Electrodynamic Boundary Conditions

Five times now this track has stood at an interface and matched fields across it: the electrostatic pillbox that found $\sigma/\varepsilon_0$ , the dielectric version with $\mathbf{D}$ , the magnetostatic loop that found $\mu_0 K$ , and the $\mathbf{H}$ -field rules for magnetized media. Every one of those arguments assumed nothing was changing. Now everything changes — Faraday's law threads EMF through every loop, the displacement current threads every surface — and the question becomes urgent: do the matching rules survive? A changing magnetic field drives circulating electric fields; surely that can break the continuity of $E_\parallel$ across a boundary? It cannot, and the reason is a one-line scaling argument that this article makes precise: the new time-derivative terms are fluxes through areas that vanish as the pillbox and loop collapse onto the surface, while surface charge and surface current are genuine singular contributions that survive. The boundary conditions of full electrodynamics are, form for form, the boundary conditions of statics — only now $\sigma_f$ and $\mathbf{K}_f$ oscillate, and the fields they stitch together obey wave equations instead of Laplace's. That survival is not a curiosity; it is the working capital of the entire second half of electromagnetism. Reflection and refraction at a glass surface, the mode structure of every waveguide and laser cavity, the current sheet that guards a metal mirror, the antenna boundary problem — each is two bulk solutions plus these four stitching rules. Learn the four-pack once, with an honest proof, and the optics chapters ahead become bookkeeping.

Key takeaways

At any interface, apply the integral Maxwell equations to a collapsing pillbox and loop: $\hat{\mathbf{n}}\cdot(\mathbf{D}_1-\mathbf{D}_2)=\sigma_f$ , $\hat{\mathbf{n}}\cdot(\mathbf{B}_1-\mathbf{B}_2)=0$ , $\hat{\mathbf{n}}\times(\mathbf{E}_1-\mathbf{E}_2)=\mathbf{0}$ , $\hat{\mathbf{n}}\times(\mathbf{H}_1-\mathbf{H}_2)=\mathbf{K}_f$ .
The time-dependent terms $-\partial\mathbf{B}/\partial t$ and $+\partial\mathbf{D}/\partial t$ contribute fluxes proportional to the collapsing area — they vanish in the limit, so the conditions keep their static form even in full electrodynamics.
Only free surface charge $\sigma_f$ and free surface current $\mathbf{K}_f$ appear; bound charge and magnetization currents are already inside $\mathbf{D}$ and $\mathbf{H}$ .
Charge conservation lives on the surface too: the mismatch of normal current feeding an interface drives $\partial\sigma_f/\partial t$ plus the surface divergence of $\mathbf{K}_f$ .
A perfect conductor collapses the four rules to $E_\parallel=0$ , $B_\perp=0$ , with $\sigma_f$ and $\mathbf{K}_f$ absorbing the normal $\mathbf{D}$ and tangential $\mathbf{H}$ — the boundary condition of mirrors, waveguides, and cavities.
KEY IDEA: as the pillbox and loop collapse onto the surface, everything proportional to volume or area dies; only true surface singularities — $\sigma_f$ and $\mathbf{K}_f$ — survive to set the jumps.

The core insight

Shrink a pillbox or loop onto the interface: volume terms and the flux of any bounded time-varying field vanish with the collapsing region, so only surface charge $\sigma_f$ (for normal $\mathbf{D}$ ) and surface current $\mathbf{K}_f$ (for tangential $\mathbf{H}$ ) can produce jumps — the electrodynamic boundary conditions are the static ones, worn by time-dependent fields.

Where this lives today

The hottest current application of these equations is, literally, engineered violations of their innocent-looking defaults. A metasurface is a sheet of sub-wavelength resonators designed so that the effective $\sigma_f$ and $\mathbf{K}_f$ it presents are programmable functions of position and polarization — in the trade these are the generalized sheet transition conditions, this article's equations (1)–(5) with designer right-hand sides. Impose a phase gradient through the sheet and the "reflected" beam steers where geometry says it shouldn't; that is how flat metalenses in consumer phone cameras (shipping since the early 2020s) focus light with zero curved glass, and how satellite reflectarrays shape beams with printed patches instead of parabolic dishes. The same equations run silently inside every electromagnetic solver. FDTD and finite-element codes spend most of their cleverness at interfaces — snapping the update equations to honor $E_\parallel$ continuity and $D_\perp$ jumps across material cells — and the perfect-conductor package (6) is the default wall in every cavity, filter, and antenna simulation. When a 5G engineer meshes a phone chassis or a physicist tunes a superconducting qubit cavity, the four-pack is the part of Maxwell they are trusting the software to enforce exactly.

The physical picture

An interface is where the differential Maxwell equations lose their footing. Cross from glass into air and $\varepsilon$ jumps; deposit charge on the surface and $\rho$ becomes a delta function; run current in a thin sheet and $\mathbf{J}$ does too. Derivatives of discontinuous fields are not defined at the surface — but the integral forms of Maxwell's equations never flinch, because integrals are perfectly happy with jumps. So the strategy, inherited from every static chapter and now applied with time running: draw a tiny closed surface or loop that straddles the boundary, write the integral law, and let the region collapse onto the interface. Whatever survives the collapse is the boundary condition. Two geometries do all the work. The pillbox — a coin lying flat on the interface, one face in each medium — feeds the two divergence laws and interrogates normal components. As its thickness $h\to 0$ , the side wall's flux dies, ordinary volume charge $\rho_f\, (\text{area}\times h)$ dies, and only a genuine surface charge $\sigma_f\,(\text{area})$ survives inside. The loop — a thin rectangle with long sides parallel to the surface, one in each medium — feeds the two curl laws and interrogates tangential components. As its thickness collapses, the short sides' line integrals die and any flux through the loop dies with the area; only a surface current threading the loop survives.

Figure

The two probes of an interface — a collapsing pillbox reads the normal components, a collapsing loop reads the tangential ones

The asymmetry among the four fields is physics, not convention. Magnetic field lines have no endpoints — $\nabla\cdot\mathbf{B}=0$ holds exactly, statics or not — so a pillbox can never trap magnetic charge and $B_\perp$ can never jump, across any interface, at any frequency. Electric field lines end on charge, so $D_\perp$ jumps exactly where free charge piles up. Tangentially the curl laws rule: there is no such thing as a "surface source of circulation" for $\mathbf{E}$ (charge does not appear in Faraday's law), but a free current sheet is precisely a surface source of circulation for $\mathbf{H}$ . What is genuinely new in electrodynamics is what does not happen. Faraday's law adds $-\mathrm{d}\Phi_B/\mathrm{d}t$ to the loop's ledger; the Ampère–Maxwell law adds the displacement flux. Both are fluxes through the collapsing loop, proportional to its dying area. The static answers survive untouched — a scaling argument this article promotes to a theorem below, because the whole edifice of wave optics leans on it. This is exactly the technology of the earlier interfaces — the pillbox and loop of Boundary Conditions on E and V, the free-charge accounting of Boundary Conditions for D and E, the current-sheet bookkeeping of Boundary Conditions for the Magnetostatic Field and Magnetic Boundary Conditions — inherited whole, and certified against time.

The four conditions, assembled

Each condition is one integral law applied to one probe. Gauss's law for $\mathbf{D}$ , written on the pillbox with $\hat{\mathbf{n}}$ the unit normal pointing from medium 2 into medium 1, keeps only the two flat faces and the enclosed free surface charge:

\hat{\mathbf{n}}\cdot(\mathbf{D}_1-\mathbf{D}_2)=\sigma_f \tag{1}

where: $\mathbf{D}_i$ is the displacement in medium $i$ , and $\sigma_f$ counts free surface charge only (C/m²). Bound charge is already accounted for inside $\mathbf{D}$ ; that is the entire point of using $\mathbf{D}$ here. Where nobody has deposited free charge — a clean glass–air surface — $D_\perp$ is continuous even though $E_\perp$ jumps by the polarization ledger $\sigma_b/\varepsilon_0$ . The same pillbox fed with $\oint\mathbf{B}\cdot\mathrm{d}\mathbf{a}=0$ has nothing to enclose, ever:

\hat{\mathbf{n}}\cdot(\mathbf{B}_1-\mathbf{B}_2)=0 \tag{2}

where: the zero is exact and unconditional — it is the no-monopole law read as a boundary condition. Of the four rules this is the only one with no possible exception; it holds across ferromagnets, superconductor surfaces, and the oscillating interfaces of optics alike. The loop fed with Faraday's law gives, once the flux term dies in the collapse (Derivation A),

\hat{\mathbf{n}}\times(\mathbf{E}_1-\mathbf{E}_2)=\mathbf{0} \tag{3}

where: the cross product with $\hat{\mathbf{n}}$ extracts the tangential part: every tangential component of $\mathbf{E}$ crosses every interface unbroken. This is the condition that will force reflected waves to exist at a glass surface — an incident wave alone cannot satisfy (3), so the boundary manufactures companions. And the loop fed with the Ampère–Maxwell law keeps only the free surface current threading it:

\hat{\mathbf{n}}\times(\mathbf{H}_1-\mathbf{H}_2)=\mathbf{K}_f \tag{4}

where: $\mathbf{K}_f$ is free surface current density (A/m), tangent to the interface, and the displacement-current contribution has died with the loop area exactly as Faraday's flux did in (3). Magnetization surface currents $\mathbf{K}_b=\mathbf{M}\times\hat{\mathbf{n}}$ never appear — they are $\mathbf{H}$ 's internal business. One table to navigate them:

component	condition	jump source	can it ever jump without one?
$D_\perp$	$D_{1\perp}-D_{2\perp}=\sigma_f$	free surface charge	no
$B_\perp$	$B_{1\perp}=B_{2\perp}$	—	never, at any frequency
$\mathbf{E}_\parallel$	$\mathbf{E}_{1\parallel}=\mathbf{E}_{2\parallel}$	—	never (bounded fields)
$\mathbf{H}_\parallel$	$\hat{\mathbf{n}}\times(\mathbf{H}_1-\mathbf{H}_2)=\mathbf{K}_f$	free surface current	no

The remaining pair — $E_\perp$ and $B_\parallel$ — are not independent conditions; in linear media they follow from (1)–(4) through the constitutive relations $\mathbf{D}=\varepsilon\mathbf{E}$ , $\mathbf{H}=\mathbf{B}/\mu$ . That is why four equations, not eight, make the problem well posed.

Derivation AWhy time dependence changes nothing in the matching rules

We want to prove the claim the whole article rests on: applied to a collapsing pillbox and loop, the full time-dependent Maxwell equations yield boundary conditions identical in form to the static ones. The strategy is a scaling audit — write each integral law on a probe of lateral size $\ell$ and thickness $h$ , track how every term scales as $h\to 0$ at fixed $\ell$ , and keep only the survivors. The one thing to watch is the hypothesis doing the real work: the bulk fields and their time derivatives stay bounded near the surface; only $\sigma_f$ and $\mathbf{K}_f$ are allowed to be singular.

Step 1. The pillbox and Gauss's law for $\mathbf{D}$ . The closed-surface flux has three pieces: top face $\approx D_{1\perp}\,A$ , bottom face $\approx -D_{2\perp}\,A$ (outward normals oppose), and side wall of area $\sim(\text{perimeter})\times h$ , which is bounded by $|\mathbf{D}|_{\max}\cdot(\text{const})\,\ell h \to 0$ . The enclosed free charge is $Q_f = \sigma_f A + \rho_f A h$ ; the volume piece dies with $h$ . Dividing by $A$ : $D_{1\perp}-D_{2\perp}=\sigma_f$ . Time never entered — Gauss's law has no time-derivative term to begin with.

Step 2. The same pillbox with $\oint\mathbf{B}\cdot\mathrm{d}\mathbf{a}=0$ . Identical accounting with zero on the right: $B_{1\perp}=B_{2\perp}$ . Again no time term exists to worry about.

Step 3. The loop and Faraday's law — the step that could have failed. Take the rectangle with long sides $\ell$ (one in each medium, along an arbitrary tangential direction $\hat{\mathbf{t}}$ ) and short sides $h$ . The circulation is $(E_{1t}-E_{2t})\,\ell + (\text{short sides})$ , and the short-side contributions are bounded by $|\mathbf{E}|_{\max}\,h\to 0$ . Faraday's right-hand side is the new ingredient: $-\frac{\mathrm{d}}{\mathrm{d}t}\int_S\mathbf{B}\cdot\mathrm{d}\mathbf{a}$ , a flux through the loop's spanning area $S=\ell h$ . Bound it honestly: $\bigl|\dot\Phi_B\bigr| \le |\partial\mathbf{B}/\partial t|_{\max}\,\ell h$ . As long as $\partial\mathbf{B}/\partial t$ is bounded near the surface — true for every physical interface, where fields are finite — this term is proportional to $h$ and dies in the collapse. Why this is the crux: the induced-EMF term is what distinguishes electrodynamics from statics inside this argument, and it loses not because it is small but because it is an area quantity being interrogated by a probe whose area is going to zero. Surface current survives the same limit because a current sheet delivers finite current through the loop no matter how thin the loop becomes — it is a line-density singularity, not a bounded bulk density.

Step 4. Dividing by $\ell$ and letting the tangential direction $\hat{\mathbf{t}}$ range over the surface: $E_{1\parallel}=E_{2\parallel}$ , i.e. $\hat{\mathbf{n}}\times(\mathbf{E}_1-\mathbf{E}_2)=\mathbf{0}$ — equation (3), with full time dependence running.

Step 5. The loop and Ampère–Maxwell. Same rectangle, applied to $\mathbf{H}$ : circulation $(H_{1t}-H_{2t})\,\ell$ plus dying short sides; right-hand side $I_{f,\text{enc}} + \frac{\mathrm{d}}{\mathrm{d}t}\int_S \mathbf{D}\cdot\mathrm{d}\mathbf{a}$ . The displacement flux is bounded by $|\partial\mathbf{D}/\partial t|_{\max}\,\ell h\to 0$ — Maxwell's own term eliminates itself from its own boundary condition. The enclosed free current is $I_{f,\text{enc}} = \mathbf{K}_f\cdot(\hat{\mathbf{t}}\times\hat{\mathbf{n}})\,\ell + J_f\,\ell h$ ; the bulk piece dies, the sheet piece survives. Rearranging the geometry factors gives $\hat{\mathbf{n}}\times(\mathbf{H}_1-\mathbf{H}_2)=\mathbf{K}_f$ — equation (4).

Step 6. Collect the survivors: equations (1)–(4), formally identical to the static set. Time dependence now lives in the symbols — $\sigma_f(t)$ , $\mathbf{K}_f(t)$ , and bulk fields obeying wave equations — but not in the matching rules. set all time derivatives to zero and the argument reduces verbatim to the electrostatic and magnetostatic derivations of the earlier boundary-condition articles; the bound "flux $\le$ max-derivative × area" is the standard estimate in Griffiths §7.3.6 and Jackson §I.5, and fails only for idealized sheets where $\partial\mathbf{B}/\partial t$ is itself a surface delta — a case flagged in the FAQ.

Charge conservation moves onto the surface

The four-pack stitches fields. But an interface that holds charge and carries current has bookkeeping of its own: charge arriving from the bulk must either pile up on the surface or flow away along it. Continuity — the law that opens the next chapter as the template of all conservation laws (Charge Conservation and the Continuity Equation) — acquires a surface edition, and the boundary conditions had better be consistent with it.

\hat{\mathbf{n}}\cdot(\mathbf{J}_1-\mathbf{J}_2)= -\frac{\partial\sigma_f}{\partial t}-\nabla_s\!\cdot\mathbf{K}_f \tag{5}

where: $\hat{\mathbf{n}}\cdot(\mathbf{J}_1-\mathbf{J}_2)$ is the net volume current leaving the interface into the bulk, $\nabla_s\cdot$ is the two-dimensional divergence within the surface, and the equation reads: current delivered to the sheet either charges it up ( $\partial\sigma_f/\partial t$ ) or spreads sideways ( $\nabla_s\cdot\mathbf{K}_f$ ). It is not an independent postulate — it follows from the four-pack plus the bulk Maxwell equations, which is the consistency check performed next.

Go deeper · Derivation B · 1 minThe surface continuity equation from the boundary conditions

Derivation BThe surface continuity equation from the boundary conditions

We derive equation (5) — charge bookkeeping on the interface — directly from conditions (1) and (4), demonstrating that Maxwell's boundary conditions carry conservation of charge built in, exactly as the bulk equations do (the bulk version is the Maxwell-consistency identity of The Continuity Equation, the E6 twin, recalled here rather than re-derived). The strategy: take the surface divergence of the tangential- $\mathbf{H}$ condition, use the bulk Ampère–Maxwell law on each side, and watch the time derivative of condition (1) appear.

Step 1. Apply $\nabla_s\cdot$ to (4). A standard vector identity for fields evaluated on the surface gives $\nabla_s\cdot\bigl[\hat{\mathbf{n}}\times(\mathbf{H}_1-\mathbf{H}_2)\bigr] = -\hat{\mathbf{n}}\cdot\bigl[\nabla\times(\mathbf{H}_1-\mathbf{H}_2)\bigr]$ — the surface divergence of a rotated tangential field is (minus) the normal component of its curl.

Step 2. Substitute the bulk Ampère–Maxwell law $\nabla\times\mathbf{H}_i=\mathbf{J}_{f,i}+\partial\mathbf{D}_i/\partial t$ on each side of the interface:

\nabla_s\cdot\mathbf{K}_f = -\hat{\mathbf{n}}\cdot(\mathbf{J}_{f,1}-\mathbf{J}_{f,2}) - \hat{\mathbf{n}}\cdot\frac{\partial}{\partial t}(\mathbf{D}_1-\mathbf{D}_2).

Step 3. The last term is the time derivative of condition (1): $\hat{\mathbf{n}}\cdot\partial_t(\mathbf{D}_1-\mathbf{D}_2)=\partial\sigma_f/\partial t$ . Substituting and rearranging:

\frac{\partial\sigma_f}{\partial t}+\nabla_s\cdot\mathbf{K}_f = -\,\hat{\mathbf{n}}\cdot(\mathbf{J}_{f,1}-\mathbf{J}_{f,2}),

which is equation (5). Why the signs make sense: with $\hat{\mathbf{n}}$ pointing into medium 1, $\hat{\mathbf{n}}\cdot\mathbf{J}_1>0$ means current escaping upward and $\hat{\mathbf{n}}\cdot\mathbf{J}_2>0$ means current arriving from below; their difference is the net loss rate, and the surface pays for it out of stored charge or lateral flow.

Step 4. Read the limits. Statics: $\partial\sigma_f/\partial t=0$ , and any current delivered must flow away along the sheet or continue through — the steady-state contract. Insulator on both sides ( $\mathbf{J}=0$ , $\mathbf{K}_f=0$ ): $\sigma_f$ is frozen, the electret case. A charging capacitor plate: all three terms alive at once, audited numerically in Worked Example 2. equation (5) is the standard interfacial continuity condition (Zangwill §2.3.3); integrating it over any patch of surface and applying the two-dimensional divergence theorem reproduces the patch's charge budget, and setting the bulk currents to zero recovers conservation of total surface charge.

The perfect conductor: the four-pack at full compression

Let medium 2 be an ideal conductor — conductivity so large that any interior field is destroyed by charge rearrangement faster than every other timescale in the problem. Then $\mathbf{E}_2=\mathbf{B}_2=0$ throughout, and the general conditions collapse into the most-used boundary condition in electromagnetic engineering:

E_{1\parallel}=0,\qquad B_{1\perp}=0,\qquad D_{1\perp}=\sigma_f,\qquad \hat{\mathbf{n}}\times\mathbf{H}_1=\mathbf{K}_f \tag{6}

where: the first pair are constraints the exterior field must obey — electric field meets a perfect conductor head-on (normally), magnetic field grazes it (tangentially) — and the second pair are readouts: whatever normal $\mathbf{D}$ and tangential $\mathbf{H}$ the exterior solution delivers, the surface manufactures exactly the $\sigma_f$ and $\mathbf{K}_f$ needed to terminate them. The conductor is a perfectly obedient source, and that obedience is what makes mirror and waveguide problems solvable: you never solve for the wall charges — you read them off afterward.

Figure

A perfect conductor forces E to arrive normally and H to run tangentially; induced sheets sigma and K terminate them

Go deeper · Derivation C · 2 minThe perfect-conductor package and what it buys

Derivation CThe perfect-conductor package and what it buys

The goal is the boundary-condition set (6) governing every metallic mirror, waveguide wall, and resonant cavity, obtained by feeding "medium 2 has no fields" into the general four-pack. The one honest subtlety is why the interior fields vanish for a perfect conductor even in electrodynamics — statics argued it once from equilibrium, but with fields oscillating we must argue from response speed, and that argument sets the limits of the idealization.

Step 1. Why $\mathbf{E}_2=0$ survives time dependence. In a conductor of conductivity $\sigma_c$ , unbalanced interior charge decays as $e^{-t/\tau}$ with $\tau=\varepsilon/\sigma_c$ — the relaxation clock derived in the E4 Continuity Equation article and recalled here by name. For copper $\tau\sim10^{-19}$ s, absurdly faster than any optical period ( $\sim10^{-15}$ s). "Perfect conductor" is the idealization $\tau\to0$ : the charges win every race, and the interior field is annulled at every instant. (Where the race gets close — real metals probed near their plasma frequency — the idealization fails; that honesty is deferred to the E8 conductor articles by name.)

Step 2. Why $\mathbf{B}_2=0$ needs its own argument. Faraday's law with $\mathbf{E}_2=0$ forces only $\partial\mathbf{B}_2/\partial t=0$ — a frozen field, not necessarily zero. For the oscillating fields of electrodynamics the constant is zero (a wave has no DC component), which is why the perfect-mirror condition is stated for time-varying fields; a genuinely static $\mathbf{B}$ passes through an ideal Ohmic conductor undisturbed. (Expelling static field as well is superconductivity — the Meissner effect, a different law recalled from the magnetostatic boundary-conditions article.)

Step 3. Feed $\mathbf{E}_2=\mathbf{B}_2=\mathbf{D}_2=\mathbf{H}_2=0$ into (1)–(4), one line each: tangential $\mathbf{E}$ continuity gives $\hat{\mathbf{n}}\times\mathbf{E}_1=0$ , so the exterior electric field arrives purely normally. Normal $\mathbf{B}$ continuity gives $\hat{\mathbf{n}}\cdot\mathbf{B}_1=0$ : purely tangential. Condition (1) becomes a readout, $\sigma_f=D_{1\perp}$ ; condition (4) becomes $\mathbf{K}_f=\hat{\mathbf{n}}\times\mathbf{H}_1$ .

Step 4. What it buys. Two homogeneous constraints on the exterior solution are exactly what a well-posed cavity or waveguide problem needs: $E_\parallel=0$ and $B_\perp=0$ on the walls discretize the field patterns into countable modes with cutoff frequencies — machinery built in the E8 and E11 chapters by name. The inhomogeneous pair then prices the hardware: $\mathbf{K}_f$ is the current the wall must carry, and in a real conductor that sheet spreads over a skin depth and dissipates — the bridge to Eddy Currents two articles ahead, and to surface impedance in E8. the package reproduces the electrostatic conductor rules ( $E_\parallel=0$ makes the surface an equipotential; $\sigma_f=\varepsilon_0 E_\perp$ ) when frequencies go to zero, and matches Jackson §I.5 and §8.1; Step 2's caveat is confirmed by the standard result that static magnetic fields penetrate normal metals freely.

Limiting cases

case	conditions reduce to	meaning
statics ( $\partial_t=0$ )	same four equations	the collapse audit passes trivially; matches all four earlier BC articles
two linear dielectrics, $\sigma_f=0$	$\varepsilon_1E_{1\perp}=\varepsilon_2E_{2\perp}$ , $\mathbf{E}_{1\parallel}=\mathbf{E}_{2\parallel}$	field-line refraction $\tan\theta_1/\tan\theta_2=\varepsilon_1/\varepsilon_2$ , owned by Boundary Conditions for D and E
two linear magnetic media, $\mathbf{K}_f=0$	$B_{1\perp}=B_{2\perp}$ , $B_{1\parallel}/\mu_1=B_{2\parallel}/\mu_2$	lines bend toward high $\mu$ — the flux-guiding of Magnetic Boundary Conditions
perfect conductor	$E_{1\parallel}=0$ , $B_{1\perp}=0$ , sheets absorb the rest	mirrors, waveguides, cavities — equation (6)
charge- and current-free interface	all tangential components of $\mathbf{E}$ , $\mathbf{H}$ continuous	the starting point of Fresnel's equations (E8, by name)
vacuum both sides, no sheets	all components continuous	no interface at all — the consistency floor

Common trapFree means free — the four-pack's most common wrong answer

$D_{1\perp}-D_{2\perp}=\sigma_f$ uses only free surface charge; the total normal- $\mathbf{E}$ jump is $(\sigma_f+\sigma_b)/\varepsilon_0$ and the two ledgers agree only after the polarization bookkeeping (owned by Boundary Conditions for D and E) is folded in. Likewise $\hat{\mathbf{n}}\times(\mathbf{H}_1-\mathbf{H}_2)=\mathbf{K}_f$ sees only free sheet current — magnetization currents $\mathbf{K}_b=\mathbf{M}\times\hat{\mathbf{n}}$ jump tangential $\mathbf{B}$ but never tangential $\mathbf{H}$ . Writing $\sigma_{\rm tot}$ or $\mathbf{K}_{\rm tot}$ on the right-hand sides is the single most reliable way to fail an exam problem on a dielectric or magnetic interface.

Worked examples

1 · A microwave mirror, audited by the four-pack (confidence builder)

A plane wave of frequency $f=10$ GHz and incident amplitude $E_0=100$ V/m strikes a perfect-conductor sheet at normal incidence. Find the surface charge and surface current the mirror must supply, and the location of the first electric-field node.

Step 1. The boundary condition does the optics. At the wall, $E_\parallel=0$ ; at normal incidence the incident field is entirely tangential, so a reflected wave with tangential amplitude $-E_0$ at the surface is forced into existence — reflection is not an extra postulate but the four-pack's demand.

Step 2. Surface charge: at normal incidence $\mathbf{E}$ has no normal component anywhere, so $\sigma_f = D_\perp = 0$ . The mirror reflects this wave without any charge layer at all.

Step 3. Surface current: incident and reflected magnetic fields constructively interfere at the wall (each of amplitude $E_0/c$ ), so the tangential field just outside is $B_{\text{tot}}=2E_0/c = 2\times100/(3.00\times10^8) = 6.7\times10^{-7}$ T. Equation (6): $K_f = H_{1\parallel} = B_{\text{tot}}/\mu_0 = 6.67\times10^{-7}/1.257\times10^{-6} = \mathbf{0.53\ A/m}$ , oscillating at 10 GHz, perpendicular to $\mathbf{B}$ in the surface. This sheet is the physical reflector: it radiates the wave that cancels the field behind the mirror and doubles as the reflected wave in front.

Step 4. Standing-wave anatomy: total tangential $\mathbf{E}$ vanishes at the wall and at every half wavelength in front of it; $\lambda = c/f = 3.00$ cm, so nodes sit at $z = 0, 1.5, 3.0,\dots$ cm. (Your microwave oven's cold spots are this boundary condition, at 2.45 GHz and $\lambda/2 = 6.1$ cm — an anchor recomputed in Energy Density of the Electromagnetic Field.)

2 · A charging capacitor plate — the surface continuity equation with live numbers (exam level)

A circular plate of radius $R=5.0$ cm is fed a steady current $I=0.50$ A at its center by a wire; the plate charges uniformly (the standard idealization, shared with the worked example of The Displacement Current). Find the surface current distribution $\mathbf{K}_f(s)$ on the plate and verify equation (5) pointwise.

Step 1. Uniform charging means $\partial\sigma_f/\partial t = I/\pi R^2 = 0.50/(\pi\times0.0025) = \mathbf{64\ C\,m^{-2}s^{-1}}$ — the same number that drives $\partial E/\partial t = 7.2\times10^{12}$ V/m·s in the gap in that article's audit.

Step 2. Current arriving at the center must spread radially while depositing charge. Conservation on a disk of radius $s$ : current still flowing past radius $s$ equals what was delivered minus what has been parked, $2\pi s\,K(s) = I - \pi s^2\,(\partial\sigma_f/\partial t) = I(1 - s^2/R^2)$ . Hence

K(s)=\frac{I}{2\pi s}\left(1-\frac{s^2}{R^2}\right),

falling from the wire-fed center to exactly zero at the rim: $K(R/2) = \frac{0.50}{2\pi\times0.025}(1-0.25) = \mathbf{2.4\ A/m}$ , $K(R)=0$ — no current leaks off the edge, as it must not.

Step 3. Verify the surface continuity equation pointwise. The surface divergence in polar coordinates: $\nabla_s\cdot\mathbf{K} = \frac{1}{s}\frac{\partial}{\partial s}(sK) = \frac{1}{s}\frac{\partial}{\partial s}\!\left[\frac{I}{2\pi}\Bigl(1-\frac{s^2}{R^2}\Bigr)\right] = -\frac{I}{\pi R^2}$ , uniform over the plate. Then $\partial\sigma_f/\partial t + \nabla_s\cdot\mathbf{K}_f = \frac{I}{\pi R^2} - \frac{I}{\pi R^2} = 0$ , matching the right side of (5): away from the feed point no volume current leaves the plate face. Equation (5) balances identically, at every radius.

Step 4. The sheet $K(s)$ is not bookkeeping decoration: it produces a real tangential- $\mathbf{H}$ jump across the plate, $\Delta H_\parallel = K(s)$ — at $s=R/2$ a field step of $2.4$ A/m, i.e. $\mu_0\Delta H = 3.0$ μT, measurable with a loop probe. Boundary conditions are laboratory statements.

Origins · 1 minArnold Sommerfeld and how this was discovered

Arnold Sommerfeld

1868–1951 · Munich — the theorist who made boundaries rigorous

Sommerfeld turned boundary conditions from a matching recipe into mathematics you could trust. His 1896 solution of diffraction by a half-plane was the first exact solution of Maxwell's equations honoring the boundary conditions on a metallic edge — no approximations, and a shock to physicists who assumed such problems were forever approximate. His 1909 analysis of radio waves over a lossy earth introduced the surface-wave and boundary-impedance ideas that antenna engineering still uses, and his "radiation condition" — fields at infinity must carry energy outward — completed the boundary data electrodynamics needs to be well posed. Through his Munich seminar (Heisenberg, Pauli, Bethe, Debye were students) and his Lectures on Theoretical Physics, the pillbox-and-loop discipline of this article became the standard it is today. Nominated for the Nobel Prize a record 84 times, he never won; his boundary conditions are in every electromagnetic solver on Earth.

History

1873

— Maxwell's Treatise poses field matching at material interfaces in integral form

1885

— Heaviside's vector recast makes the four-condition package explicit and teachable

1896

— Sommerfeld solves the perfectly conducting half-plane exactly: boundary conditions can be honored, not just approximated

1909

— Sommerfeld's lossy-earth propagation paper founds boundary-impedance methods for radio

1912

— the radiation condition completes the boundary data for exterior problems

1966

— Yee's FDTD algorithm builds the staggered grid around exactly these continuity rules

2011

— generalized sheet transition conditions launch metasurface optics: designer σ_f and K_f

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.3.6 and §9.3 (boundary conditions, reflection) · Jackson, Classical Electrodynamics, 3rd ed., §I.5 and §7.3 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 10–11 · Zangwill, Modern Electrodynamics, §2.3.3 and Ch. 17. The normal-incidence Fresnel amplitude is $r=(n_1-n_2)/(n_1+n_2)$ and the skin depth is $\delta=\sqrt{2/(\mu\sigma\omega)}$ ; the field-line refraction relation $\tan\theta_1/\tan\theta_2=\varepsilon_1/\varepsilon_2$ follows directly from tangential continuity. Worked numbers: mirror $K_f = 2E_0/\mu_0 c = 0.53$ A/m at $E_0=100$ V/m; charging plate $\partial\sigma_f/\partial t = I/\pi R^2 = 64$ C m⁻²s⁻¹ and $K(R/2)=2.4$ A/m for $I=0.5$ A, $R=5$ cm.

Builds on: The Displacement Current · Faraday's Law of Induction · The Continuity Equation.

Leads to: Fresnel Equations · Waveguides and Cavities · The Skin Effect.

Faraday's law says a changing B drives circulating E — why doesn't that break the continuity of tangential E at a surface?

Because the circulation Faraday drives is proportional to the flux through the circuit you test with. The boundary-condition loop has area $\ell h$ with $h\to0$ : however violently $\mathbf{B}$ oscillates, a bounded $\partial\mathbf{B}/\partial t$ through a vanishing area drives vanishing EMF. Induced electric fields are real and non-conservative — over finite loops (the betatron, the transformer) they do everything the induction articles promised — but they cannot create a step in $E_\parallel$ across a surface, because a step would show up in an infinitesimally thin loop, and that loop encloses no flux.

Are there really only four independent boundary conditions? What about normal E and tangential B?

At a linear-media interface, four is the full set. $E_\perp$ 's jump follows from $D_\perp$ 's via $\mathbf{D}=\varepsilon\mathbf{E}$ , and $B_\parallel$ 's from $H_\parallel$ 's via $\mathbf{H}=\mathbf{B}/\mu$ — writing all eight would double-count. The deep counting: each medium's Maxwell equations propagate the fields once initial and boundary data are set, and the four matching rules are exactly the data an interface must supply, no more. In wave problems you will often use just the two tangential conditions (on $\mathbf{E}$ and $\mathbf{H}$ ) — for source-free interfaces the normal pair is then automatic, which is why Fresnel derivations look like they "forgot" half the conditions. They didn't; the other half is redundant there.

Do these conditions hold if the boundary itself is moving?

Not in this form. The derivation planted the pillbox and loop on a stationary surface; a moving interface drags flux through the probe at a rate that no longer vanishes, and the conditions pick up velocity terms (e.g. tangential $\mathbf{E}$ continuity becomes continuity of $\mathbf{E}+\mathbf{v}\times\mathbf{B}$ 's tangential part — the co-moving field). Reflection off a moving mirror adds Doppler shifts on top. The clean way to handle motion is to transform to the boundary's rest frame, apply this article's conditions, and transform back — machinery that belongs to relativistic electrodynamics (E10) by name.

The collapse argument assumed bounded time derivatives. What if the surface itself carries an oscillating current sheet — isn't dB/dt singular there?

Sharp catch, and it is exactly the fine print in Derivation A. An idealized sheet makes $\mathbf{B}$ discontinuous, so $\partial\mathbf{B}/\partial t$ contains a surface delta and the innocent bound "max derivative × area" needs care. The resolution: the delta lives in the surface itself, and the loop's spanning area straddles it symmetrically — the flux contribution still scales with $h$ once you integrate across the sheet, so the conclusion stands for any sheet of finite total strength. What genuinely breaks the argument is a double layer (a dipole sheet of current or charge), which carries a derivative-of-delta singularity; those need extra jump terms, and metasurface theory (the generalized sheet transition conditions in the today-section) is precisely the systematic bookkeeping for them.

Where does the surface charge on a mirror come from in the femtoseconds an optical field gives it?

From conduction electrons shifting a fraction of an atomic spacing. The estimate is the reassuring part: a surface charge $\sigma_f=\varepsilon_0 E$ with $E\sim10^3$ V/m needs $\sigma_f/e \sim 5\times10^{10}$ electrons per m² — about one electron per ten million surface atoms, displaced by picometers. Metals rearrange charge on the relaxation clock $\tau=\varepsilon_0/\sigma_c\sim10^{-19}$ s, orders faster than the $10^{-15}$ s optical period, so the sheet keeps up effortlessly. The idealization only fails approaching the plasma frequency (ultraviolet for most metals), where the electrons can no longer follow and metals turn transparent — the honest boundary of "perfect conductor," handled in the E8/E11 conductor articles.

Is sigma_f on a perfect conductor a cause of the field or a consequence of it?

Both, self-consistently — but for problem-solving, treat it as a consequence. The exterior solution is fixed by the homogeneous conditions ( $E_\parallel=0$ , $B_\perp=0$ ) plus sources away from the wall; then $\sigma_f=D_\perp$ and $\mathbf{K}_f=\hat{\mathbf{n}}\times\mathbf{H}$ are read off as outputs. Physically, of course, those sheets are what enforce the homogeneous conditions in the first place — the induced current radiates exactly the field that cancels the interior and builds the reflected wave. The method-of-images articles ran on the same logic: solve with the constraint, harvest the charges afterward.

How do these four conditions produce the laws of optics?

By starving the problem of alternatives. Put an incident plane wave against an interface and try to satisfy continuous tangential $\mathbf{E}$ and $\mathbf{H}$ with the incident wave alone: impossible — the ledger doesn't balance at any instant. Add a reflected and a transmitted wave and demand the balance hold at every point of the surface and every time: the phases must then agree everywhere, which forces equal frequencies and matched tangential wavevectors (Snell's law and the law of reflection — the kinematics, derived in the E8 oblique-incidence article by name), and the surviving amplitude conditions become two linear equations whose solutions are the Fresnel coefficients. Every mirror, lens coating, and fiber facet is these four equations, iterated.