Related Concepts: 4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem | 4.04 Dynamic Boundary Conditions for Electromagnetic Fields | 5.02 Plane Waves in Lossless vs. Lossy Media & Skin Depth Calculations | 3.03 Ampere’s Circuital Law & Vector Magnetic Potential (A)

5.01 Wave Equations & Helmholtz Equations in Source-Free Media

Core Idea

Maxwell’s four coupled first-order equations can be untangled into single second-order equations in one variable at a time. The result has the exact mathematical form of a classical wave equation — which proves that electromagnetic disturbances propagate through space at speed $u=1/\sqrt{\mu ϵ}$. In free space that number comes out to $3\times 10^8$ m/s, which is the speed of light. Maxwell’s discovery that light is an electromagnetic wave was a consequence of this algebra, not an experiment.

The single most-tested item in Chapter 5

“Deduce the homogeneous wave equations for both scalar and vector potentials” has appeared in 2015, 2017, 2021, 2022, 2023, 2024 and 2025 — seven of eleven papers, always 10–13 marks. It was completely absent from your previous notes. Section 3 fixes that.

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1. Master Derivation: The 3D Vector Wave Equations for $\vec{E}$ and $\vec{H}$

[PYQ: 2018, 2019, 2024, 2025]

1.1 The source-free starting point

A source-free simple medium means ${\rho }_v=0$, $\vec{J}=0$, and the medium is linear, isotropic and homogeneous with constants $ϵ,\mu$. Maxwell’s equations reduce to:

\nabla \cdot \vec{E} = 0 \tag{1} \nabla \cdot \vec{H} = 0 \tag{2} \nabla \times \vec{E} = -\mu\frac{\partial \vec{H}}{\partial t} \tag{3} \nabla \times \vec{H} = \epsilon\frac{\partial \vec{E}}{\partial t} \tag{4}

Why we must decouple them

Equations (3) and (4) are coupled: solving for $\vec{E}$ requires knowing $\vec{H}$, and vice versa. The standard trick is to take the curl of one and substitute the other, producing a single second-order equation in one unknown.

1.2 The decoupling

Take the curl of Faraday’s law (3):

\nabla \times (\nabla \times \vec{E}) = -\mu\frac{\partial}{\partial t}(\nabla \times \vec{H}) \tag{5}

Apply the vector identity to the left-hand side:

\nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2\vec{E} \tag{6}

Substitute $\nabla \cdot \vec{E}=0$ from (1), so the first term dies. Substitute (4) into the right-hand side of (5):

$-{\nabla }^2\vec{E}=-\mu \frac{\partial }{\partial t}\left(ϵ\frac{\partial \vec{E}}{\partial t}\right)$

Key Result — 3D Homogeneous Vector Wave Equations

$\boxed{{\nabla }^2\vec{E}-\mu ϵ\frac{{\partial }^2\vec{E}}{\partial t^2}=0}$ and, by taking the curl of (4) instead and repeating identically, $\boxed{{\nabla }^2\vec{H}-\mu ϵ\frac{{\partial }^2\vec{H}}{\partial t^2}=0}$

1.3 Extracting the wave speed

Compare with the classical wave equation for any disturbance $\psi$:

${\nabla }^2\psi -\frac{1}{u^2}\frac{{\partial }^2\psi }{\partial t^2}=0$

Matching coefficients gives $\frac{1}{u^2}=\mu ϵ$, so:

$\boxed{u=\frac{1}{\sqrt{\mu ϵ}}}[\text{m/s}]c=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}\approx 3\times 10^8[\text{m/s}]$

Key Exam Checkpoint

Two PYQ variants ask for this result under different wording:

• “Show that ${\nabla }^{2}E-\frac{1}{c^2}\frac{{\partial }^{2}E}{\partial t^2}=0$ using Maxwell’s equations” [PYQ: 2018] — do the derivation in free space so $\mu ϵ\to {\mu }_0{ϵ}_0=1/c^2$.
• “Show that the electromagnetic field vector travels with speed $1/\sqrt{\mu ϵ}$” [PYQ: 2019, 2024] — do the same derivation, then explicitly perform the coefficient-matching step above. The matching is where the marks are.

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2. Time-Harmonic Fields & the Helmholtz Equations

[PYQ: 2016, 2021, 2023, 2025]

Almost all engineering fields are time-harmonic — they vary sinusoidally as $e^{j\omega t}$. In the phasor domain:

$\frac{\partial }{\partial t}\to j\omega \frac{{\partial }^2}{\partial t^2}\to (j\omega {)}^2=-{\omega }^2$

Substituting into the wave equations:

${\nabla }^2{\vec{E}}_s-\mu ϵ(-{\omega }^2){\vec{E}}_s=0$

Key Result — Homogeneous Vector Helmholtz Equations

$\boxed{{\nabla }^2{\vec{E}}_s+k^2{\vec{E}}_s=0}\boxed{{\nabla }^2{\vec{H}}_s+k^2{\vec{H}}_s=0}$ where the wave number is $k=\omega \sqrt{\mu ϵ}=\frac{\omega }{u}=\frac{2\pi }{\lambda }[\text{rad/m}]$

"Explain the term wave number" [PYQ: 2021]

The wave number $k$ (also called the phase constant $\beta$ in a lossless medium) is the spatial frequency of the wave — the phase shift in radians undergone by the wave per metre travelled. It is the spatial analogue of angular frequency $\omega$, which measures phase shift per second. $k=2\pi /\lambda$ says the phase advances by $2\pi$ over exactly one wavelength.

"Significance of the Helmholtz equation" [PYQ: 2023]

The Helmholtz equation removes time from the problem entirely. A messy partial differential equation in four variables $(x,y,z,t)$ becomes a purely spatial equation whose solutions are the standing/travelling field patterns of the structure. It is the master equation from which plane waves, waveguide modes and antenna radiation patterns are all obtained.

Form in a lossy medium [PYQ: 2018 — "Write the form of Helmholtz's equation in long medias"]

In a lossy medium ($\sigma \ne 0$) the same procedure with $\vec{J}=\sigma \vec{E}$ gives ${\nabla }^2{\vec{E}}_s-{\gamma }^2{\vec{E}}_s=0,\gamma =\sqrt{j\omega \mu (\sigma +j\omega ϵ)}=\alpha +j\beta$ The real wave number $k$ is replaced by the complex propagation constant $\gamma$. Full treatment in 5.02 Plane Waves in Lossless vs. Lossy Media & Skin Depth Calculations.

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3. Master Derivation: Homogeneous Wave Equations for Scalar and Vector Potentials

[PYQ: 2015, 2017, 2021, 2022, 2023, 2024, 2025] — ⭐⭐⭐⭐⭐ the highest-frequency question in Chapter 5

3.1 What the potentials are, and why we use them

[PYQ: 2024 — “Explain scalar and vector potentials along with their significances”]

Vector magnetic potential $\vec{A}$

Since $\nabla \cdot \vec{B}=0$ always, and the divergence of any curl is identically zero, $\vec{B}$ can always be written as the curl of some vector field: $\vec{B}=\nabla \times \vec{A}[\text{Wb/m}]$ Significance: $\vec{A}$ replaces three coupled components of $\vec{B}$ with a single quantity that automatically satisfies $\nabla \cdot \vec{B}=0$. It also has a direct integral solution in terms of the source current, which $\vec{B}$ does not.

Electric scalar potential $V$

Substituting $\vec{B}=\nabla \times \vec{A}$ into Faraday’s law: $\nabla \times \vec{E}=-\frac{\partial }{\partial t}(\nabla \times \vec{A})⟹\nabla \times \left(\vec{E}+\frac{\partial \vec{A}}{\partial t}\right)=0$ A curl-free field is the gradient of a scalar, so: $\vec{E}+\frac{\partial \vec{A}}{\partial t}=-\nabla V⟹\boxed{\vec{E}=-\nabla V-\frac{\partial \vec{A}}{\partial t}}$ Significance: $V$ generalises the electrostatic potential. Note the second term — in electrodynamics the electric field has a second source beyond charge separation: a time-varying vector potential. This term is induction.

Why bother with potentials at all?

Solving for $\vec{E}$ and $\vec{H}$ directly means six coupled scalar unknowns. Solving for $V$ and $\vec{A}$ means four, and each satisfies its own independent wave equation once the Lorentz gauge is applied. Radiation and antenna theory are built entirely on $V$ and $\vec{A}$.

3.2 Deriving the non-homogeneous wave equations

Step 1 — start from Ampère–Maxwell. In a simple medium:

$\nabla \times \vec{H}=\vec{J}+ϵ\frac{\partial \vec{E}}{\partial t}$

Substitute $\vec{H}=\vec{B}/\mu =(\nabla \times \vec{A})/\mu$ and $\vec{E}=-\nabla V-\partial \vec{A}/\partial t$:

$\nabla \times (\nabla \times \vec{A})=\mu \vec{J}+\mu ϵ\frac{\partial }{\partial t}\left(-\nabla V-\frac{\partial \vec{A}}{\partial t}\right)$

Step 2 — expand the curl-of-curl using $\nabla \times (\nabla \times \vec{A})=\nabla (\nabla \cdot \vec{A})-{\nabla }^2\vec{A}$:

$\nabla (\nabla \cdot \vec{A})-{\nabla }^2\vec{A}=\mu \vec{J}-\mu ϵ\nabla \frac{\partial V}{\partial t}-\mu ϵ\frac{{\partial }^2\vec{A}}{\partial t^2}$

Step 3 — regroup:

\nabla^2\vec{A} - \mu\epsilon\frac{\partial^2\vec{A}}{\partial t^2} = -\mu\vec{J} + \nabla\left(\nabla\cdot\vec{A} + \mu\epsilon\frac{\partial V}{\partial t}\right) \tag{7}

Step 4 — apply the Lorentz gauge. $\vec{A}$ is not yet unique: only its curl has been fixed, so we are free to choose its divergence. Choose:

Lorentz Gauge Condition

$\boxed{\nabla \cdot \vec{A}+\mu ϵ\frac{\partial V}{\partial t}=0}$

This annihilates the entire bracketed term in (7), leaving:

\boxed{\;\nabla^2\vec{A} - \mu\epsilon\frac{\partial^2\vec{A}}{\partial t^2} = -\mu\vec{J}\;} \tag{8}

Step 5 — repeat for $V$. Substitute $\vec{E}=-\nabla V-\partial \vec{A}/\partial t$ into Gauss’s law $\nabla \cdot \vec{E}={\rho }_v/ϵ$:

$-{\nabla }^{2}V-\frac{\partial }{\partial t}(\nabla \cdot \vec{A})=\frac{{\rho }_v}{ϵ}$

Use the Lorentz gauge to replace $\nabla \cdot \vec{A}=-\mu ϵ\partial V/\partial t$:

$-{\nabla }^{2}V+\mu ϵ\frac{{\partial }^{2}V}{\partial t^2}=\frac{{\rho }_v}{ϵ}$

\boxed{\;\nabla^2 V - \mu\epsilon\frac{\partial^2 V}{\partial t^2} = -\frac{\rho_v}{\epsilon}\;} \tag{9}

Equations (8) and (9) are the non-homogeneous wave equations for the vector and scalar potentials. [PYQ: 2016 — “Using Lorentz’s gauge deduce the nonhomogeneous wave equation for vector potential $\vec{A}$ and scalar potential $V$“]

3.3 The homogeneous case — and when it becomes non-homogeneous

[PYQ: 2015, 2021 — “How would these equations turn out to be non-homogeneous?“]

In a source-free region (${\rho }_v=0$, $\vec{J}=0$), the right-hand sides vanish:

Homogeneous Wave Equations for the Potentials

$\boxed{{\nabla }^{2}V-\mu ϵ\frac{{\partial }^{2}V}{\partial t^2}=0}\boxed{{\nabla }^2\vec{A}-\mu ϵ\frac{{\partial }^2\vec{A}}{\partial t^2}=0}$

	Homogeneous	Non-homogeneous
Condition	${\rho }_v=0$ and $\vec{J}=0$	${\rho }_v\ne 0$ or $\vec{J}\ne 0$
Region	Source-free space, away from all charges/currents	Inside or on the sources themselves
Scalar equation	${\nabla }^{2}V-\mu ϵ{\partial }^{2}V/\partial t^2=0$	${\nabla }^{2}V-\mu ϵ{\partial }^{2}V/\partial t^2=-{\rho }_v/ϵ$
Vector equation	${\nabla }^2\vec{A}-\mu ϵ{\partial }^2\vec{A}/\partial t^2=0$	${\nabla }^2\vec{A}-\mu ϵ{\partial }^2\vec{A}/\partial t^2=-\mu \vec{J}$
Describes	Free propagation of an existing wave	Generation/radiation of a wave by a source

The answer to "how do they become non-homogeneous", in one sentence

The equations become non-homogeneous the moment the region of interest contains sources — a non-zero volume charge density ${\rho }_v$ or current density $\vec{J}$. The source terms $-{\rho }_v/ϵ$ and $-\mu \vec{J}$ then appear as forcing functions on the right-hand side, converting a free-propagation problem into a radiation problem.

graph TD
    M["Maxwell's Equations"] --> P1["B = ∇×A"]
    M --> P2["E = −∇V − ∂A/∂t"]
    P1 --> S["Substitute into<br/>Ampère–Maxwell and Gauss"]
    P2 --> S
    S --> G["Apply LORENTZ GAUGE<br/>∇·A + με ∂V/∂t = 0"]
    G --> NH["NON-HOMOGENEOUS<br/>∇²A − με∂²A/∂t² = −μJ<br/>∇²V − με∂²V/∂t² = −ρᵥ/ε"]
    NH -->|"set ρᵥ = 0, J = 0<br/>(source-free region)"| H["HOMOGENEOUS<br/>∇²A − με∂²A/∂t² = 0<br/>∇²V − με∂²V/∂t² = 0"]
    NH -->|"solve"| R["RETARDED POTENTIALS<br/>see §4"]

Exam efficiency — one derivation covers seven years

Write §3.2 once and you have simultaneously answered the 2015, 2017, 2021, 2022, 2023, 2024 and 2025 versions plus the 2016 Lorentz-gauge variant. The only thing that changes between years is whether you stop at the homogeneous form or continue to the non-homogeneous discussion.

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4. Retarded Potentials — Fields Travel, They Don’t Teleport

[PYQ: 2018 — 12 Marks] — “Starting from homogeneous wave equation, show that the scalar potential at a distance $\bar{R}$ from the surface at time $t$ depends on the value of the charge density at an earlier time $(t-\bar{R}/u)$.”

The physical point

In electrostatics, $V=Q/4\pi ϵR$ — if the charge changes, the potential everywhere changes instantly. That violates causality. The wave equation fixes this: the potential at distance $R$ reflects what the source was doing a travel-time $R/u$ earlier.

Step 1. Away from the source, the scalar potential obeys the homogeneous wave equation:

${\nabla }^{2}V-\mu ϵ\frac{{\partial }^{2}V}{\partial t^2}=0$

Step 2. For a point source at the origin the problem is spherically symmetric, so $V=V(R,t)$ only, and the Laplacian in spherical coordinates reduces to:

${\nabla }^{2}V=\frac{1}{R^2}\frac{\partial }{\partial R}\left(R^2\frac{\partial V}{\partial R}\right)=\frac{{\partial }^{2}V}{\partial R^2}+\frac{2}{R}\frac{\partial V}{\partial R}$

$⟹\frac{{\partial }^{2}V}{\partial R^2}+\frac{2}{R}\frac{\partial V}{\partial R}-\mu ϵ\frac{{\partial }^{2}V}{\partial t^2}=0$

Step 3. Substitute $V(R,t)=\frac{1}{R}\bar{u}(R,t)$. The derivatives are:

$\frac{\partial V}{\partial R}=-\frac{\bar{u}}{R^2}+\frac{1}{R}\frac{\partial \bar{u}}{\partial R},\frac{{\partial }^{2}V}{\partial R^2}=\frac{2\bar{u}}{R^3}-\frac{2}{R^2}\frac{\partial \bar{u}}{\partial R}+\frac{1}{R}\frac{{\partial }^2\bar{u}}{\partial R^2},\frac{{\partial }^{2}V}{\partial t^2}=\frac{1}{R}\frac{{\partial }^2\bar{u}}{\partial t^2}$

Substituting and simplifying, every term with $1/R^2$ and $1/R^3$ cancels, leaving the 1-D wave equation:

$\frac{{\partial }^2\bar{u}}{\partial R^2}-\mu ϵ\frac{{\partial }^2\bar{u}}{\partial t^2}=0$

Step 4. Its general solution is a pair of counter-travelling waves:

$\bar{u}(R,t)=f\left(t-\frac{R}{u}\right)+g\left(t+\frac{R}{u}\right),u=\frac{1}{\sqrt{\mu ϵ}}$

The inward wave $g(t+R/u)$ would arrive before the source acted, violating causality, so it is discarded:

$V(R,t)=\frac{1}{R}f\left(t-\frac{R}{u}\right)$

Step 5. Fix $f$ by matching the static limit. As $u\to \infty$ (or for a static charge), this must reduce to $V=Q(t)/4\pi ϵR$:

$\frac{1}{R}f(t)=\frac{Q(t)}{4\pi ϵR}⟹f(t)=\frac{Q(t)}{4\pi ϵ}$

Key Result — Retarded Scalar Potential

$\boxed{V(R,t)=\frac{Q\left(t-R/u\right)}{4\pi ϵR}}$ and for a continuous charge distribution, $V(R,t)=\frac{1}{4\pi ϵ}{\int }_{V^{\prime }}\frac{{\rho }_v\left(t-R/u\right)}{R}d{v}^{\prime }$ By the identical argument, the retarded vector potential is $\vec{A}(R,t)=\frac{\mu }{4\pi }{\int }_{V^{\prime }}\frac{\vec{J}\left(t-R/u\right)}{R}d{v}^{\prime }$

The sentence that earns the concluding mark

“The potential observed at distance $R$ and time $t$ is set not by what the source is doing now, but by what it was doing at the earlier time $t-R/u$ — the retardation time needed for the disturbance to travel the distance $R$ at finite speed $u$. Electromagnetic information propagates; it does not act at a distance.”

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5. High-Yield Theorem: Duality of Source-Free Solutions

[PYQ: 2024, 2025 — 10 Marks]

Statement

Show that if $\vec{E}$ and $\vec{H}$ are solutions of source-free Maxwell’s equations in a simple medium characterised by $ϵ$ and $\mu$, then so are ${\vec{E}}^{\prime }=\eta \vec{H}$ and ${\vec{H}}^{\prime }=-\vec{E}/\eta$, where $\eta =\sqrt{\mu /ϵ}$.

Proof — verify all four source-free Maxwell equations.

(a) Divergence equations:

$\nabla \cdot {\vec{E}}^{\prime }=\nabla \cdot (\eta \vec{H})=\eta (\nabla \cdot \vec{H})=0✓$ $\nabla \cdot {\vec{H}}^{\prime }=\nabla \cdot \left(-\frac{\vec{E}}{\eta }\right)=-\frac{1}{\eta }(\nabla \cdot \vec{E})=0✓$

(b) A useful identity. Since $\eta =\sqrt{\mu /ϵ}$:

$\eta ϵ=\sqrt{\frac{\mu }{ϵ}}\cdot ϵ=\sqrt{\mu ϵ}\text{and}\frac{\mu }{\eta }=\frac{\mu }{\sqrt{\mu /ϵ}}=\sqrt{\mu ϵ}⟹\boxed{\eta ϵ=\frac{\mu }{\eta }}$

(c) Faraday’s law — must show $\nabla \times {\vec{E}}^{\prime }=-j\omega \mu {\vec{H}}^{\prime }$:

$\nabla \times {\vec{E}}^{\prime }=\nabla \times (\eta \vec{H})=\eta (\nabla \times \vec{H})=\eta (j\omega ϵ\vec{E})=j\omega (\eta ϵ)\vec{E}=j\omega \frac{\mu }{\eta }\vec{E}=-j\omega \mu \left(-\frac{\vec{E}}{\eta }\right)=-j\omega \mu {\vec{H}}^{\prime }✓$

(d) Ampère’s law — must show $\nabla \times {\vec{H}}^{\prime }=j\omega ϵ{\vec{E}}^{\prime }$:

$\nabla \times {\vec{H}}^{\prime }=\nabla \times \left(-\frac{\vec{E}}{\eta }\right)=-\frac{1}{\eta }(\nabla \times \vec{E})=-\frac{1}{\eta }(-j\omega \mu \vec{H})=j\omega \frac{\mu }{\eta }\vec{H}=j\omega ϵ(\eta \vec{H})=j\omega ϵ{\vec{E}}^{\prime }✓$

All four equations are satisfied, so $({\vec{E}}^{\prime },{\vec{H}}^{\prime })$ is a valid electromagnetic field. Q.E.D.

What this actually means — the duality principle

The transformation $\vec{E}\to \eta \vec{H}$, $\vec{H}\to -\vec{E}/\eta$ is a 90° rotation in field space that leaves Maxwell’s equations invariant. It is the reason electric and magnetic phenomena mirror each other so precisely in source-free regions, and it lets antenna engineers convert a known dipole solution into a known slot solution for free.

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6. Common Mistakes That Cost Marks

Avoid these

1. Forgetting to state the source-free condition (${\rho }_v=0$, $\vec{J}=0$) before dropping $\nabla (\nabla \cdot \vec{E})$. That term only vanishes because of Gauss’s law in a source-free region.
2. Deriving only the $\vec{E}$ equation. Every PYQ asks for “both” — write the $\vec{H}$ result too, even if you just state that the procedure is identical.
3. Confusing the two families of wave equations. ${\nabla }^2\vec{E}-\mu ϵ{\partial }_t^2\vec{E}=0$ is for fields; ${\nabla }^{2}V-\mu ϵ{\partial }_t^{2}V=-{\rho }_v/ϵ$ is for potentials. The seven-year question asks for the potential family. Read the question.
4. Skipping the Lorentz gauge justification. You must say that $\nabla \cdot \vec{A}$ is still free to choose because only $\nabla \times \vec{A}$ was fixed by $\vec{B}=\nabla \times \vec{A}$.
5. Writing $k=\omega \mu ϵ$ instead of $k=\omega \sqrt{\mu ϵ}$.
6. In the retarded-potential proof, keeping the $g(t+R/u)$ term. You must discard it and explicitly cite causality.

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7. PYQ Bank — Verbatim Questions & Answer Plans

Q1 — Homogeneous potential wave equations [PYQ: 2017, 2022, 2023, 2025 — 10/13 Marks] ⭐⭐⭐⭐⭐

“Deduce the homogeneous wave equations for both scalar and vector potentials.”

Answer plan: §3.1 (define $\vec{A}$ and $V$, derive $\vec{E}=-\nabla V-\partial \vec{A}/\partial t$) → §3.2 Steps 1–5 (substitute into Maxwell, apply Lorentz gauge) → set ${\rho }_v=\vec{J}=0$ to reach the boxed homogeneous pair in §3.3.

Q2 — Homogeneous + how they become non-homogeneous [PYQ: 2015 — 12 Marks]

“Deduce the homogeneous wave equations for both scalar and vector potentials. How would these equations turn out to be non-homogeneous?”

Answer plan: Q1 answer, then the §3.3 comparison table plus the one-sentence conclusion about source terms.

Q3 — "Write and analyze" phrasing [PYQ: 2021 — 10 Marks]

“Write and analyze the homogeneous wave equation for both scalar and vector potentials. Explain how these equations would turn out to be nonhomogeneous.”

Answer plan: Same as Q2. “Analyze” means: identify the equation as a wave equation, extract $u=1/\sqrt{\mu ϵ}$ from the coefficient, and comment that both potentials propagate at the same finite speed.

Q4 — Potentials + significance + wave equations [PYQ: 2024 — 13 Marks]

“Explain scalar and vector potentials along with their significances. Deduce the homogeneous wave equations for both potentials.”

Answer plan: Spend the first third on §3.1 — the two abstract callouts on what $V$ and $\vec{A}$ are and why they are worth using — then the standard derivation. The significance is separately marked here.

Q5 — Lorentz gauge, non-homogeneous [PYQ: 2016 — 12 Marks]

“Using Lorentz’s gauge deduce the nonhomogeneous wave equation for vector potential $\vec{A}$ and scalar potential V.”

Answer plan: §3.2 in full, and stop at equations (8) and (9) — do not set the sources to zero. Make the Lorentz gauge condition prominent and explain why the gauge freedom exists.

Q6 — General wave equation for E and H → Helmholtz [PYQ: 2025 — 13 Marks]

“Derive the general wave equation for $\vec{E}$ and $\vec{H}$ and convert them to Helmholtz’s equations for sinusoidal time dependence.”

Answer plan: §1 in full for both $\vec{E}$ and $\vec{H}$, then §2 for the phasor conversion. Define $k$ and give its units.

Q7 — Helmholtz from Maxwell [PYQ: 2016 — 10 Marks]

“Starting from Maxwell’s equations obtain homogeneous vector Helmholtz’s equation.”

Answer plan: §1 then §2, compressed — the target is the Helmholtz form, so move quickly through the time-domain step.

Q8 — State Helmholtz + wave number [PYQ: 2021 — 05 Marks]

“State the homogeneous vector Helmholtz’s equation and explain the term ‘wave number’.”

Answer plan: The two boxed Helmholtz equations from §2 plus the “wave number” abstract callout. No derivation needed at 5 marks.

Q9 — Significance of Helmholtz [PYQ: 2023 — 06 Marks]

“Write and explain the significance of the Helmholtz’s equation.”

Answer plan: State the equation, then the §2 significance callout — emphasise that it eliminates time and turns the problem into a purely spatial one.

Q10 — Helmholtz in lossy media [PYQ: 2018 — 05 Marks]

“Write the form of Helmholtz’s equation in long medias.” (reads as “lossy media”)

Answer plan: ${\nabla }^2{\vec{E}}_s-{\gamma }^2{\vec{E}}_s=0$ with $\gamma =\sqrt{j\omega \mu (\sigma +j\omega ϵ)}=\alpha +j\beta$; define $\alpha$ and $\beta$.

Q11 — Free-space wave equation [PYQ: 2017, 2021 — 08/11 Marks]

“Deduce the fundamental equation for free space propagation.”

Answer plan: §1 with $\mu \to {\mu }_0$, $ϵ\to {ϵ}_0$, concluding with $c=1/\sqrt{{\mu }_0{ϵ}_0}\approx 3\times 10^8$ m/s and the remark that this is the speed of light.

Q12 — Speed of the field vector [PYQ: 2019, 2024 — 10 Marks]

“Show that the electromagnetic field vector travels with speed $1/\sqrt{\mu ϵ}$ through the derivation of homogeneous vector wave equation for source free fields in simple media.”

Answer plan: §1.2 then §1.3. The coefficient-matching against the classical wave equation is the mark-bearing step — show it explicitly.

Q13 — Show ${\nabla }^{2}E-\frac{1}{c^2}{\partial }_t^{2}E=0$ [PYQ: 2018 — 08 Marks]

“Show that ${\nabla }^{2}E-\frac{1}{C^2}\frac{{\partial }^{2}E}{\partial t^2}=0$ using Maxwell’s equation.”

Answer plan: §1.2 in free space; finish by noting ${\mu }_0{ϵ}_0=1/c^2$.

Q14 — Duality of solutions [PYQ: 2024, 2025 — 10 Marks]

“Show that if $\vec{E}$ and $\vec{H}$ are solutions of source free Maxwell’s equation in simple medium characterized by $ϵ$ and $\mu$, then so also $E^{\prime }$ and $H^{\prime }$, where $E^{\prime }=\eta H$; $H^{\prime }=-\frac{E}{\eta }$; where $\eta =\sqrt{\mu /ϵ}$.”

Answer plan: §5 — all four checks. The identity $\eta ϵ=\mu /\eta$ in step (b) is what makes the algebra work; derive it explicitly rather than asserting it.

Q15 — Retarded scalar potential [PYQ: 2018 — 12 Marks]

“Starting from homogeneous wave equation, show that the scalar potential at a distance $\bar{R}$ from the surface at time t depends on the value of the charge density at an earlier time $(t-\bar{R}/u)$.”

Answer plan: §4 Steps 1–5. The two mark-critical moves are (i) the substitution $V=\bar{u}/R$ that collapses the spherical Laplacian to a 1-D wave equation, and (ii) discarding the inward wave on causality grounds.

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8. Self-Check Before Moving On

• Can you decouple Maxwell’s curl equations into the vector wave equation for $\vec{E}$, naming the identity used?
• Can you extract $u=1/\sqrt{\mu ϵ}$ by coefficient-matching?
• Can you convert to the Helmholtz form and define the wave number three ways ($\omega \sqrt{\mu ϵ}$, $\omega /u$, $2\pi /\lambda$)?
• Can you derive $\vec{E}=-\nabla V-\partial \vec{A}/\partial t$ from $\vec{B}=\nabla \times \vec{A}$ and Faraday’s law?
• Can you state the Lorentz gauge and explain why you are allowed to impose it?
• Can you write both non-homogeneous potential wave equations and say exactly what makes them non-homogeneous?
• Can you run the retarded-potential derivation and justify discarding the inward solution?
• Can you verify all four Maxwell equations for the duality transformation?

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Source: 05 electromagnetic_waves_master_notes.md §1 (master dump), ECE 2105 Syllabus Weeks 10–11, PYQ bank 2015–2025, Cheng Ch. 8, 03 solution to em eqns.pdf, Masuk sir-2309008.pdf