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