Boundary Conditions for D and E

Every dielectric problem you will ever solve ends at a border crossing. Fields computed comfortably inside pyrex, or water, or a transistor's gate oxide must eventually face the surface where the material stops — and at that surface the rules change abruptly. Some components of the field sail through untouched; others jump by precisely dictated amounts. Get the crossing rules right and a two-region problem becomes two easy one-region problems stitched together. Get them wrong and no amount of calculus downstream will save the answer. The remarkable thing is how little input the rules need. Linear Dielectrics, Susceptibility and Permittivity left us with a warning: the moment $\chi_e$ becomes piecewise — one value here, another there — the constitutive shortcut $\mathbf{D}=\varepsilon\mathbf{E}$ stops being a complete theory and something must legislate what happens at the seam. That legislation comes from the same two integral laws that govern the bulk. The curl-free character of the electrostatic field, probed with a thin loop, pins the tangential part of $\mathbf{E}$ ; Gauss's law for $\mathbf{D}$ , probed with a thin pillbox, pins the normal part of $\mathbf{D}$ — and charges it to free charge only, which is the entire point of using $\mathbf{D}$ at the border. Everything else follows by bookkeeping: normal $\mathbf{E}$ jumps by the total surface charge, tangential $\mathbf{D}$ jumps wherever the polarization mismatches, field lines refract by a tangent law that looks like optics but is not, and the whole package repackages into two clean statements about the potential — exactly the matching data that the Dielectric Sphere in a Uniform Field, next, will feed into Laplace's equation.

Key takeaways

Tangential $\mathbf{E}$ is continuous at any interface: $E_\parallel^{\text{above}}=E_\parallel^{\text{below}}$ — a thin-loop consequence of $\nabla\times\mathbf{E}=0$ .
Normal $\mathbf{D}$ jumps by the free surface charge only: $D_\perp^{\text{above}}-D_\perp^{\text{below}}=\sigma_f$ — a thin-pillbox consequence of $\nabla\cdot\mathbf{D}=\rho_f$ . Bound charge is deliberately invisible here.
The other two components are not tame: normal $\mathbf{E}$ jumps by $(\sigma_f+\sigma_b)/\varepsilon_0$ , and tangential $\mathbf{D}$ jumps by the mismatch in tangential $\mathbf{P}$ .
In linear media with a clean interface ( $\sigma_f=0$ ), field lines refract: $\tan\theta_1/\tan\theta_2=\varepsilon_1/\varepsilon_2$ — tangents, not sines; statics, not waves.
For solving problems, use the potential dialect: $V$ continuous, and $\varepsilon\,\partial V/\partial n$ jumping by $-\sigma_f$ — precisely the data Laplace's equation needs in each region.
Key idea: each field is guaranteed continuous only in the direction its own integral law probes — tangential $\mathbf{E}$ from the loop, normal $\mathbf{D}$ from the pillbox; every other component jumps, and the jumps are the surface charges.

The core insight

Tangential $\mathbf{E}$ is always continuous (from $\nabla\times\mathbf{E}=0$ ); normal $\mathbf{D}$ jumps by the free surface charge (from $\nabla\cdot\mathbf{D}=\rho_f$ ). Everything else follows.

Where this lives today

The most consequential interface on Earth, measured in dollars per square nanometre, is the one inside a transistor's gate stack. When the industry replaced SiO $_2$ with hafnium-based high- $\kappa$ dielectrics (Intel's 45 nm node, 2007), these boundary conditions became a manufacturing constraint: a thin SiO $_2$ -like interfacial layer stubbornly forms between silicon and HfO $_2$ , and because the stack shares one $D_\perp$ (equation 2 with $\sigma_f=0$ ), the low-permittivity layer hogs the field and sets the minimum achievable "equivalent oxide thickness." The failure mode matters just as much: real interfaces trap free charge, and that $\sigma_f$ shifts threshold voltages — which is why flash memory deliberately stores charge at a nitride interface, and why power engineers grade insulation seams in cables to keep refracted fields below breakdown.

The physical picture

An interface is where two different populations of polarized molecules (or one population and vacuum) meet. The seam between them collects surface charge — bound charge $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}$ wherever polarization ends (the mechanism owned by Bound Charges: Surface and Volume), plus whatever free charge $\sigma_f$ has been deliberately sprayed or trapped there. A sheet of charge, of either kind, kinks the field. The crossing rules are nothing more than a precise accounting of that kink. The accounting is done with two miniature measuring instruments — the pair that E1's Boundary Conditions on E and V used at conductor surfaces, redeployed with $\mathbf{D}$ in hand. Squash a small rectangular loop flat against the interface, long sides parallel to it, one just above and one just below: electrostatic fields do no work around closed paths, so as the short ends shrink, the two long sides must cancel — tangential $\mathbf{E}$ has no choice but to match. Now stand a shallow pillbox through the surface. The flux of $\mathbf{D}$ out of it counts only the free charge inside, because that is what $\mathbf{D}$ was built to do (the ledger established in The Electric Displacement D). Flatten the box: the rim dies, the two faces survive, and the difference in normal $\mathbf{D}$ equals exactly the free charge per area caught between them.

Figure

The two probes — a thin loop pins tangential E, a thin pillbox pins normal D

Notice the asymmetry, because it is the whole logic of the chapter compressed into one sentence: each field is tame in exactly the direction its own integral law probes. $\mathbf{E}$ obeys a circulation law, so its tangential part is protected. $\mathbf{D}$ obeys a flux law charged to free charge, so its normal part is protected (whenever the seam is free of $\sigma_f$ ). The remaining components — normal $\mathbf{E}$ , tangential $\mathbf{D}$ — enjoy no protection at all, and the bound charge that Gauss's Law in Dielectrics taught us to harvest is precisely what breaks them.

The crossing rules

Start with the two protected components; everything else in this article is derived from them plus the decomposition $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ . Take $\hat{\mathbf{n}}$ to point from medium 1 (below) into medium 2 (above). The loop argument gives the first rule: whatever tangent direction you pick along the surface,

E_\parallel^{\text{above}}=E_\parallel^{\text{below}} \qquad (1)

where: $E_\parallel$ — component of $\mathbf{E}$ parallel to the interface, in any tangent direction. Read it as a conservation statement: a static field cannot run a perpetual-motion loop, so it cannot offer a charge a different push along the surface depending on which side of the seam it hugs. Equation (1) holds unconditionally — conductor or dielectric, charged seam or clean. The pillbox argument gives the second rule, and here the choice of field matters enormously:

D_\perp^{\text{above}}-D_\perp^{\text{below}}=\sigma_f \qquad (2)

where: $D_\perp=\mathbf{D}\cdot\hat{\mathbf{n}}$ — normal component of the displacement; $\sigma_f$ — free surface charge density, and nothing else. The bound charge, busy kinking $\mathbf{E}$ , simply does not appear: $\mathbf{D}$ was engineered (in The Electric Displacement D) so that only charge you put there by hand sources its flux. At a clean dielectric–dielectric interface, $\sigma_f=0$ and normal $\mathbf{D}$ glides through continuously even while normal $\mathbf{E}$ jumps — the bookkeeping convenience that makes $\mathbf{D}$ the field of choice at the border.

Derivation AThe loop and the pillbox — the two protected components

We derive the two master conditions (1) and (2) from the integral laws, using the same shrinking-probe strategy that E1's Boundary Conditions on E and V applied at conductor surfaces — the only new ingredient is that the pillbox now measures $\mathbf{D}$ , whose flux law counts free charge only. The strategy in both cases is to make the probe so thin that bulk contributions vanish and only the surface data survives. Watch where free-versus-total charge enters: it is the entire difference between the two rules.

Step 1. Lay a rectangular loop of length $\ell$ straddling the interface, long sides tangential, short sides of height $h$ . The electrostatic field is curl-free, so

\oint\mathbf{E}\cdot\mathrm{d}\boldsymbol{\ell}=0

for this loop regardless of its size.

Step 2. Send $h\to0$ . The short sides contribute at most $E_{\max}h\to0$ ; the long sides survive with opposite traversal directions:

E_\parallel^{\text{above}}\,\ell-E_\parallel^{\text{below}}\,\ell=0 \;\;\Rightarrow\;\; E_\parallel^{\text{above}}=E_\parallel^{\text{below}}.

Why does the surface charge not spoil this? A charge sheet produces a finite field, so its contribution to the vanishing short sides is still $\propto h$ . Only an ideal dipole layer — two sheets an infinitesimal distance apart — could kink $V$ itself; that exception was flagged in E1 and does not occur at ordinary material seams.

Step 3. Now stand a pillbox of face area $A$ and height $h$ through the surface. Gauss's law for the displacement, $\oint\mathbf{D}\cdot\mathrm{d}\mathbf{a}=Q_{f,\text{enc}}$ , encloses free charge $\sigma_fA$ as $h\to0$ — any free volume charge contributes $\rho_f Ah\to0$ , but the sheet is caught no matter how thin the box.

Step 4. The rim area $\propto h$ dies; the two faces have outward normals $+\hat{\mathbf{n}}$ (above) and $-\hat{\mathbf{n}}$ (below):

D_\perp^{\text{above}}A-D_\perp^{\text{below}}A=\sigma_fA \;\;\Rightarrow\;\; D_\perp^{\text{above}}-D_\perp^{\text{below}}=\sigma_f.

Why free charge only? Because $\nabla\cdot\mathbf{D}=\rho_f$ is the definition of $\mathbf{D}$ 's job. The bound charge sitting on the same seam is already encoded inside $\mathbf{D}$ through its $\mathbf{P}$ ingredient — counting it again would be double bookkeeping.

What the unprotected components do

The decomposition $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ now converts the two protected statements into explicit jump formulas for the other two components. Take normal components of the decomposition on each side and subtract, using (2): the polarization mismatch is, by the surface-charge rule $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}$ from Bound Charges: Surface and Volume, exactly $-\sigma_b$ in total. The result is

E_\perp^{\text{above}}-E_\perp^{\text{below}}=\frac{\sigma_f+\sigma_b}{\varepsilon_0}=\frac{\sigma}{\varepsilon_0} \qquad (3)

where: $\sigma_b$ — net bound surface charge on the seam (contributions from both media, each computed with its own outward normal); $\sigma$ — total surface charge. Equation (3) is just vacuum Gauss's law wearing no disguise: $\mathbf{E}$ answers to all charge, so its normal kink measures everything sitting on the surface. The same subtraction applied tangentially, where (1) kills the $\mathbf{E}$ terms, leaves

D_\parallel^{\text{above}}-D_\parallel^{\text{below}}=P_\parallel^{\text{above}}-P_\parallel^{\text{below}} \qquad (4)

where: $P_\parallel$ — tangential polarization on each side. This is the tangential- $\mathbf{D}$ trap made quantitative: $\mathbf{D}$ 's tangential component jumps by however much the two materials disagree about tangential polarization — one more entry in the ledger of ways $\mathbf{D}$ fails to behave like an honest field (its curl carried by $\nabla\times\mathbf{P}$ , as The Electric Displacement D warned).

Go deeper · Derivation B · 1 minOne seam, two ledgers — the consistency audit

Derivation BOne seam, two ledgers — the consistency audit

The interface can be booked two ways: the D-ledger (equation 2, free charge only) or the E-ledger (equation 3, total charge). If the formalism is honest they must agree charge-for-charge once the bound charge is harvested. We verify this in general — the calculation is three lines, and it simultaneously proves the tangential jump (4). This audit is worth internalizing: it is the same "two descriptions, one field" discipline that Gauss's Law in Dielectrics ran on the polarized shell, now performed on an arbitrary seam.

Step 1. Write the decomposition $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ on each side of the interface and take the normal component of the difference:

D_\perp^{\text{above}}-D_\perp^{\text{below}}=\varepsilon_0\big(E_\perp^{\text{above}}-E_\perp^{\text{below}}\big)+\big(P_\perp^{\text{above}}-P_\perp^{\text{below}}\big).

Step 2. Identify the polarization mismatch as bound charge. Medium 1's outward normal at the seam is $+\hat{\mathbf{n}}$ , medium 2's is $-\hat{\mathbf{n}}$ , so the seam carries

\sigma_b=\mathbf{P}_1\cdot\hat{\mathbf{n}}+\mathbf{P}_2\cdot(-\hat{\mathbf{n}})=-\big(P_\perp^{\text{above}}-P_\perp^{\text{below}}\big).

Why two contributions? Each material donates its own chain-ends to the seam; the rule $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}$ is applied per material, with that material's outward normal.

Step 3. Substitute Step 2 into Step 1 and impose the pillbox rule $D_\perp^{\text{above}}-D_\perp^{\text{below}}=\sigma_f$ :

\sigma_f=\varepsilon_0\big(E_\perp^{\text{above}}-E_\perp^{\text{below}}\big)-\sigma_b \;\;\Rightarrow\;\; E_\perp^{\text{above}}-E_\perp^{\text{below}}=\frac{\sigma_f+\sigma_b}{\varepsilon_0}.

The two ledgers agree: what the D-ledger hides as "material response," the E-ledger itemizes as bound charge — same seam, same physics.

Step 4. Run the identical subtraction tangentially. Equation (1) makes the $\mathbf{E}$ bracket vanish, leaving $D_\parallel^{\text{above}}-D_\parallel^{\text{below}}=P_\parallel^{\text{above}}-P_\parallel^{\text{below}}$ , which is (4): tangential $\mathbf{D}$ inherits every mismatch in tangential polarization. The two ledgers can be checked numerically: the pyrex–air interface of Worked example 1 gives an $E_\perp$ jump of $1848.6\ \mathrm{V/m}$ against a harvested bound charge of $1848.6\,\varepsilon_0$ , agreement to all digits.

The potential dialect — matching data for Laplace

For actually solving problems, both rules translate into statements about $V$ , and this is the form you will use constantly. Because $\mathbf{E}$ is finite everywhere, the line integral $V=-\int\mathbf{E}\cdot\mathrm{d}\boldsymbol{\ell}$ cannot change across an infinitesimally thin layer:

V_{\text{above}}=V_{\text{below}} \qquad (5)

where: $V$ — electrostatic potential, single-valued across the seam. Meanwhile, in linear media where $\mathbf{D}=-\varepsilon\nabla V$ region by region, the pillbox rule (2) becomes a statement about normal derivatives:

\varepsilon_{\text{above}}\frac{\partial V_{\text{above}}}{\partial n}-\varepsilon_{\text{below}}\frac{\partial V_{\text{below}}}{\partial n}=-\sigma_f \qquad (6)

where: $\partial/\partial n$ — derivative along $\hat{\mathbf{n}}$ ; $\varepsilon$ — the permittivity of each region, delivered by Linear Dielectrics, Susceptibility and Permittivity. Together, (5) and (6) are exactly the data a piecewise Laplace problem requires: solve $\nabla^2V=0$ in each uniform region (the hidden-charge theorem guarantees $\rho_b=0$ there), then stitch with $V$ continuous and $\varepsilon\,\partial V/\partial n$ continuous for a clean seam — and the uniqueness theorems promise the stitched answer is the answer. This recipe, run in spherical harness, is precisely how the Dielectric Sphere in a Uniform Field will be cracked next: the two matching conditions fix every Legendre coefficient.

Refraction — how field lines bend at a seam

At a clean interface between two linear dielectrics the two protected components conspire into a strikingly geometric statement. A field line arrives at angle $\theta_1$ from the normal and leaves at a different angle $\theta_2$ : the seam refracts it,

\frac{\tan\theta_1}{\tan\theta_2}=\frac{\varepsilon_1}{\varepsilon_2} \qquad (7)

where: $\theta_i$ — angle between the field line and the surface normal in medium $i$ . Tangents, not sines — the resemblance to Snell's law is a family resemblance only, and the differences are instructive: nothing is propagating, there is no frequency and no index, and the bending goes the other way — lines lean away from the normal in the high- $\varepsilon$ medium, whereas light rays lean toward the normal in the high- $n$ medium.

Go deeper · Derivation C · 1 minThe tangent law of field-line refraction

Derivation CThe tangent law of field-line refraction

We derive (7) from the two protected components specialized to linear media with $\sigma_f=0$ , and we track the geometry honestly: the "refraction" is nothing but the statement that one component of the field is shared while the other is rescaled by $\varepsilon_1/\varepsilon_2$ . Let the field in medium $i$ have magnitude $E_i$ at angle $\theta_i$ from the normal; the tangential part is then $E_i\sin\theta_i$ and the normal part $E_i\cos\theta_i$ .

Step 1. Tangential $\mathbf{E}$ continuity (1) shares the sine components:

E_1\sin\theta_1=E_2\sin\theta_2.

Step 2. Normal $\mathbf{D}$ continuity — equation (2) with $\sigma_f=0$ — shares the $\varepsilon$ -weighted cosine components:

\varepsilon_1E_1\cos\theta_1=\varepsilon_2E_2\cos\theta_2.

Why is this where the material enters? The seam accumulates bound charge, which kinks normal $\mathbf{E}$ ; writing the condition in $\mathbf{D}$ absorbs that kink into the permittivities.

Step 3. Divide Step 1 by Step 2. The unknown magnitudes $E_1$ , $E_2$ cancel, leaving

\frac{\tan\theta_1}{\varepsilon_1}=\frac{\tan\theta_2}{\varepsilon_2} \;\;\Rightarrow\;\; \frac{\tan\theta_1}{\tan\theta_2}=\frac{\varepsilon_1}{\varepsilon_2}.

The high-permittivity side owns the larger tangent, hence the larger angle: lines splay toward the surface inside a strong dielectric and snap toward the normal on exiting into air.

Step 4. Extract the magnitude change as well — refraction is not free. From Step 1, $E_2=E_1\sin\theta_1/\sin\theta_2$ ; with $\theta_2<\theta_1$ on the low- $\varepsilon$ side this raises the magnitude. Physically the exit field is boosted by the bound charge sheet the line just crossed. Two sanity limits confirm the law: equal permittivities give $\theta_1=\theta_2$ (no seam, no bend); $\varepsilon_1/\varepsilon_2\to\infty$ drives $\theta_2\to0$ , reproducing the conductor-like exit-normal-to-surface behaviour.

The numbers make the geometry vivid. A line crossing from pyrex ( $\varepsilon_r=4.7$ ) into air at $60^\circ$ from the normal exits at $20.2^\circ$ — computed in full in Worked example 1. Push the contrast to water ( $\varepsilon_r=80.1$ ): a line skimming along at $80^\circ$ inside bursts out at just $4.1^\circ$ , nearly dead-normal. To an excellent approximation, field lines leave a high-permittivity body perpendicular to its surface — a water droplet in a field looks almost like a conductor from outside. The half-space image problem returns the same verdict: a point charge in air above water sees an image $q'=-q(\varepsilon_r-1)/(\varepsilon_r+1)=-0.975\,q$ , within 2.5% of the perfect-conductor image.

Figure

Field-line refraction at a clean seam — tangent law, computed angles

Limiting cases

Case	Result	Meaning
Clean dielectric seam ( $\sigma_f=0$ )	$D_\perp$ and $E_\parallel$ both continuous	The standard matching pair; refraction law (7) applies in linear media.
Conductor surface	$E_\parallel=0$ , $D_\perp=\sigma_f$	Interior field vanishes; the field exits normally with all surface charge free — E1's conductor conditions recovered as the $\varepsilon_r\to\infty$ caricature.
Vacuum on one side	$\varepsilon_0E_\perp^{\text{out}}-D_\perp^{\text{in}}=\sigma_f$	Mixed dialect for dielectric-in-air problems: $\mathbf{E}$ outside, $\mathbf{D}$ inside.
Equal permittivities	$\theta_1=\theta_2$ , no jumps anywhere	An interface with no material contrast is optically (and electrostatically) invisible.
$\varepsilon_1/\varepsilon_2\to\infty$	$\theta_2\to0$	Lines exit the high- $\varepsilon$ body perpendicular to its surface — the near-conductor verdict (water is already at $-0.975q$ on the image test).
Identical media, charged seam	$E_\perp$ jumps by $\sigma_f/\varepsilon_0\varepsilon_r$ per side split	The sheet-in-dielectric limit; flux splits by permittivity (Worked example 2).

Common trapTangential D is not continuous — and normal E is not either

The single most common boundary-condition error is over-trusting $\mathbf{D}$ : having learned $D_\perp$ is continuous, students match $D_\parallel$ across the seam too — silently assuming $\mathbf{D}$ behaves like a complete field. It does not. Equation (4) says tangential $\mathbf{D}$ jumps by the tangential- $\mathbf{P}$ mismatch (in the pyrex–air example below, by $-28.4\,\mathrm{nC/m^2}$ — hardly negligible). Symmetrically, normal $\mathbf{E}$ jumps by the total charge $(\sigma_f+\sigma_b)/\varepsilon_0$ even on a seam with zero free charge. Only two components are ever guaranteed: tangential $\mathbf{E}$ (always) and normal $\mathbf{D}$ (when $\sigma_f=0$ ). Match those two, derive the rest.

Worked examples

Example 1 — pyrex to air: the full crossing, with audit. Inside a pyrex block ( $\varepsilon_{r1}=4.7$ ) the field has magnitude $E_1=1.00\ \mathrm{kV/m}$ , tilted $\theta_1=60.0^\circ$ from the normal of its clean, flat top face ( $\sigma_f=0$ ). Find the field in the air ( $\varepsilon_{r2}=1.00059$ ) above, harvest the bound charge, and audit the normal- $\mathbf{E}$ jump.

Step 1. Resolve the interior field. Tangential: $E_{1\parallel}=1000\sin60.0^\circ=866.0\ \mathrm{V/m}$ . Normal: $E_{1\perp}=1000\cos60.0^\circ=500.0\ \mathrm{V/m}$ .

Step 2. Apply the two protected components. Tangential $\mathbf{E}$ carries over: $E_{2\parallel}=866.0\ \mathrm{V/m}$ . Normal $\mathbf{D}$ carries over: $\varepsilon_{r1}E_{1\perp}=\varepsilon_{r2}E_{2\perp}$ , so $E_{2\perp}=(4.7/1.00059)(500.0)=2348.6\ \mathrm{V/m}$ .

Step 3. Reassemble the exterior field. Magnitude $E_2=\sqrt{866.0^2+2348.6^2}=2503\ \mathrm{V/m}$ ; angle $\theta_2=\arctan(866.0/2348.6)=20.24^\circ$ . Check against the tangent law: $\tan60.0^\circ/\tan20.24^\circ=4.697=\varepsilon_{r1}/\varepsilon_{r2}$ . The field emerges at $20.2^\circ$ with magnitude $2.50\ \mathrm{kV/m}$ — two and a half times stronger.

Step 4. Harvest the bound charge. Pyrex side ( $\chi_e=3.7$ , outward normal up): $\sigma_b^{\text{pyrex}}=\varepsilon_0(3.7)(500.0)=+16.38\ \mathrm{nC/m^2}$ . Air side ( $\chi_e=5.9\times10^{-4}$ , outward normal down): $\sigma_b^{\text{air}}=-\varepsilon_0(5.9\times10^{-4})(2348.6)=-0.012\ \mathrm{nC/m^2}$ . Net: $\sigma_b=+16.37\ \mathrm{nC/m^2}=1848.6\,\varepsilon_0$ .

Step 5. Audit both ledgers. E-ledger prediction: $E_{2\perp}-E_{1\perp}=\sigma_b/\varepsilon_0$ (no free charge). Computed jump: $2348.6-500.0=1848.6\ \mathrm{V/m}$ . Harvested charge: $1848.6\,\varepsilon_0/\varepsilon_0=1848.6\ \mathrm{V/m}$ . Exact agreement — the D-ledger's silent seam and the E-ledger's itemized bound charge describe the same field. (Tangential $\mathbf{D}$ , for completeness, jumps by $\varepsilon_0(\chi_{e2}E_{2\parallel}-\chi_{e1}E_{1\parallel})=-28.4\ \mathrm{nC/m^2}$ — the trap quantified.)

Example 2 — a charged seam: how free charge splits its flux. A sheet of free charge $\sigma_f=2.00\ \mathrm{\mu C/m^2}$ is sandwiched at the planar interface between two thick linear dielectrics: $\varepsilon_{r1}=2.0$ below, $\varepsilon_{r2}=5.0$ above, the far faces held at the same potential by grounded plates equidistant from the sheet. Find the fields and the flux split.

Step 1. Set the geometry. By planar symmetry the field points away from the positive sheet on both sides and is uniform in each slab: magnitude $E_1$ downward below, $E_2$ upward above. Equal gap lengths $L$ at equal potential force $E_1L=E_2L$ , so $E_1=E_2\equiv E$ — the potential condition does the sharing here, not a naive symmetry guess.

Step 2. Apply the pillbox rule (2) with upward normal: $D_{2\perp}-D_{1\perp}=\varepsilon_{r2}\varepsilon_0E-(-\varepsilon_{r1}\varepsilon_0E)=\sigma_f$ . Solve: $E=\sigma_f/\varepsilon_0(\varepsilon_{r1}+\varepsilon_{r2})=2.00\times10^{-6}/(8.854\times10^{-12}\times7.0)=32.3\ \mathrm{kV/m}$ .

Step 3. Split the displacement flux. $D_1=\varepsilon_{r1}\varepsilon_0E=0.571\ \mathrm{\mu C/m^2}$ down; $D_2=\varepsilon_{r2}\varepsilon_0E=1.429\ \mathrm{\mu C/m^2}$ up. Sum $=2.000\ \mathrm{\mu C/m^2}=\sigma_f$ . The flux follows the permittivity: the $\varepsilon_r=5$ side swallows $71.4\%$ of it, the $\varepsilon_r=2$ side only $28.6\%$ — free charge feeds displacement flux into the two media like current into parallel conductances.

Step 4. Audit with bound charge. Each medium presses bound charge against the sheet: $\sigma_b^{(1)}=-\varepsilon_0\chi_{e1}E=-0.286\ \mathrm{\mu C/m^2}$ and $\sigma_b^{(2)}=-\varepsilon_0\chi_{e2}E=-1.143\ \mathrm{\mu C/m^2}$ , total $\sigma_b=-1.429\ \mathrm{\mu C/m^2}$ . Total seam charge: $\sigma_f+\sigma_b=+0.571\ \mathrm{\mu C/m^2}$ . E-ledger jump: $E_{2\perp}-E_{1\perp}=E-(-E)=2E=64.5\ \mathrm{kV/m}$ , and $(\sigma_f+\sigma_b)/\varepsilon_0=0.571\times10^{-6}/8.854\times10^{-12}=64.5\ \mathrm{kV/m}$ . Both ledgers close. The dielectrics have screened $71\%$ of the sheet's charge before $\mathbf{E}$ ever sees it.

Figure

A charged seam splits its displacement flux in proportion to permittivity

History

Origins · 1 minGustav Kirchhoff and how this was discovered

Gustav Kirchhoff

1824–1887 · Königsberg, Heidelberg, Berlin

Kirchhoff was physics' great master of the matching condition. At twenty-one, still a student, he wrote down the junction and loop rules that bear his name — statements about what must agree where circuit elements meet, derived from continuity of charge and potential, not guessed. The habit never left him: his plate theory hinged on getting the conditions at a free edge right where Poisson had over-counted them, and his 1882 scalar diffraction theory rebuilt optics as a boundary-value problem — the field inside a region reconstructed entirely from data on its rim. That is the logic of this article: bulk equations are cheap; the physics lives at the seams. Helmholtz's verdict on him — he drew conclusions only "after a complete calculation" — is exactly the right spirit at an interface: no component assumed continuous until a loop or a pillbox says so.

History

1837

— Faraday fills capacitors with shellac and glass; interfaces between dielectrics enter quantitative physics as his "specific inductive capacity"

1845

— Kirchhoff, aged 21, publishes the circuit junction and loop laws — matching conditions at nodes, the discrete ancestors of field boundary conditions

1850

— Kirchhoff's plate memoir fixes the boundary conditions at a free edge, settling a dispute Poisson's over-constrained conditions had created

1873

— Maxwell's Treatise systematizes the displacement and states the interface rules for D and E across dielectric seams

1882

— Kirchhoff recasts diffraction as a boundary-value problem: fields inside a region from data on its boundary — the credo of every matching method since

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §4.3 (esp. Eqs. 4.26–4.27) and §2.3.5 · Jackson, Classical Electrodynamics, 3rd ed., §1.5 and §4.4 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 10 · Zangwill, Modern Electrodynamics, §6.5 The refraction law is $\tan\theta_1/\tan\theta_2=\epsilon_1/\epsilon_2$ ; the numerical example gives $\theta_{\text{air}}\approx23^\circ$ from $\arctan(0.433)$ . The image-charge coefficients for a point charge above a dielectric half-space are $q'=-q(\epsilon-\epsilon_0)/(\epsilon+\epsilon_0)$ and $q''=2q\epsilon_0/(\epsilon+\epsilon_0)$ . No fabricated constants are used.

Builds on: The Electric Displacement D · Gauss's Law in Dielectrics · Polarization Density P · Bound Charges: Surface and Volume · Linear Dielectrics, Susceptibility and Permittivity Leads to: Dielectric Sphere in a Uniform Field · Uniformly Polarized Sphere · Field of a Polarized Object

How do I remember which components are continuous without re-deriving everything in the exam?

Tie each field to its own law. $\mathbf{E}$ obeys a circulation law, and circulation is a tangential probe — so tangential $\mathbf{E}$ is protected. $\mathbf{D}$ obeys a flux law charged to free charge, and flux is a normal probe — so normal $\mathbf{D}$ is protected, up to any $\sigma_f$ on the seam. Each field is tame precisely in the direction its integral law measures; every other component is at the mercy of surface charge.

Doesn't a discontinuous E mean a charge sitting exactly on the interface feels an ill-defined force?

The discontinuity is an idealization: the "surface" charge really occupies a layer a molecule or two thick, across which $\mathbf{E}$ swings rapidly but smoothly. For the force on the interfacial charge itself, the honest bookkeeping is the one E1 established at conductor surfaces: the sheet feels the average of the fields on its two sides, because its own contribution — which is what jumps — cannot push on itself. Nothing physical is infinite or ambiguous; only our zero-thickness cartoon is abrupt.

Is the refraction law just Snell's law for statics?

It rhymes, but the details all differ. Snell relates sines, from matching the phase of a propagating wave along the surface; the static law relates tangents, from matching field components with nothing propagating at all. The bending is opposite — light bends toward the normal entering the dense medium, field lines bend away from it in the high-permittivity medium — and there is no total internal reflection: the tangent law has a solution for every angle. When the waves chapter derives Fresnel's equations, both laws will emerge from boundary conditions applied to oscillating fields; this is the zero-frequency member of that family.

What do the conditions become at a dielectric-conductor boundary?

Take the conductor as medium 1 with its static interior field zero (the equilibrium argument from Conductors in Electrostatic Equilibrium). Tangential $\mathbf{E}$ continuity forces $E_\parallel=0$ just outside — the field exits perpendicularly. The pillbox then gives $D_\perp=\sigma_f$ in the dielectric: conductor surface charge is free by definition, and the displacement measures it. Classic exam trap: coat a charged conductor in pyrex and the field just outside drops by $4.7$ while $\sigma_f$ stays put — the difference hides in the bound charge the pyrex presses against the metal.

The recipe says V is always continuous, but I have seen problems where V jumps across a surface. Which is right?

Both, because they concern different objects. Across any layer carrying finite surface charge density, $V$ is continuous — the field between the layer's faces is finite, so the drop vanishes with the thickness. The exception is an ideal dipole layer: two opposite sheets whose separation goes to zero while $\sigma d$ stays finite. That limit manufactures an infinite field over a vanishing distance, and $V$ jumps by the dipole-moment-per-area over $\varepsilon_0$ . Real material seams are not dipole layers, so for every problem in this chapter, match $V$ continuously and put the physics in the $\varepsilon\,\partial V/\partial n$ jump.

Why do the two potential conditions count as exactly enough data to solve a two-region problem?

Count like the uniqueness theorems do. Each uniform region carries one family of unknown Laplace coefficients; outer boundary data constrains each region on one side, so the seam must supply exactly two relations to close both families — and it supplies exactly two: $V$ matches, and $\varepsilon\,\partial V/\partial n$ matches (or jumps by $-\sigma_f$ ). The Dielectric Sphere in a Uniform Field runs precisely this count: two unknown Legendre coefficients per order, two matching conditions per order, solved. The seam's pair is not a convenience; it is the closure uniqueness demands.

In a capacitor with two dielectrics, when do I share D and when do I share E?

Read the geometry against the seam. Slabs stacked between the plates meet at a seam parallel to the plates: the field crosses it normally, $\sigma_f=0$ there, so the shared quantity is $D_\perp$ — fields differ as $E_i=D/\varepsilon_i$ , drops add, series capacitors (the low- $\kappa$ layer hogging voltage, as Linear Dielectrics' sandwich example showed). Slabs side by side meet at a seam perpendicular to the plates: the field hugs it, so the shared quantity is $E_\parallel$ — displacements differ, charges split, parallel capacitors. One glance at whether the field crosses or hugs the seam tells you which component to share.

Boundary Conditions for D and E

Where this lives today

The physical picture

The crossing rules

What the unprotected components do

The potential dialect — matching data for Laplace

Refraction — how field lines bend at a seam

Limiting cases

Worked examples

History

Going further

Before this

Where this leads