Boundary Conditions for the Magnetostatic Field

Every field theory in this track earns its crossing rules. Electrostatics earned them in E1: at a charged surface, $\vec E$ 's perpendicular component jumps by $\sigma/\varepsilon_0$ while the tangential part glides through untouched — two rules, pillbox and loop, that powered every boundary-value method afterwards. This article derives the magnetic mirror image, and the mirror swaps everything: at a current sheet, it is the perpendicular component of $\vec B$ that is always continuous — no exceptions, anywhere in the universe, because $\nabla\cdot\vec B=0$ tolerates no surface "magnetic charge" — and the tangential component that jumps, by exactly $\mu_0K$ , at right angles to the surface current that causes it. One compact vector equation holds both rules: $\vec B_{\rm above}-\vec B_{\rm below}=\mu_0\,\vec K\times\hat n$ . The chapter has been paying into this account all along. The infinite sheet's $\pm\mu_0K/2$ fields (the Ampère's-law trio) differ across the sheet by $\mu_0K$ — that was the boundary condition in disguise. The solenoid's interior $\mu_0nI$ dropping to exterior zero across a winding carrying $K=nI$ — the same rule, curved. This article turns those instances into the general theorem, adds the matching conditions for the vector potential ( $\vec A$ continuous, its normal derivative jumping), and then extracts the rule's mechanical dividend: the force per area on a current sheet is $\vec K\times\vec B_{\rm avg}$ — the average of the two sides, never the field of either side alone — which packages into the tidiest formula in magnet engineering, the magnetic pressure $B^2/2\mu_0$ . A 1.5-tesla MRI solenoid pushes outward on its own windings at nine atmospheres; the crossing rules are where that number comes from. The version of these rules for magnetized matter — where $\vec H$ takes over and permeabilities refract field lines — belongs to the next chapter and is deferred to it by name. Here: vacuum, currents, and the exact geometry of the jump.

Key takeaways

Perpendicular rule (from $\nabla\cdot\vec B=0$ , pillbox): $B^\perp$ is continuous across any surface — always, since nature supplies no magnetic surface charge.
Tangential rule (from Ampère's law, loop): $\vec B^\parallel$ jumps by $\mu_0K$ across a surface current, in the direction perpendicular to $\vec K$ ; the component of $\vec B^\parallel$ along $\vec K$ is continuous.
One packaging holds all of it: $\vec B_{\rm above}-\vec B_{\rm below}=\mu_0\,\vec K\times\hat n$ , with $\hat n$ the normal from "below" to "above."
The vector potential matches smoothly in value — $\vec A$ continuous across the sheet — while its normal derivative jumps: $\partial\vec A_{\rm above}/\partial n-\partial\vec A_{\rm below}/\partial n=-\mu_0\vec K$ .
Force on the sheet: the sheet cannot push itself — use the average field, $\vec f=\vec K\times\vec B_{\rm avg}$ (per area); for a solenoid this becomes the outward magnetic pressure $P=B^2/2\mu_0$ .
Audits already banked: infinite sheet ( $\pm\mu_0K/2$ : jump $\mu_0K$ ✓) and solenoid wall ( $\mu_0nI\to0$ across $K=nI$ ✓).
KEY IDEA: $\nabla\cdot\vec B=0$ makes $B^\perp$ continuous; $\nabla\times\vec B=\mu_0\vec J$ makes $\vec B^\parallel$ jump by $\mu_0\vec K\times\hat n$ — the entire discontinuity is tangential and set by the surface current.

The core insight

$\nabla\cdot\vec B=0$ makes $B^\perp$ continuous; $\nabla\times\vec B=\mu_0\vec J$ makes $\vec B^\parallel$ jump by $\mu_0\vec K\times\hat n$ — the entire discontinuity lives in the tangential component and is set by the surface current.

Where this lives today

The most spectacular current sheet in the neighborhood is the magnetopause — the boundary where Earth's field meets the shocked solar wind. It is, to good approximation, this article's equation (2) at planetary scale: the field magnitude and direction jump across a layer a few hundred kilometers thick carrying $\sim10$ – $100$ mA/m, and the layer sits where it does because the force balance of the worked examples holds — magnetic pressure $B^2/2\mu_0$ inside balancing solar-wind ram pressure outside. NASA's MMS mission flies four spacecraft in tetrahedron formation through this sheet repeatedly, measuring $\vec B$ on both sides and inverting the jump for $\vec K$ — literally worked example 2, with the data arriving at 100 samples per second. The same inversion runs at the heliospheric current sheet, at Jupiter's magnetodisc, and in every laboratory plasma whose diagnostics report field jumps. In the superconducting world the boundary conditions run in reverse: instead of currents dictating jumps, demanded fields dictate currents. The Meissner state enforces $\vec B=0$ inside and hence $B^\perp=0$ at the surface; the material manufactures whatever screening $\vec K$ equation (2) requires to cancel the interior field — up to the critical current where it no longer can. Magnet builders live in the force corollary daily: the $B^2/2\mu_0$ pressure sets the mechanical budgets of MRI solenoids (≈9 atm at 1.5 T), of the 8+ tesla LHC dipoles (hundreds of atmospheres, reacted by stainless collars), and of every pulsed-field lab, where reusable 60–100 tesla magnets take gigapascal pressures shot after shot (1.4 GPa at 60 T, 4.0 GPa at 100 T) only because CuNb conductor and Zylon reinforcement carry the hoop stress — it is the single-turn-coil technique, above ~100 T, that sacrifices its coil on every shot.

Origins · 1 minOliver Heaviside and how this was discovered

Oliver Heaviside

1850–1925 · London and Devon — the self-taught engineer who gave Maxwell's theory its shape

Heaviside left school at sixteen, learned physics as a telegraph operator, and never held an academic post — and modern electromagnetism is written in his notation. Maxwell's Treatise sprawled across twenty equations in potentials and quaternion fragments; in 1885 Heaviside compressed it to the four vector field equations every student now learns, championing $\vec E$ and $\vec B$ (and the divergence/curl grammar this article's derivations run on) against both the quaternionists and the potential-purists — "the duplex equations," he called them, with surface discontinuity conditions worked out as carefully as the bulk laws, because telegraphy lives at boundaries: cable sheaths, sea returns, the interfaces where signals reflect and leak. His operational calculus solved the resulting transmission problems decades before mathematicians licensed it ("Shall I refuse my dinner because I do not fully understand the process of digestion?"). He predicted the ionospheric layer that bears half his name (Kennelly–Heaviside), coined "inductance," "impedance," and "permeability," and died poor, cantankerous, and posthumously indispensable — the patron physicist of every field jump computed with a pillbox and a loop.

History

1865

— Maxwell's dynamical theory contains the field equations in potential form; surface conditions live implicitly in the mathematics.

1873

— The Treatise systematizes electromagnetic boundary problems; Kelvin-style pillbox-and-loop arguments spread through the literature.

1885

— Heaviside recasts Maxwell into the four vector equations and makes interface conditions explicit working tools of cable telegraphy.

1933

— Meissner and Ochsenfeld discover superconductors expel magnetic flux: boundary conditions enforced by matter itself, screening currents on demand.

2015

— The four MMS spacecraft begin flying through the magnetopause current sheet, inverting measured field jumps for K — the crossing rules as a daily space-physics instrument.

The physical picture

Boundary conditions are field equations read at point-blank range. The differential laws $\nabla\cdot\vec B=0$ and $\nabla\times\vec B=\mu_0\vec J$ hold everywhere, but at an idealized surface current — a solenoid's winding sheet, a superconductor's screening layer, the magnetopause — "everywhere" includes a two-dimensional layer where $\vec J$ is infinite (finite current in zero thickness). The differential laws don't fail there; they integrate into jump statements, exactly as Gauss's law did at a charged surface in E1. The method is the same two-instrument kit, and using it here is deliberate reuse: a vanishingly thin pillbox interrogates the divergence law, a vanishingly thin Amperian loop interrogates the curl law, and between them they extract everything a surface can do to a field. What the two laws permit is beautifully complementary to the electric case, and the swap is worth internalizing as physics rather than memorizing as table entries. Electric fields begin on charges, so a charge sheet can start new perpendicular field — hence the $\sigma/\varepsilon_0$ jump in $E^\perp$ . Magnetic fields begin nowhere — the no-monopole law — so no sheet of anything can create or destroy perpendicular $\vec B$ : flux that enters from below must exit above, period. What a current sheet can do is deflect field sideways: it is a source of curl, and curl lives in the tangential components. The jump is therefore purely tangential, perpendicular to $\vec K$ (the loop derivation shows why), and proportional to how much current the sheet carries. A field line crossing a current sheet keeps its normal component and gets its tangential component sheared — it kinks, like a light ray refracting, with $\mu_0K$ as the shear strength.

Figure

The two instruments at a current sheet — pillbox reads the divergence law, loop reads the curl law

The chapter's inventory already contains both rules incognito. The Ampère's-law trio produced a flat sheet whose field flips from $+\mu_0K/2$ to $-\mu_0K/2$ across it: tangential jump $\mu_0K$ , no perpendicular component at all. It produced a solenoid whose axial field falls from $\mu_0nI$ inside to zero outside across the winding — a sheet with $K=nI$ : tangential jump $\mu_0K$ again, while the radial (perpendicular) component stays zero-to-zero. Two geometries, one theorem waiting to be stated. Now we state it.

The crossing rules, derived

Set the stage: a surface carries current density $\vec K$ (amperes per meter of width); $\hat n$ is the unit normal pointing from the "below" side to the "above" side; fields immediately adjacent to the surface are $\vec B_{\rm above}$ and $\vec B_{\rm below}$ .

Derivation AThe perpendicular rule — no-monopoles as a boundary condition

The goal is the first crossing rule, $B^\perp_{\rm above}=B^\perp_{\rm below}$ , valid across any surface whatsoever — current-carrying or not, in vacuum or (looking ahead) in matter. The strategy is the E1 pillbox verbatim, applied to the magnetic Gauss law; the interest is in what doesn't appear on the right-hand side.

Step 1. Deploy the pillbox. Straddle the surface with a thin cylinder: faces of area $a$ parallel to the surface (one just above, one just below), wall of height $\epsilon\to0$ . Integrate $\nabla\cdot\vec B=0$ over its volume and apply the divergence theorem:

\oint\vec B\cdot\mathrm{d}\vec a =B^\perp_{\rm above}\,a-B^\perp_{\rm below}\,a+(\text{wall})=0 .

Step 2. Kill the wall, read the faces. The wall's area $\propto\epsilon\to0$ and $\vec B$ stays bounded, so the wall contributes nothing. What remains:

B^\perp_{\rm above}=B^\perp_{\rm below} \qquad (1)

where: both sides are evaluated at the same surface point. Compare the electric pillbox at this exact step: there, the enclosed surface charge $\sigma a$ survived the squeeze and produced the jump $\sigma/\varepsilon_0$ . Here the right-hand side is the no-monopole zero — there is no magnetic $\sigma$ to enclose — and the rule is unconditional. Even the most violent surface current cannot make flux appear or vanish: whatever threads into the sheet from one side threads out the other. Why this rule is deeper than its derivation: (1) is the boundary-value form of "field lines never end." It is why flux is a robust, countable commodity (the flux through a superconducting ring is conserved to one part in many billions), and it survives unchanged into the magnetized-matter chapter, where the tangential rules get renegotiated with $\vec H$ but $B^\perp$ 's continuity is untouchable. *Both known cases confirm rule (1): the solenoid's radial $B$ is zero on both sides of the winding (trivially continuous), and the flat sheet's field is purely tangential so $B^\perp=0=0$ on both sides. The geometry is identical to the electric pillbox — the only difference is that the source term is zero here, exactly as $\nabla\cdot\vec B=0$ versus $\nabla\cdot\vec E=\rho/\varepsilon_0$ dictates.*

Go deeper · Derivation B · 2 minThe tangential rule and the vector packaging

Derivation BThe tangential rule and the vector packaging

The goal is the second rule — the tangential field jumps by $\mu_0K$ , perpendicular to $\vec K$ — and its assembly with (1) into the single vector statement $\vec B_{\rm above}-\vec B_{\rm below}=\mu_0\vec K\times\hat n$ . The strategy runs the thin Amperian loop in its two independent orientations (the E1 tangential loop, upgraded: there the circulation was zero and tangential $\vec E$ glided through; here the loop can enclose surface current, and the enclosed amount depends on the loop's orientation relative to $\vec K$ — that orientation-dependence is what makes the jump directional).

Step 1. Loop perpendicular to K. Take a thin rectangle straddling the surface: long sides of length $\ell$ parallel to the surface and perpendicular to $\vec K$ , short sides $\epsilon\to0$ . Ampère's law: the circulation is $(B^\parallel_{\perp K,\rm above}-B^\parallel_{\perp K,\rm below})\,\ell$ (short sides die with $\epsilon$ ), and the loop's spanning ribbon is pierced by the sheet along a line of length $\ell$ , catching current $K\ell$ :

B^{\parallel}_{\perp K,\rm above}-B^{\parallel}_{\perp K,\rm below}=\mu_0 K .

The tangential component perpendicular to the current jumps by $\mu_0K$ — sign fixed by the right-hand rule (circulate so the loop's normal runs along $\vec K$ ).

Step 2. Loop parallel to K. Rotate the rectangle 90°, long sides now along $\vec K$ : the spanning ribbon now runs parallel to the current flow and catches none of it — $I_{\rm enc}=0$ :

B^{\parallel}_{\parallel K,\rm above}=B^{\parallel}_{\parallel K,\rm below} .

The tangential component along the current is continuous. The jump is not merely tangential — it is tangential and transverse to $\vec K$ : three constraints from two loops and a pillbox, which is all three components of $\vec B$ accounted for.

Step 3. Package. All three statements assemble into one vector equation. Claim:

\vec B_{\rm above}-\vec B_{\rm below}=\mu_0\,\vec K\times\hat n . \qquad (2)

Audit the claim against each rule: $\vec K\times\hat n$ is perpendicular to $\hat n$ — purely tangential, so the normal components match ✓ (rule 1); it is perpendicular to $\vec K$ — no jump along the current ✓ (Step 2); and its magnitude is $K$ — jump $\mu_0K$ transverse to the current ✓ (Step 1), with the right-hand orientation built into the cross product. Why the direction had to be $\vec K\times\hat n$ : the sheet is a source of curl, and curl deflects field around the current — just as a wire's field circles the wire, a sheet's contribution runs transverse to $\vec K$ , oppositely on the two sides. Equation (2) is the local, exact version of that picture, valid on curved surfaces (the loop and pillbox only ever probe a vanishing neighborhood, where every smooth surface is flat). *Two explicit checks confirm equation (2). Flat sheet: $\vec K=K\hat x$ , $\hat n=\hat z$ , so $\mu_0K(\hat x\times\hat z)=-\mu_0K\hat y$ , which matches $-\frac{\mu_0K}{2}\hat y-(+\frac{\mu_0K}{2}\hat y)$ from the Ampère's-law trio. Solenoid wall: $\vec K=nI\hat\phi$ , $\hat n=\hat s$ , giving $\mu_0nI(\hat\phi\times\hat s)=-\mu_0nI\hat z$ for outside-minus-inside — exactly the $\mu_0nI\to0$ drop across the winding.*

Before the potential-matching and force derivations, the mechanical picture they lead to — the field as compressed gas inside its own winding:

Figure

The solenoid as a pressure vessel — the field pushes its winding outward at B squared over two mu0

Go deeper · Derivation C · 2 minMatching the vector potential — and the force on the sheet

Derivation CMatching the vector potential — and the force on the sheet

The goal is twofold: the crossing rules for $\vec A$ (continuous in value, kinked in slope), which downstream boundary-value work needs; and the mechanical dividend — the force per unit area on a current sheet, $\vec f=\vec K\times\vec B_{\rm avg}$ , with its celebrated corollary $P=B^2/2\mu_0$ . The strategy for $\vec A$ : apply the same two instruments to $\vec A$ 's own defining properties ( $\nabla\cdot\vec A=0$ in Coulomb gauge; $\oint\vec A\cdot\mathrm{d}\vec\ell=\Phi$ ). For the force: the E1 average-field argument, recalled by name and re-run with Lorentz in place of Coulomb.

Step 1. A is continuous. Normal component: the Coulomb-gauge condition $\nabla\cdot\vec A=0$ pillboxes exactly like (1) — $A^\perp$ continuous. Tangential components: run a thin loop and use the flux meaning — $\oint\vec A\cdot\mathrm{d}\vec\ell=\Phi_{\rm enc}$ , and the flux through a loop of vanishing area is zero because $\vec B$ , though discontinuous, stays bounded (a finite jump is not a delta function). So tangential $\vec A$ matches too:

\vec A_{\rm above}=\vec A_{\rm below} \qquad (3)

— the potential crosses every current sheet without a step, which is what the solenoid example found by construction (both branches meeting at $\mu_0nIR/2$ ).

Step 2. Its slope kinks. The field jump (2) must come from somewhere, and with $\vec A$ continuous along the surface, all tangential derivatives match — the discontinuity is forced into the normal derivative. Writing $\vec B=\nabla\times\vec A$ and isolating the normal-derivative terms against (2):

\frac{\partial\vec A_{\rm above}}{\partial n}-\frac{\partial\vec A_{\rm below}}{\partial n}=-\mu_0\vec K \qquad (4)

— a kink of magnitude $\mu_0K$ , antiparallel to the current. (Solenoid audit: $\partial_sA_\phi$ is $+\mu_0nI/2$ inside and $-\mu_0nI/2$ outside at the wall — kink $-\mu_0nI=-\mu_0K$ ✓.)

Step 3. The sheet cannot push itself — average the field. What force does the field exert on the sheet? The naive $\vec K\times\vec B$ is ill-posed: which $\vec B$ ? The field is different on the two sides, and part of it is the sheet's own doing. Split the field at the surface into external plus self: $\vec B_{\rm above/below}=\vec B_{\rm ext}\pm\tfrac12\mu_0\vec K\times\hat n$ — the self-field is the flat-sheet $\pm\mu_0K/2$ contribution (locally, every smooth sheet is flat), antisymmetric across the sheet; the external part is whatever both sides share. The sheet feels only $\vec B_{\rm ext}$ (no self-force — the wires article's theorem, localized), and the antisymmetric self-part cancels in the two-side average:

\vec B_{\rm ext}=\frac{\vec B_{\rm above}+\vec B_{\rm below}}{2}\equiv\vec B_{\rm avg} \;\;\Longrightarrow\;\; \vec f=\vec K\times\vec B_{\rm avg} \quad(\text{force per unit area}). \qquad (5)

This is the magnetic twin of E1's rule that a charged conductor's surface feels $\sigma\vec E_{\rm avg}$ — same logic, same average, recalled rather than reinvented.

Step 4. The pressure corollary. Solenoid: $\vec K=nI\hat\phi$ , $\vec B_{\rm avg}=\tfrac12(\mu_0nI+0)\hat z=\tfrac12\vec B_{\rm in}$ . Then $\vec f=nI\,\hat\phi\times\tfrac12\mu_0nI\,\hat z=\tfrac12\mu_0n^2I^2\,\hat s$ — radially outward, magnitude

P=\frac{(\mu_0nI)^2}{2\mu_0}=\frac{B_{\rm in}^2}{2\mu_0} : \qquad (6)

the field pushes the winding apart with a pressure equal to its own energy density (a coincidence with deep roots — E7's stress tensor owns the general statement). The winding of a magnet is a pressure vessel with the field as its compressed gas. Equation (6) can be reached two independent ways: the average-field force $K\cdot B_{\rm in}/2=\mu_0n^2I^2/2$ and the pressure formula $B_{\rm in}^2/2\mu_0$ agree identically — for example, $n=1000$ m $^{-1}$ , $I=100$ A gives 6.28 kPa by both routes. Anchor values: 1.5 T yields $9.0\times10^5$ Pa $\approx$ 8.8 atm; 30 T yields $3.6\times10^8$ Pa $\approx$ 3500 atm, past the yield strength of soft copper.

Limiting cases

case	result	meaning
no surface current ( $\vec K=0$ )	$\vec B$ fully continuous	ordinary interfaces are magnetically invisible in vacuum
flat infinite sheet	$\pm\tfrac12\mu_0\vec K\times\hat n$ each side	the trio's result = pure self-field, external zero
solenoid wall	axial drop $\mu_0nI$ , radial continuous	(2) curved; outward pressure $B^2/2\mu_0$
superconductor surface (Meissner)	$B^\perp=0$ and interior $\vec B=0$ ; $\vec K$ adjusts to enforce it	boundary condition run in reverse: field dictates the screening current
magnetized media interface	$B^\perp$ continuous still; tangential rule rewritten with $\vec H$ and free current	deferred by name to the magnetic-matter chapter (its BC article)
field line crossing a sheet	normal component kept, tangential sheared by $\mu_0K$	the "kink" — refraction-like, with $\mu_0K$ the shear

Common trapWhich field pushes the sheet? Never the field on one side — always the average

The single most common error at surfaces, electric or magnetic: computing the force on a current sheet (or winding, or plasma boundary) as $\vec K\times\vec B$ using the field on one side. The field adjacent to the sheet includes the sheet's own contribution ( $\pm\mu_0K/2$ locally), and no object pushes itself — using the one-sided field silently adds or subtracts a self-force and typically doubles or zeroes the answer. The correct instrument is the two-side average, $\vec f=\vec K\times\tfrac12(\vec B_{\rm above}+\vec B_{\rm below})$ — the same average-field rule E1 established for $\sigma$ on conductors, and the reason the solenoid's pressure is $B^2/2\mu_0$ rather than $B^2/\mu_0$ . Second trap, one door over: these vacuum rules do not survive into magnetized matter unedited — at a material interface the bound surface currents are part of $\vec K$ , and the practical tangential condition is rewritten with $\vec H$ and free currents only (next chapter's boundary-conditions article). $B^\perp$ 's continuity, alone, crosses that door unchanged.

Worked examples

1 · Auditing the solenoid wall, then pricing it (confidence builder)

A solenoid has $n=1000$ turns/m at $I=100$ A ( $K=nI=10^5$ A/m, $B_{\rm in}=0.126$ T, $B_{\rm out}=0$ ). Check both crossing rules at the winding and find the force per area on it.

Step 1. Perpendicular rule. The radial component is zero inside and outside (the solenoid derivation's symmetry interrogation): $0=0$ ✓ — continuity holds vacuously but instructively: no flux escapes sideways through the winding.

Step 2. Tangential rule. With $\hat n=\hat s$ (outward) and $\vec K=K\hat\phi$ : $\mu_0\vec K\times\hat n=\mu_0K(\hat\phi\times\hat s)=-\mu_0K\hat z$ . And indeed $\vec B_{\rm out}-\vec B_{\rm in}=0-\mu_0nI\hat z=-\mu_0K\hat z$ ✓. The entire $0.126$ T drop is the boundary condition, executed by the winding.

Step 3. The force. Average field at the winding: $\tfrac12(0.126+0)=0.0628$ T, axial. Force per area: $f=KB_{\rm avg}=10^5\times0.0628$ = 6.3 kPa, radially outward — and the pressure route $B^2/2\mu_0=(0.126)^2/(2\times4\pi\times10^{-7})$ gives the same 6.3 kPa ✓. Modest here (a fifteenth of an atmosphere); the point is the audit — two independent routes, one number.

Step 4. Scale it. The pressure grows as $B^2$ : at 1.5 T (MRI main coil) it is $9\times10^5$ Pa ≈ 8.8 atm — the winding is permanently pressurized like a truck tire stack; at 30 T (record-class resistive magnets) it is $3.6\times10^8$ Pa ≈ 3500 atm, beyond copper's yield strength — which is why such magnets are built from dispersion-strengthened conductors and massive reinforcement, and why "magnet engineering" is substantially stress engineering.

2 · Reading a current sheet from field data (exam level)

Magnetometers report, just above a planar current layer at $z=0$ : $\vec B_{\rm above}=(2.0,\,1.0,\,3.0)$ mT; just below: $\vec B_{\rm below}=(-0.5,\,1.0,\,3.0)$ mT. Find $\vec K$ , verify consistency, and compute the force per area on the layer.

Step 1. Consistency audit first. Perpendicular ( $z$ ) components: $3.0=3.0$ mT ✓ — the data respect rule (1); if they didn't, no current sheet could explain them (that would take magnetic charge). The $y$ components also match — so the jump is purely along $x$ , magnitude $\Delta B_x=2.5$ mT.

Step 2. Invert the packaging. With $\hat n=\hat z$ : $\mu_0\vec K\times\hat z=(2.5,\,0,\,0)$ mT. Since $\hat y\times\hat z=\hat x$ , the current runs along $\hat y$ : $K=\Delta B_x/\mu_0=2.5\times10^{-3}/(4\pi\times10^{-7})$ = $2.0\times10^3$ A/m along $\hat y$ — the sheet carries two kiloamperes per meter of width, and its direction is perpendicular to the jump it causes, as the cross product demands.

Step 3. The kink, visualized. The tangential field swings from $\arctan(2.0/3.0)=34°$ tilt on top to $\arctan(-0.5/3.0)=-9.5°$ below (angles from the normal direction in the $xz$ -plane): the field line crossing the sheet keeps its vertical component and shears sideways — the refraction-like kink.

Step 4. Force on the layer. $\vec B_{\rm avg}=(0.75,\,1.0,\,3.0)$ mT. $\vec f=\vec K\times\vec B_{\rm avg}=2.0\times10^3\,\hat y\times(0.75\hat x+1.0\hat y+3.0\hat z)\times10^{-3}=(6.0,\,0,\,-1.5)$ N/m² — magnitude 6.2 N/m², pushing the sheet along $+x$ and squeezing it down: current layers in magnetized environments are load-bearing structures, which is precisely how the magnetopause (below) stands its ground.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §5.4.2 (Eqs. 5.74–5.76) · Jackson, Classical Electrodynamics, 3rd ed., §5.8–5.12 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 6 · Zangwill, Modern Electrodynamics, §10.3 The sheet self-field is $\tfrac12\mu_0K$ and the magnetic pressure is $B^2/2\mu_0$ ; $\mu_0=4\pi\times10^{-7}$ T·m/A (pre-2019 exact value). The infinite sheet's field derivation is recalled here as the self-field; media/ $\vec H$ refraction conditions are deferred to the magnetic-matter chapter. Worked numbers: 6.3 kPa by two routes; 8.8 atm at 1.5 T, 3500 atm at 30 T; $K=2.0$ kA/m and $f=6.2$ N/m² from the field-data inversion.

Builds on: Divergence of B and Absence of Magnetic Monopoles · Ampere's Law · Magnetic Vector Potential · Boundary Conditions for E and V

Leads to: Magnetization · The Auxiliary Field H · Magnetic Boundary Conditions

Why does the electric field jump in its perpendicular component but the magnetic field in its tangential one?

Because the jumps inherit the geometry of the source laws. Electric fields diverge from charges: a surface charge is a plane of field-line beginnings, and beginnings show up as a step in the component leaving the surface — perpendicular. Magnetic fields curl around currents: a surface current is a plane of circulation, and circulation shows up as a shear in the components along the surface — tangential, transverse to the current, opposite on the two sides (picture the flat sheet's field, $-\hat y$ above and $+\hat y$ below for $\vec K$ along $\hat x$ ). Each field's other component is protected by the law that has zero on its right side: no magnetic charge protects $B^\perp$ ; no electrostatic circulation protects $E^\parallel$ . The four rules are the four differential laws, read at a surface.

Are idealized surface currents ever real, or is K always an approximation?

Two genuinely distinct cases. In ordinary conductors, $K$ is always shorthand for a volume current in a thin layer — a solenoid's winding, a busbar's skin-depth layer at AC — and the "jump" is really a steep ramp across the layer's thickness; the idealization is excellent whenever you observe from distances large compared with that thickness (exactly like the charged-surface idealization in E1). In superconductors, the current genuinely flows in a layer of the London penetration depth, tens of nanometers — still finite, but so thin that for any classical purpose the surface-current description is essentially exact, and it is self-adjusting: the material generates whatever $\vec K$ the boundary condition $\vec B_{\rm inside}=0$ demands. Plasma current sheets (magnetopause, reconnection layers) sit between: hundreds of kilometers thick, yet thin compared to the systems they bound, and routinely modeled with equation (2).

The field jumps across the sheet, so which value does the sheet itself experience?

Neither — the arithmetic mean, and the warning box's logic is worth internalizing as a general principle. Decompose the field at the surface into the sheet's own contribution (locally $\pm\tfrac12\mu_0\vec K\times\hat n$ , antisymmetric — it flips across the sheet) plus everything external (symmetric — the same on both sides, since external sources produce smooth fields there). A body exerts no net force on itself, so only the external part pushes, and averaging the two sides is precisely the operation that deletes the antisymmetric self-part and keeps the symmetric external part. The same average appeared for surface charge in E1 and will appear for magnetized surfaces in the next chapter: whenever a source layer sits in its own discontinuity, average across it.

How do these conditions actually get used to solve problems, the way the electric ones powered image charges and separation of variables?

Identically — they are the glue conditions of magnetostatic boundary-value problems. In current-free regions one can introduce a magnetic scalar potential (flagged in the vector-potential article) or use $\vec A$ ; solve Laplace-type equations region by region; then stitch the regions together by demanding $B^\perp$ continuous and the tangential condition with whatever $\vec K$ lives on the interface. The magnetized-sphere problem of the coming chapter — the magnetic twin of E3's uniformly polarized sphere — is solved exactly this way, with (2) supplying the coefficients matching that $\sigma_b$ supplied electrically. The $\vec A$ -continuity conditions (3)–(4) run the same program when currents thread the region and a scalar potential is illegal. Every technique from E2 transfers; only the glue formulas changed.

Is the magnetic pressure B²/2μ0 a real pressure, like a gas pushing on a wall?

Mechanically indistinguishable from one, with one refinement. Normal to the field lines, a magnetic field pushes exactly like a gas at pressure $B^2/2\mu_0$ — the solenoid winding genuinely bears 8.8 atm at 1.5 T, measurable as hoop strain. The refinement: along the field lines the same quantity acts as a tension — field lines pull like stretched cords — so the full stress state is anisotropic (pressure across, tension along), packaged by the Maxwell stress tensor that the conservation-laws chapter constructs. The pair explains at a stroke why parallel field lines repel sideways (pressure) and why bent lines straighten (tension) — the two intuitions plasma physicists reason with daily, and both are the boundary-condition force rule (5) applied to the right surfaces.

What happens to these rules at the surface of magnetized material, where there is no free current at all?

They hold verbatim — with the bound currents included in $\vec K$ . A uniformly magnetized bar carries a genuine surface current $\vec K_b=\vec M\times\hat n$ (the counting argument of the next article), and equation (2) applied with that $\vec K_b$ gives the correct field jump at the magnet's surface; nothing in the derivations cared whether electrons drift through a lattice or circulate in atoms. The practical problem is that $\vec K_b$ is usually unknown until the field is solved — chicken and egg — which is why the matter chapter introduces $\vec H$ : its tangential condition involves only free surface current (often zero), turning the implicit problem into an explicit one. The division of labor: $\vec B$ -rules are always true and fundamental; $\vec H$ -rules are the workhorse when matter hides its currents. Both appear, derived, in that chapter's boundary-conditions article.

Do the crossing rules change for curved surfaces like the solenoid's cylinder?

No — and the reason is worth stating once, cleanly. The pillbox and loop are local instruments: their dimensions shrink to zero, and in that limit every smooth surface is indistinguishable from its tangent plane (curvature enters only at the next order in the instruments' size, and vanishes with them). So (1), (2), and the $\vec A$ -conditions hold pointwise on any smooth surface, with $\hat n$ and $\vec K$ the local normal and current density — exactly how we audited the cylindrical solenoid wall with a formula derived in flat geometry. What curvature does affect is the fields' subsequent behavior away from the surface (a curved sheet focuses or spreads its field), but that is the bulk problem's business, not the crossing rules'. Only at genuinely singular geometry — edges, corners, cusps — do the surface rules give way to the sharper local analyses E2 previewed for conductors.