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