10.02.04 · em-sr / magnetostatics

Boundary conditions at interfaces: normal and tangential components of E, D, B, H

shipped3 tiersLean: none

Anchor (Master): Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 1.4-1.5; Stratton, Electromagnetic Theory (1941), Ch. 1-3

Intuition Beginner

When an electromagnetic field crosses from one material to another (say, from glass to air), the field changes abruptly at the boundary. But not all components change in the same way. The key principle is:

Normal components (perpendicular to the surface) of D and B are continuous across the boundary (unless there is surface charge or surface monopoles).
Tangential components (parallel to the surface) of E and H are continuous across the boundary (unless there is surface current).

Think of it this way: the field lines of D (representing free charge effects) bend at the interface but do not break — they continue across, just at a different angle. Similarly, the field lines of B never have endpoints (no magnetic monopoles), so the normal component of B must be continuous.

For the tangential components, the electric field E along the surface cannot jump, because that would require infinite energy (it would mean a voltage difference over zero distance). The same argument applies to H unless there is a surface current flowing on the boundary.

Visual Beginner

Component	Condition	Exception
Normal $D_{n}$	Continuous	Jump if surface charge $σ_{f}$ present
Normal $B_{n}$	Always continuous	(No magnetic monopoles)
Tangential $E_{t}$	Continuous	Always
Tangential $H_{t}$	Continuous	Jump if surface current $K_{f}$ present

Worked example Beginner

A uniform electric field $E_{0} = E_{0} \hat{z}$ exists in vacuum. A large flat dielectric slab ( $ϵ_{r} = 4$ ) is placed perpendicular to the field. Find the fields inside the dielectric.

The field is entirely normal to the surface (the $z$ -direction is perpendicular to the slab). The boundary condition for the normal component of $D$ is: $D_{1}^{n} = D_{2}^{n}$ (no free surface charge). In vacuum, $D_{1} = ϵ_{0} E_{0}$ . Inside the dielectric, $D_{2} = ϵ E_{2} = 4 ϵ_{0} E_{2}$ . Setting them equal: $ϵ_{0} E_{0} = 4 ϵ_{0} E_{2}$ , so $E_{2} = E_{0} /4$ . The electric field is reduced by a factor of 4 inside the dielectric. The bound surface charge on the dielectric creates an opposing field that partially cancels the applied field.

Check your understanding Beginner

Formal definition Intermediate+

At an interface between media 1 and 2 with unit normal $\hat{n}$ pointing from 1 to 2, the four boundary conditions are:

\hat{n} \cdot (D_{2} - D_{1}) = σ_{f}

\hat{n} \cdot (B_{2} - B_{1}) = 0

\hat{n} \times (E_{2} - E_{1}) = 0

\hat{n} \times (H_{2} - H_{1}) = K_{f}

where $σ_{f}$ is the free surface charge density and $K_{f}$ is the free surface current density.

Derivation (pillbox method for normal D). Construct a Gaussian pillbox of area $Δ a$ and vanishing height straddling the interface. Apply Gauss's law: $\oint D \cdot d a = Q_{f, enc}$ . As the height shrinks to zero, the side contribution vanishes, leaving $D_{2}^{n} Δ a - D_{1}^{n} Δ a = σ_{f} Δ a$ , hence $\hat{n} \cdot (D_{2} - D_{1}) = σ_{f}$ .

Derivation (amperian loop for tangential E). Construct a rectangular loop of length $Δ l$ and vanishing height straddling the interface. Apply Faraday's law: $\oint E \cdot d l = - d Φ_{B} / d t$ . As the height shrinks to zero, the area vanishes, so the flux term vanishes, leaving $(E_{2}^{t} - E_{1}^{t}) Δ l = 0$ , hence $E_{1}^{t} = E_{2}^{t}$ .

In linear media. For linear, isotropic media with $D = ϵ E$ and $B = μ H$ , the boundary conditions become:

ϵ_{1} E_{1}^{n} = ϵ_{2} E_{2}^{n} (if σ_{f} = 0)

B_{1}^{n} = B_{2}^{n}

E_{1}^{t} = E_{2}^{t}

\frac{B _{1}^{t}}{μ _{1}} = \frac{B _{2}^{t}}{μ _{2}} (if K_{f} = 0)

The field lines refract at the interface. For $E$ , the angles $θ_{1}, θ_{2}$ with respect to the normal satisfy $ϵ_{1} tan θ_{2} = ϵ_{2} tan θ_{1}$ . For $B$ , the analogous relation is $μ_{1} tan θ_{2} = μ_{2} tan θ_{1}$ .

Key derivation Intermediate+

Derivation (Refraction of field lines at a dielectric interface).

Consider an interface between two linear dielectrics with permittivities $ϵ_{1}$ and $ϵ_{2}$ . The electric field in each medium has normal and tangential components. From the boundary conditions: $E_{1}^{t} = E_{2}^{t}$ and $ϵ_{1} E_{1}^{n} = ϵ_{2} E_{2}^{n}$ .

Let $θ_{1}$ and $θ_{2}$ be the angles the fields make with the normal. Then $E_{1}^{t} = E_{1} sin θ_{1}$ , $E_{1}^{n} = E_{1} cos θ_{1}$ , and similarly for medium 2. From the tangential condition: $E_{1} sin θ_{1} = E_{2} sin θ_{2}$ . From the normal condition: $ϵ_{1} E_{1} cos θ_{1} = ϵ_{2} E_{2} cos θ_{2}$ . Dividing:

\frac{tan θ _{1}}{ϵ _{1}} = \frac{tan θ _{2}}{ϵ _{2}}

This is the law of refraction for electric field lines. When $ϵ_{2} > ϵ_{1}$ , the field bends toward the normal (like light entering a denser medium). The field lines crowd together in the higher- $ϵ$ medium because $D$ is continuous (same number of lines) but $E$ is smaller.

Bridge. The boundary conditions are the foundation for solving every electromagnetic problem involving material interfaces: dielectric interfaces 10.04.03, conductor boundaries 10.01.03, waveguide walls 10.04.04, and cavity surfaces 10.04.05. The foundational insight is that Maxwell's equations in differential form are valid only in regions where the material properties are smooth; at interfaces, the differential equations must be supplemented by the boundary (jump) conditions. The central message is that the boundary conditions are not additional postulates but direct consequences of the integral form of Maxwell's equations applied to infinitesimal volumes and loops straddling the interface. Putting these together, the four conditions (two for normal components, two for tangential) provide exactly the constraints needed to determine the field on both sides of the interface given the source distribution, and they generalise to time-dependent problems without change (the time-derivative terms vanish as the pillbox or loop height goes to zero).

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Derive the boundary condition for the tangential component of $H$ from Ampere's law applied to an amperian loop straddling the interface.

Hint

Apply $\oint H \cdot d l = I_{f, enc}$ to a rectangle of length $ℓ$ and vanishing height. The enclosed free current is $K_{f} ℓ$ where $K_{f}$ is the surface current density.

Answer

Construct a rectangular loop of length $ℓ$ straddling the interface, with sides parallel and perpendicular to the surface. As the height goes to zero: $\oint H \cdot d l = H_{2}^{t} ℓ - H_{1}^{t} ℓ = I_{f, enc}$ . The enclosed free current is $K_{f} ℓ$ (surface current times length), where $K_{f}$ is the component of surface current perpendicular to the loop. In vector form: $\hat{n} \times (H_{2} - H_{1}) = K_{f}$ .

If there is no free surface current ( $K_{f} = 0$ ), then $H_{1}^{t} = H_{2}^{t}$ : the tangential component of $H$ is continuous. In linear media with $B = μ H$ : $B_{1}^{t} / μ_{1} = B_{2}^{t} / μ_{2}$ .

Exercise 4 (medium, symbolic).

A uniform magnetic field $B_{0} = B_{0} \hat{y}$ makes a 45-degree angle with the normal to a flat interface between two magnetic materials with $μ_{1} = 2 μ_{0}$ and $μ_{2} = 6 μ_{0}$ . Find the direction and magnitude of $B_{2}$ .

Hint

The angle with the normal satisfies $μ_{1} tan θ_{2} = μ_{2} tan θ_{1}$ . The normal component of $B$ is continuous; the tangential component of $H$ is continuous.

Answer

$θ_{1} = 4 5^{\circ}$ , so $tan θ_{1} = 1$ . From the refraction law: $μ_{1} tan θ_{2} = μ_{2} tan θ_{1}$ , giving $2 μ_{0} tan θ_{2} = 6 μ_{0} \cdot 1$ , so $tan θ_{2} = 3$ , $θ_{2} = arctan 3 \approx 71. 6^{\circ}$ . The field bends toward the surface in the higher- $μ$ medium.

Normal component: $B_{1}^{n} = B_{0} cos 4 5^{\circ} = B_{0} / 2$ . Since $B_{n}$ is continuous: $B_{2}^{n} = B_{0} / 2$ . Then $B_{2}^{t} = B_{2}^{n} tan θ_{2} = 3 B_{0} / 2$ . So $B_{2} = B_{0} / 2 1 + 9 = B_{0} 10/2 = B_{0} 5$ , directed at $71. 6^{\circ}$ from the normal. The field magnitude increased because $H$ is continuous tangentially and $μ_{2} > μ_{1}$ .

Exercise 5 (hard, symbolic).

A surface current $K = K \hat{y}$ flows on the flat $x$ - $y$ plane ( $z = 0$ ). Find the magnetic field above and below the current sheet using boundary conditions. Assume vacuum on both sides.

Hint

By symmetry and Ampere's law, $B_{x}$ is the only component and it jumps by $μ_{0} K$ across the sheet. Use $\hat{n} \times (B_{2} / μ_{0} - B_{1} / μ_{0}) = K$ .

Answer

For an infinite current sheet in the $x$ - $y$ plane with $K = K \hat{y}$ : by the right-hand rule and symmetry, $B$ is in the $\hat{x}$ direction, with $B_{x} > 0$ for $z < 0$ and $B_{x} < 0$ for $z > 0$ (or vice versa). The normal component ( $B_{z}$ ) is continuous and equals zero by symmetry. The tangential boundary condition: $\hat{n} \times (B_{2} / μ_{0} - B_{1} / μ_{0}) = K$ , with $\hat{n} = \hat{z}$ . So $\hat{z} \times (B_{2} - B_{1}) / μ_{0} = K \hat{y}$ . If $B_{1} = B_{1} \hat{x}$ (below, $z < 0$ ) and $B_{2} = B_{2} \hat{x}$ (above, $z > 0$ ): $\hat{z} \times (B_{2} - B_{1}) \hat{x} / μ_{0} = (B_{2} - B_{1}) \hat{y} / μ_{0} = K \hat{y}$ . So $B_{2} - B_{1} = μ_{0} K$ . By symmetry, $B_{1} = - B_{2}$ , giving $2 B_{2} = μ_{0} K$ , $B_{2} = μ_{0} K /2$ and $B_{1} = - μ_{0} K /2$ . Above: $B = \frac{μ _{0} K}{2} \hat{x}$ , below: $B = - \frac{μ _{0} K}{2} \hat{x}$ .

Exercise 6 (hard, symbolic).

Show that the boundary conditions are equivalent to the requirement that Maxwell's equations hold in the weak (distributional) sense. Explain the connection to the pillbox and loop constructions.

Hint

In the weak formulation, the fields are distributions and the derivatives include delta-function contributions at interfaces. The pillbox and loop constructions extract the coefficients of these delta functions.

Answer

In the distributional formulation, a field that is smooth except at an interface $S$ has derivatives that contain delta-function terms on $S$ . For example, if $D$ has a jump $[D] = D_{2} - D_{1}$ across $S$ , then $div D$ (in the distributional sense) equals the smooth divergence plus $\hat{n} \cdot [D] δ_{S}$ where $δ_{S}$ is the surface delta function.

Gauss's law in the distributional sense: $div D = ρ_{f}$ (distributionally). Matching the delta-function terms: $\hat{n} \cdot [D] δ_{S} = σ_{f} δ_{S}$ , giving $\hat{n} \cdot [D] = σ_{f}$ . This is exactly the pillbox boundary condition.

Similarly, $curl E = - \partial B / \partial t$ distributionally. The curl of a field with a tangential jump contains $\hat{n} \times [E] δ_{S}$ . Since $\partial B / \partial t$ has no delta-function part (B has no normal jump), we get $\hat{n} \times [E] = 0$ : the tangential E boundary condition from the amperian loop.

The pillbox and loop constructions are simply the physical realisations of extracting the distributional (delta-function) content of Maxwell's equations at discontinuities.

Exercise 7 (medium, symbolic).

A dielectric sphere of radius $R$ and permittivity $ϵ$ is placed in a uniform external field $E_{0}$ . State the boundary conditions at $r = R$ and explain how they determine the field inside.

Hint

At $r = R$ : $E_{θ}$ is continuous (tangential), $ϵ E_{r}$ is continuous (normal D, no free charge). These two conditions determine the two unknown coefficients in the interior solution $E_{in} = E_{0} 3/ (ϵ / ϵ_{0} + 2)$ .

Answer

By symmetry, the field inside is uniform and parallel to $E_{0}$ : $E_{in} = E_{in} \hat{z}$ . In spherical coordinates at $r = R$ , the tangential component is $E_{θ}$ and the normal component is $E_{r}$ .

Tangential condition: $E_{θ}^{out} (R) = E_{θ}^{in} (R)$ . Normal condition: $ϵ_{0} E_{r}^{out} (R) = ϵ E_{r}^{in} (R)$ (no free surface charge).

The exterior field is $E_{out} = E_{0} + E_{dipole}$ where the dipole field depends on the polarisation. Applying both conditions simultaneously gives $E_{in} = 3 ϵ_{0} / (ϵ + 2 ϵ_{0}) \cdot E_{0}$ . For $ϵ \to ϵ_{0}$ (no sphere): $E_{in} = E_{0}$ . For $ϵ \to \infty$ (conducting sphere): $E_{in} = 0$ .

The two boundary conditions provide exactly the right number of constraints to determine the two unknown quantities: the interior field strength and the induced dipole moment.

Lean formalization Intermediate+

Mathlib has differential forms and Sobolev space theory but not the electromagnetic boundary conditions, the pillbox/loop constructions, or the distributional interpretation of interface conditions. The surface charge and surface current sources as distributions are absent. lean_status: none.

Advanced results Master

Boundary conditions in the differential-forms language. Maxwell's equations in the language of differential forms are $d F = 0$ and $d * F = J$ where $F$ is the field 2-form and $J$ is the current 3-form. At a material interface, these equations must hold distributionally. The condition $d F = 0$ implies that the tangential components of $E$ and $H$ (the pullback of $F$ to the interface) are continuous. The condition $d * F = J$ implies that the normal components of $D$ and $B$ (the pullback of $\astF$ ) jump by the surface sources. This formulation is coordinate-free and manifestly Lorentz-invariant.

Interface conditions for nonlinear and anisotropic media. For nonlinear media ( $D$ depends nonlinearly on $E$ ), the boundary conditions still hold at the level of $D$ and $E$ : $D_{n}$ and $E_{t}$ are continuous. However, the constitutive relation $D (E)$ is different on each side, so the field on one side is a nonlinear function of the field on the other. For anisotropic media ( $D = ϵ \cdot E$ with tensor $ϵ$ ), the refraction law becomes more complex: $ϵ_{ij} E_{j} n_{i}$ is continuous (normal $D$ ) and $E_{i} - (E_{j} n_{j}) n_{i}$ is continuous (tangential $E$ ).

Leontovich boundary conditions (impedance boundary conditions). For a good (but not perfect) conductor, the fields penetrate a skin depth $δ = 2/ (μ σ ω)$ . The Leontovich condition approximates the conductor by a surface impedance: $E_{t} = Z_{s} \hat{n} \times H_{t}$ where $Z_{s} = (1 + i) / (σ δ) = μ ω / (2 σ) (1 + i)$ is the surface impedance. This avoids solving for the fields inside the conductor.

Synthesis. The boundary conditions at material interfaces are the essential bridge between Maxwell's equations in bulk matter and the solution of practical electromagnetic problems. The foundational insight is that the four boundary conditions (two normal, two tangential) are direct consequences of the integral form of Maxwell's equations and provide exactly the constraints needed to match solutions across interfaces. The central message is that the boundary conditions are universal: they hold for static and time-dependent fields, for linear and nonlinear media, and in any coordinate system. Putting these together, the boundary conditions are the load-bearing structure for every electromagnetic boundary-value problem: dielectrics 10.04.03, conductors 10.01.03, waveguides 10.04.04, cavities 10.04.05, and scattering problems 10.07.06.

Full proof set Master

Proposition (The four boundary conditions). The boundary conditions at a material interface are necessary and sufficient for Maxwell's equations to hold in the presence of discontinuities.

Proof (necessity). Each boundary condition is derived by applying the corresponding integral law to an infinitesimal volume or loop straddling the interface:

$\hat{n} \cdot [D] = σ_{f}$ : Apply $\oint_{S} D \cdot d a = Q_{f, enc}$ to a pillbox of vanishing height.
$\hat{n} \cdot [B] = 0$ : Apply $\oint_{S} B \cdot d a = 0$ to a pillbox of vanishing height.
$\hat{n} \times [E] = 0$ : Apply $\oint_{C} E \cdot d l = - d / d t \int_{S} B \cdot d a$ to a loop of vanishing height (the flux term vanishes).
$\hat{n} \times [H] = K_{f}$ : Apply $\oint_{C} H \cdot d l = I_{f, enc} + d / d t \int_{S} D \cdot d a$ to a loop of vanishing height (the displacement current term vanishes).

Sufficiency. The boundary conditions ensure that Maxwell's equations hold in the weak (distributional) sense at the interface, which together with the equations in the bulk gives a well-posed problem. $□$

Connections Master

Electrostatics 10.01.03 uses the E and D boundary conditions to solve conductor and dielectric problems.
Dielectrics 10.01.04 relate D and E through the permittivity; the boundary conditions link fields across interfaces.
Plane waves 10.04.03 use all four boundary conditions to derive Fresnel reflection/transmission coefficients.
Waveguides 10.04.04 require the tangential E and normal B conditions at conducting walls.
Magnetic materials 10.02.03 add the B-H relation and the boundary conditions for H.
Covariant EM 10.06.01 expresses the boundary conditions in a manifestly Lorentz-invariant form.

Historical & philosophical context Master

The boundary conditions were implicit in Maxwell's original formulation (1865) but were first stated explicitly by Oliver Heaviside in the 1880s as part of his vector-calculus reformulation of Maxwell's theory. Heaviside recognised that the differential form of Maxwell's equations requires smooth material properties, and that the interface conditions must be stated separately.

The pillbox and amperian loop constructions became standard pedagogical tools in the 20th century, appearing in every EM textbook from Abraham and Becker (1932) through Griffiths (2017). The distributional interpretation (boundary conditions as delta-function terms in the differential equations) was made precise by Laurent Schwartz in his theory of distributions (1950).

The Leontovich impedance boundary condition (1940s) was an important practical advance: it allowed engineers to treat conducting surfaces (like the walls of a waveguide) without solving for the fields inside the conductor, greatly simplifying antenna and waveguide design.

Bibliography Master

Heaviside, O., Electromagnetic Theory, Vols. I-III (Electrician Publishing, 1893-1912).
Stratton, J. A., Electromagnetic Theory (McGraw-Hill, 1941).
Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge, 2017).
Leontovich, M. A., "On the approximate boundary conditions for electromagnetic fields on the surface of well-conducting bodies," Investigations of Radio Waves (1948).

Prerequisites

10.02.01
10.01.03
10.01.04

Used in

10.02.03
10.04.03

Tier anchors

beginner: Griffiths, Introduction to Electrodynamics, 4th ed. (2017), Ch. 4.4, 5.4.2
intermediate: Griffiths, Introduction to Electrodynamics, 4th ed. (2017), Ch. 4-5; Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 1-5
master: Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 1.4-1.5; Stratton, Electromagnetic Theory (1941), Ch. 1-3

References

Griffiths — Introduction to Electrodynamics, 4th ed. (Cambridge, 2017) · Ch. 4.4 Boundary conditions; Ch. 5.4.2 Boundary conditions for B and A
Jackson — Classical Electrodynamics, 3rd ed. (Wiley, 1999) · Ch. 1.4 Boundary conditions at a surface of discontinuity; Ch. 5.8 Magnetostatic boundary-value problems
Stratton — Electromagnetic Theory (McGraw-Hill, 1941) · Ch. 1 The electromagnetic field; boundary conditions at surfaces of discontinuity

Estimated time

beginner: 15m
intermediate: 35m
master: 55m