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