$.

The answer in one paragraph

Boundary conditions are obtained by contracting an integration loop (or pillbox) onto the interface. In that limit the enclosed area and volume both go to zero. Since $\partial \vec{B}/\partial t$ and $\partial \vec{D}/\partial t$ remain finite in any real medium, the time-derivative terms — which are the only difference between the static and dynamic Maxwell equations — integrate to zero. What survives depends solely on instantaneous field values and local surface charge/current densities, which are the same quantities that appear in statics. Hence the dynamic boundary conditions are algebraically identical to the static ones.

Key Exam Checkpoint

The mark-critical phrase is ”$\partial \vec{B}/\partial t$ and $\partial \vec{D}/\partial t$ remain finite while the enclosed area tends to zero.” Without the word finite, the argument is not rigorous — you must rule out a delta-function field.

graph TD
    A["Maxwell's integral forms"] --> B["Draw loop Δw × Δh<br/>and pillbox ΔA × Δh<br/>across the interface"]
    B --> C["Shrink Δh → 0"]
    C --> D["Enclosed area → 0<br/>Enclosed volume → 0"]
    D --> E["∂B/∂t and ∂D/∂t finite<br/>⟹ their integrals → 0"]
    E --> F["Only instantaneous field values<br/>and surface ρs, Js survive"]
    F --> G["DYNAMIC BCs ≡ STATIC BCs"]

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3. The Four General Boundary Conditions

[PYQ: 2018, 2019, 2021, 2022, 2023, 2024] — Heavily Tested

Let ${\hat{a}}_{n2}$ be the unit normal directed from medium 2 into medium 1.

3.1 Tangential Electric Field $E_t$ — from Faraday’s law

Around loop $abcd$ with $\Delta h\to 0$, the two short sides contribute nothing:

$E_{1t}\Delta w-E_{2t}\Delta w=0$

$\boxed{E_{1t}=E_{2t}}\text{vector form: }{\hat{a}}_{n2}\times ({\vec{E}}_1-{\vec{E}}_2)=0$

Meaning: the tangential electric field is always continuous across any interface, without exception.

3.2 Normal Electric Flux Density $D_n$ — from Gauss’s law

Over the pillbox with $\Delta h\to 0$, the curved side wall contributes nothing:

$D_{1n}\Delta A-D_{2n}\Delta A={\rho }_s\Delta A$

$\boxed{D_{1n}-D_{2n}={\rho }_s}\text{vector form: }{\hat{a}}_{n2}\cdot ({\vec{D}}_1-{\vec{D}}_2)={\rho }_s$

Meaning: the normal component of $\vec{D}$ jumps by exactly the free surface charge density ${\rho }_s$.

3.3 Tangential Magnetic Field $H_t$ — from Ampère’s law

$H_{1t}\Delta w-H_{2t}\Delta w=I_{enc}=J_s\Delta w$

$\boxed{H_{1t}-H_{2t}=J_s}\text{vector form: }{\hat{a}}_{n2}\times ({\vec{H}}_1-{\vec{H}}_2)={\vec{J}}_s$

Meaning: the tangential $\vec{H}$ jumps by the free surface current density ${\vec{J}}_s$. Since a genuine surface current sheet requires infinite conductivity, ${\vec{J}}_s=0$ for all real media — so tangential $\vec{H}$ is continuous everywhere except at a perfect conductor.

3.4 Normal Magnetic Flux Density $B_n$ — from Gauss’s law for magnetism

$B_{1n}\Delta A-B_{2n}\Delta A=0$

$\boxed{B_{1n}=B_{2n}}\text{vector form: }{\hat{a}}_{n2}\cdot ({\vec{B}}_1-{\vec{B}}_2)=0$

Meaning: the normal component of $\vec{B}$ is always continuous — a direct consequence of there being no magnetic monopoles to terminate flux on.

3.5 Summary Table

Component	Boundary condition	Vector form	Source equation	Always continuous?
Tangential $\vec{E}$	$E_{1t}=E_{2t}$	${\hat{a}}_{n2}\times ({\vec{E}}_1-{\vec{E}}_2)=0$	Faraday	Yes
Normal $\vec{D}$	$D_{1n}-D_{2n}={\rho }_s$	${\hat{a}}_{n2}\cdot ({\vec{D}}_1-{\vec{D}}_2)={\rho }_s$	Gauss (E)	Only if ${\rho }_s=0$
Tangential $\vec{H}$	$H_{1t}-H_{2t}=J_s$	${\hat{a}}_{n2}\times ({\vec{H}}_1-{\vec{H}}_2)={\vec{J}}_s$	Ampère–Maxwell	Only if ${\vec{J}}_s=0$
Normal $\vec{B}$	$B_{1n}=B_{2n}$	${\hat{a}}_{n2}\cdot ({\vec{B}}_1-{\vec{B}}_2)=0$	Gauss (B)	Yes

Memory hook

“Tangential E and normal B are always continuous; the other two jump by whatever free surface source is present.”

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4. Case A — Interface Between Two Lossless Media

[PYQ: 2018, 2021, 2022, 2023, 2024] — part (i) of the standard question

A lossless medium has zero conductivity: ${\sigma }_1={\sigma }_2=0$. With no conductivity there are no free charges or free currents available to accumulate on the surface:

${\rho }_s=0\text{and}{\vec{J}}_s=0$

Substituting into the four general conditions:

Component	Condition	Expanded in terms of $ϵ$, $\mu$
Tangential $\vec{E}$	$E_{1t}=E_{2t}$	—
Tangential $\vec{H}$	$H_{1t}=H_{2t}$	—
Normal $\vec{D}$	$D_{1n}=D_{2n}$	${ϵ}_1{E}_{1n}={ϵ}_2{E}_{2n}$
Normal $\vec{B}$	$B_{1n}=B_{2n}$	${\mu }_1{H}_{1n}={\mu }_2{H}_{2n}$

Result

All four field components are continuous across a lossless–lossless interface. The tangential components pass through unchanged; the normal components pass through with their flux conserved, so the field intensities $E_n$ and $H_n$ scale inversely with $ϵ$ and $\mu$ respectively.

Why this matters in Chapter 5

These are precisely the two conditions ($E_{1t}=E_{2t}$ and $H_{1t}=H_{2t}$) used at $z=0$ to derive $\Gamma =\frac{{\eta }_2-{\eta }_1}{{\eta }_2+{\eta }_1}$ and $\tau =\frac{2{\eta }_2}{{\eta }_2+{\eta }_1}$.

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5. Case B — Interface Between a Dielectric and a Perfect Conductor

[PYQ: 2018, 2021, 2022, 2023, 2024] — part (ii) of the standard question

A perfect electric conductor (PEC) has infinite conductivity, ${\sigma }_2\to \infty$.

Why all fields vanish inside a perfect conductor

If ${\vec{E}}_2$ were non-zero inside a medium with ${\sigma }_2\to \infty$, Ohm’s law $\vec{J}=\sigma \vec{E}$ would give infinite current density and infinite power dissipation $\sigma E^2$ — physically impossible. Therefore ${\vec{E}}_2=0$. Faraday’s law $\nabla \times \vec{E}=-\partial \vec{B}/\partial t$ then forces $\partial {\vec{B}}_2/\partial t=0$; since a time-varying field cannot have a static residue, ${\vec{B}}_2=0$ as well. ${\vec{E}}_2={\vec{D}}_2={\vec{H}}_2={\vec{B}}_2=0$

Substituting these zeros (medium 1 = dielectric, medium 2 = PEC):

Component	General condition	Reduces to	Physical meaning
Tangential $\vec{E}$	$E_{1t}=E_{2t}$	$\boxed{E_{1t}=0}$	$\vec{E}$ must strike a conductor perpendicularly; no tangential field can survive on its surface.
Normal $\vec{D}$	$D_{1n}-D_{2n}={\rho }_s$	$\boxed{D_{1n}={\rho }_s}$	All normal electric flux terminates on induced surface charge.
Tangential $\vec{H}$	${\hat{a}}_{n2}\times ({\vec{H}}_1-{\vec{H}}_2)={\vec{J}}_s$	$\boxed{{\hat{a}}_{n2}\times {\vec{H}}_1={\vec{J}}_s\Rightarrow H_{1t}=J_s}$	A surface current sheet flows on the conductor, numerically equal to the tangential $\vec{H}$.
Normal $\vec{B}$	$B_{1n}=B_{2n}$	$\boxed{B_{1n}=0}$	No magnetic flux can penetrate a perfect conductor; $\vec{B}$ must lie tangential to its surface.

The PEC picture in one line

At a perfect conductor: $\vec{E}$ is purely normal, $\vec{B}$ is purely tangential, and both are supported by induced surface charge ${\rho }_s$ and surface current ${\vec{J}}_s$.

graph LR
    subgraph "Case A: Lossless / Lossless"
    A1["σ₁ = σ₂ = 0<br/>⟹ ρs = 0, Js = 0"] --> A2["E1t = E2t<br/>H1t = H2t<br/>ε₁E1n = ε₂E2n<br/>μ₁H1n = μ₂H2n"]
    end
    subgraph "Case B: Dielectric / Perfect Conductor"
    B1["σ₂ → ∞<br/>⟹ E₂ = H₂ = 0"] --> B2["E1t = 0<br/>D1n = ρs<br/>H1t = Js<br/>B1n = 0"]
    end

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6. Comparison With the Static Boundary Conditions

	Electrostatics (2.04 Electrostatic Boundary Conditions, Refraction Law & Poisson-Laplace Equations)	Magnetostatics (3.04 Magnetization, Magnetic Materials, Boundary Conditions & Hall Effect)	Dynamic (this note)
Tangential	$E_{1t}=E_{2t}$	$H_{1t}-H_{2t}=J_s$	Identical to both
Normal	$D_{1n}-D_{2n}={\rho }_s$	$B_{1n}=B_{2n}$	Identical to both
Difference	—	—	None. Only the proof changes (the time-derivative terms must be shown to vanish).

Exam efficiency

If you already know the electrostatic and magnetostatic boundary conditions, you have already memorised the dynamic ones. The only new material in this note is the scaling proof of §2 and the two special-case reductions of §4 and §5.

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7. Common Mistakes That Cost Marks

Avoid these

1. Writing $E_{1n}=E_{2n}$ instead of $D_{1n}=D_{2n}$. It is the flux density that is continuous, not the field intensity. The intensities differ by the ratio ${ϵ}_2/{ϵ}_1$.
2. Writing $B_{1t}=B_{2t}$ instead of $H_{1t}=H_{2t}$. Tangential continuity belongs to $\vec{H}$, normal continuity belongs to $\vec{B}$. Mixing them up is the single most common error here.
3. Forgetting to justify ${\rho }_s=0,{\vec{J}}_s=0$ in the lossless case. The justification is $\sigma =0$ — say it.
4. Not justifying why all fields vanish inside a PEC. The Ohm’s-law argument in §5 is worth a mark.
5. Giving only the scalar forms. Several papers ask for “boundary equations for both electric field vectors and magnetic field vectors” — write the ${\hat{a}}_{n2}\times$ and ${\hat{a}}_{n2}\cdot$ vector forms too.
6. Omitting the direction convention. Always state which way ${\hat{a}}_{n2}$ points; otherwise the signs in your answer are meaningless.

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8. PYQ Bank — Verbatim Questions & Answer Plans

Q1 — Boundary equations + two special cases [PYQ: 2018, 2021, 2022, 2023, 2024 — 08/09/10 Marks] Heavily Tested

“Write down the boundary equations for both electric field vectors and magnetic field vectors. Hence derive (i) boundary conditions between two lossless media and (ii) boundary conditions between a dielectric media and perfect conductor.”

Answer plan (3 marked parts):

1. General equations — all four conditions of §3, in both scalar and vector form, with the loop/pillbox figure and the ${\hat{a}}_{n2}$ convention stated.
2. Part (i) — set ${\sigma }_1={\sigma }_2=0\Rightarrow {\rho }_s=0,{\vec{J}}_s=0$, then present the §4 table. State the conclusion: all four components continuous.
3. Part (ii) — justify ${\vec{E}}_2={\vec{H}}_2=0$ inside the PEC via Ohm’s law, then present the §5 table with the four boxed results and their physical meanings.

Q2 — Why dynamic = static [PYQ: 2020 — 10 Marks]

“Why the boundary conditions for electromagnetic fields are same to the boundary conditions for static electric and static magnetic field.”

Answer plan: The §2 scaling proof. Show the limit explicitly for both Faraday’s law and Ampère’s law, emphasise that $\partial \vec{B}/\partial t$ and $\partial \vec{D}/\partial t$ remain finite while the enclosed area vanishes, and close with the §2 summary paragraph.

Q3 — Short form [PYQ: 2019 — 06 Marks]

“Write down the boundary conditions between two electromagnetic medium.”

Answer plan: At 6 marks, the §3.5 summary table plus the four vector forms is sufficient. Add one line noting they are identical to the static conditions.

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9. Self-Check Before Moving On

• Can you draw the loop and the pillbox and say which equation each is used with?
• Can you prove that the time-derivative terms vanish as $\Delta h\to 0$, for both Faraday and Ampère?
• Can you write all four boundary conditions in scalar and vector form, with the normal convention stated?
• Can you say instantly which two components are always continuous?
• Can you reduce the general set to the lossless–lossless case, justifying ${\rho }_s={\vec{J}}_s=0$?
• Can you justify why $\vec{E}=\vec{H}=0$ inside a perfect conductor, and give all four PEC conditions?
• Can you see how $E_{1t}=E_{2t}$ and $H_{1t}=H_{2t}$ will produce $\Gamma$ and $\tau$ in Chapter 5?

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Source: 04 time_varying_fields_and_maxwells_equations.md §4 (master dump), ECE 2105 Syllabus Week 11, PYQ bank 2015–2025, Cheng Ch. 7-7, 01 Static Electric Field.pdf

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📄 Section: 00 Chapter 4 Active-Recall Diagnostic Quiz

00 Chapter 4 Active-Recall Diagnostic Quiz (Time-Varying Fields & Maxwell’s Equations)

> [!abstract] Overview

> Test your conceptual understanding and mathematical recall of charge conservation, continuity equation, Faraday’s law of induction, displacement current density J_d, complete Maxwell’s equations, and Poynting’s power flow theorem.

Question 1: Continuity Equation Differential Form

State the differential form of the continuity equation for electric current.

> [!faq]- Solution

> ∇ · J = -∂ρ_v / ∂t

> The divergence of current density equals the negative rate of change of volume charge density.

Question 2: Steady Current Case

What does the continuity equation reduce to for steady (direct) currents (∂ρ_v / ∂t = 0), and what physical law does it represent?

> [!faq]- Solution

> ∇ · J = 0

> It shows that steady electric currents are solenoidal (no sources/sinks) and represents Kirchhoff’s Current Law (KCL) in field theory (Σ I = 0).

Question 3: Faraday’s Law Integral Form

Write the integral form of Faraday’s Law of Electromagnetic Induction.

> [!faq]- Solution

> e = ∮_C E · dl = -d/dt ∫_S B · ds = -dΦ/dt

Question 4: Three Forms of Induced EMF

List the three physical mechanisms of induced EMF depending on whether the loop or magnetic field is moving/time-varying.

> [!faq]- Solution

> 1. Transformer EMF: Stationary loop in time-varying magnetic field (e_trans = -∫_S ∂B / ∂t · ds).

> 2. Motional EMF: Moving loop in static magnetic field (e_motion = ∮_C (u × B) · dl).

> 3. Combined EMF: Moving loop in time-varying magnetic field (e_total = e_trans + e_motion).

Question 5: Incompleteness of Ampere’s Static Law

Why is Ampere’s static curl equation ∇ × H = J mathematically incomplete for time-varying fields?

> [!faq]- Solution

> Taking the divergence yields ∇ · (∇ × H) = 0. But by continuity, ∇ · J = -∂ρ_v / ∂t ≠ 0 for dynamic fields, creating a mathematical contradiction unless displacement current J_d is added.

Question 6: Displacement Current Density Formula

Write the formula for displacement current density J_d in terms of electric flux density D and electric field E.

> [!faq]- Solution

> J_d = ∂D / ∂t = ε ∂E / ∂t [A/m^2]

Question 7: Maxwell’s Equations in Differential Form

List the four Maxwell’s equations in differential form for time-varying electromagnetic fields in a simple medium (ε, μ).

> [!faq]- Solution

> 1. ∇ · E = ρ_v / ε (Gauss’s Law for Electric Field)

> 2. ∇ · B = 0 (Gauss’s Law for Magnetic Field)

> 3. ∇ × E = -μ ∂H / ∂t (Faraday’s Law)

> 4. ∇ × H = J + ε ∂E / ∂t (Ampere-Maxwell Law)

Question 8: Poynting Vector Definition

Define the instantaneous Poynting Vector S and specify its SI units.

> [!faq]- Solution

> S = E × H [Watts/m^2]

> It represents the instantaneous directional power density carried by an electromagnetic wave.

Question 9: Poynting’s Theorem Physical Meaning

State the physical principle expressed by Poynting’s Theorem (∇ · S + ∂w / ∂t = -J · E).

> [!faq]- Solution

> The net outward power flux through a closed surface plus the rate of increase of stored electromagnetic energy inside equals the negative work done by the field on charges (ohmic power dissipation -J · E).

Question 10: Capacitor Conduction vs. Displacement Current

How does displacement current J_d ensure current continuity across a parallel-plate capacitor?

> [!faq]- Solution

> Inside the wires, physical conduction current I_c = dQ/dt flows. Between the insulated capacitor plates, J = 0, but time-varying electric flux ∂D / ∂t creates displacement current I_d = ∫ ∂D / ∂t ds = I_c, ensuring continuous total current across the circuit loop.