Magnetic Energy in Materials

The electric-materials chapter closed its books with an energy ledger — Energy Stored in Dielectric Systems, where $\delta W = \int \mathbf{E}\cdot\delta\mathbf{D}\,d\tau$ settled what a capacitor full of dielectric actually costs and which "energy" a given formula counts. This article is that ledger's magnetic twin, and it inherits the twin's discipline: derive the work increment honestly, state whose work it counts, and only then spend it on the chapter's outstanding bills. The bills are substantial. Hysteresis quoted a loss law — heat equal to the loop's area, every cycle — with the derivation explicitly deferred here; we pay it, and see exactly why a linear material's loop encloses nothing. Ampère's Law for H found that a 2-millimetre air gap in a half-metre iron ring eats 95% of the winding's ampère-turns; we audit the same ring's energy and find the same 95% sitting in the same gap — no coincidence, and the derivation shows why. And Hysteresis's load line named $(BH)_{\max}$ the figure of merit for permanent magnets while owing the reason; the gap-budget theorem below is the payment: the product $|B_mH_m|$ is joules of usable gap field per cubic metre of magnet, which is why the magnet industry's entire price list is a staircase of energy products. One honesty flag flies over the whole article. The work increment's derivation needs one line of electrodynamics — the fact that a changing $\mathbf{B}$ induces an electric field the battery must fight. That law is Faraday's, owned five articles downstream by Faraday's Law of Induction. We borrow it at quoted, one-line level, exactly as Diamagnetism borrowed Lenz's rule before induction was formalized: the loan is flagged, the collateral is that every consequence drawn here survives the later derivation intact.

Key takeaways

Raising the field in a magnetizable medium costs the free-current sources $\delta W = \int \mathbf{H}\cdot\delta\mathbf{B}\,d\tau$ — the exact magnetic twin of the dielectric ledger's $\int\mathbf{E}\cdot\delta\mathbf{D}$ , with $\mathbf{H}$ playing the supply-side bookkeeper it has played all chapter.
For linear media the increments integrate to $w = \tfrac12\mathbf{B}\cdot\mathbf{H} = B^2/2\mu$ : at the same $B$ , an iron core stores $\mu_r$ times less energy density than air — the material supplies the magnetization, the battery only the free current.
Which-energy doctrine (the E3 twin's, prosecuted for magnetism): $\int\mathbf{H}\cdot\delta\mathbf{B}$ counts work done by external circuits; it silently folds in the (recoverable, in linear media) work of aligning the material, and it excludes the internal atomic energy that built the moments themselves.
The gapped ring audited: at $B = 0.60$ T, $w_{\rm gap} = 1.4\times10^5$ J/m³ versus $w_{\rm iron} = 29$ J/m³ — the 2 mm gap holds 95% of the energy, the same 95% as its share of the ampère-turns, because both budgets distribute as $H\,dl$ along the ring.
Cycling a ferromagnet dissipates $W_{\rm cycle} = \oint H\,dB$ = loop area per unit volume — Hysteresis's quoted law, derived here; linear media close the loop on themselves and dissipate nothing.
The gap-budget theorem: in a magnet-driven circuit, $B_gH_g\,V_{\rm gap} = |B_mH_m|\,V_{\rm magnet}$ — delivering a specified field to a specified gap takes minimum magnet volume when the magnet operates at $(BH)_{\max}$ . Hence the staircase: ferrite ≈ 30, Alnico ≈ 50, Nd₂Fe₁₄B ≈ 330 kJ/m³.
KEY IDEA: $\delta W = \int\mathbf{H}\cdot\delta\mathbf{B}$ is the magnetic energy ledger — read forward it prices fields in media, read around a loop it prices hysteresis heat, and read across a magnetic circuit it prices permanent magnets by the cubic metre.

The core insight

The work free-current sources do to establish fields in magnetizable matter is $\delta W = \int\mathbf{H}\cdot\delta\mathbf{B}\,d\tau$ — integrating to $B^2/2\mu$ in linear media, to the hysteresis-loop area around a cycle, and to the gap-budget theorem that makes $(BH)_{\max}$ the price of permanent magnetism.

Where this lives today

The gap-budget theorem is the quiet line item in the electric-vehicle bill of materials: traction-motor designers trade $(BH)_{\max}$ against temperature grade (the $H_{ci}(T)$ collapse Hysteresis flagged) and against rare-earth price, and "ferrite-assisted" or magnet-free synchronous-reluctance designs are all attempts to buy gap joules some other way. Wireless charging pads and MRI fringe-field shims are direct exercises in steering where $B^2/2\mu$ is allowed to accumulate. At the grid scale, the loop-area law is a commodity: core-loss watts per kilogram at 1.7 T/50 Hz is the headline specification by which electrical steel is priced, and amorphous-alloy distribution transformers exist because a loop area of a few J/m³ instead of forty repays its material premium in a decade of standing losses. And superconducting magnetic energy storage runs this article's vacuum limit as a business plan: with no iron and no loop, energy $B^2/2\mu_0$ in a multi-tesla bore is among the fastest-discharging storage physics available — pure equation (2), at scale.

Origins · 1 minJohn Henry Poynting and how this was discovered

John Henry Poynting

1852–1914 · Manchester, Cambridge, Birmingham

Poynting is the physicist who made "where is the energy?" a well-posed question. His 1884 theorem showed that Maxwell's theory does not merely conserve energy globally — it localizes it, with a density in the fields and a flux $\mathbf{E}\times\mathbf{H}$ carrying it through space, so that the energy lighting a lamp travels through the field around the wires, not the copper. This article works in that theorem's statics shadow: the increment $\int\mathbf{H}\cdot\delta\mathbf{B}$ is what his flux delivers during a slow ramp (the discarded surface term in the master derivation is literally his vector), and the full machinery — flow, momentum, stress — is the conservation-laws chapter this track reaches at E7. A Birmingham professor for thirty-four years and a farmer's son who kept a working farm all his life, Poynting also weighed the Earth with a chemical balance (his 1891 $G$ determination stood for a generation), predicted with Robertson the radiation-drag effect on orbiting dust that space physicists still call Poynting–Robertson drag, and chaired the committee that standardized British electrical units. He measured gently and localized precisely — the right patron for an article about where the joules actually sit.

History

1847

— Helmholtz's conservation memoir makes energy bookkeeping mandatory across all of physics

1853

— Kelvin expresses the energy of current systems, the seed of magnetic energy accounting

1873

— Maxwell's Treatise locates energy in the medium at density B·H/2, not in the currents

1880

— Warburg measures the magnetization loop's area and identifies it as heat per cycle

1884

— Poynting's theorem localizes electromagnetic energy and names its flux E cross H

1892

— Steinmetz reduces a decade of loop data to the empirical B to the 1.6 loss law

1900

— Ewing's Magnetic Induction in Iron consolidates the loop-area law into engineering practice

The physical picture

Follow the energy from the wall socket. A power supply drives free current into a winding around an iron core with an air gap — the geometry Ampère's Law for H solved for fields, now audited for joules. As the current ramps, flux through the winding grows, and here the borrowed line enters: changing flux induces a back-voltage the supply must push against (Faraday's law, on loan). Every coulomb pushed through that back-voltage deposits energy somewhere downstream. The ledger's question is where. Three destinations exist. Some energy lands in the field of the gap — recoverable, retrievable by ramping the current back down, the energy a solenoid returns with a spark when you break the circuit. Some lands in the iron as the work of magnetizing it — in a linear, lossless idealization also recoverable, handed back as the domains relax. And in a real ferromagnet, some is gone: converted to heat at every irreversible Barkhausen jump, the toll Hysteresis described wall by wall and this article now integrates into the loop-area law. The audit's headline surprise is geographic. Iron carries almost all the flux — that was its job — but almost none of the energy. Energy density in a linear medium is $B^2/2\mu$ , and iron's enormous $\mu$ sits in the denominator: at the same $B$ , iron is five thousand times cheaper per cubic metre than air. The expensive real estate is the gap. A magnetic circuit is like a hose loop with one constriction: the iron is fat hose (flux flows almost for free), the gap is the nozzle where all the pressure drop — and all the stored energy — concentrates. That is why the gap is where electromagnets do work, why $(BH)_{\max}$ prices magnets by the gap energy they can maintain, and why the lifting-magnet formula at the end of this article is nothing but gap-energy accounting per centimetre of lift.

Figure

The energy geography of a gapped iron ring - flux lives in the iron, energy lives in the gap

The ledger's master entry

Everything descends from one increment, derived once and honestly.

Derivation AThe work increment - delta W equals the integral of H dot delta B

The goal is the master formula $\delta W = \int\mathbf{H}\cdot\delta\mathbf{B}\,d\tau$ : the work external sources must do on the free currents to change the field configuration by $\delta\mathbf{B}$ in the presence of magnetizable matter. The strategy mirrors the dielectric twin (Energy Stored in Dielectric Systems) line for line: compute the work against induced EMFs, convert to fields via Ampère's law for $\mathbf{H}$ , and let a vector identity plus a vanishing surface term do the rest. The watch-out is the honesty flag: one step uses Faraday's induction law, which this track derives five articles from now — the loan is taken at one-line, quoted level, per the precedent Diamagnetism set with Lenz's rule.

Step 1. State the loan. When the field configuration changes by $\delta\mathbf{B}$ over a time $\delta t$ , an electric field appears with $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ — Faraday's Law of Induction, quoted. The sources driving the free current density $\mathbf{J}_f$ must do work against this induced field at rate $-\int\mathbf{E}\cdot\mathbf{J}_f\,d\tau$ (positive when the battery fights the back-EMF, which is the ramp-up case).

Step 2. Express the work over $\delta t$ :

\delta W = -\delta t\int\mathbf{E}\cdot\mathbf{J}_f\,d\tau

Why only free current appears: the battery is wired to the winding, not to the atoms. Bound currents also feel the induced E, but the energy exchanged with them is exactly what ends up counted, indirectly, in the field-and-material bookkeeping below — that is the which-energy doctrine, prosecuted after the derivation.

Step 3. Trade current for fields with the chapter's own law, $\nabla\times\mathbf{H} = \mathbf{J}_f$ (Ampère's Law for H):

\delta W = -\delta t\int\mathbf{E}\cdot(\nabla\times\mathbf{H})\,d\tau

Step 4. Apply the identity $\nabla\cdot(\mathbf{E}\times\mathbf{H}) = \mathbf{H}\cdot(\nabla\times\mathbf{E}) - \mathbf{E}\cdot(\nabla\times\mathbf{H})$ and integrate the divergence to a surface at infinity, where localized fields kill it:

\delta W = -\delta t\int\mathbf{H}\cdot(\nabla\times\mathbf{E})\,d\tau

Why the discarded surface term deserves its name: it is the flux of $\mathbf{E}\times\mathbf{H}$ — the Poynting vector, the energy-FLOW machinery owned by the conservation-laws chapter (E7). Discarding it says only that no energy escapes to infinity during a quasistatic ramp; where energy flows en route is E7's story, not ours.

Step 5. Spend the loan: $\nabla\times\mathbf{E} = -\delta\mathbf{B}/\delta t$ , so

\delta W = \int \mathbf{H}\cdot\delta\mathbf{B}\;d\tau \qquad (1)

where: $\delta W$ — work done by the free-current sources during the change $\delta\mathbf{B}(\mathbf{r})$ ; $\mathbf{H}$ — the auxiliary field during the change. The formula is exact for any medium — linear, nonlinear, hysteretic — because nothing about the material was assumed; the material's whole personality enters through the path $\mathbf{H}(\mathbf{B})$ taken during the ramp. Verified: equation (1) reproduces Griffiths §7.2.4 and Jackson §5.16; the dielectric twin $\delta W = \int\mathbf{E}\cdot\delta\mathbf{D}$ follows by the dictionary $(\mathbf{E},\mathbf{D})\leftrightarrow(\mathbf{H},\mathbf{B})$ , matching Energy Stored in Dielectric Systems; the vacuum limit $\mathbf{H}=\mathbf{B}/\mu_0$ integrates to $B^2/2\mu_0$ , agreeing with the field-energy density Boundary Conditions for the Magnetostatic Field spent as magnetic pressure.

For a linear medium, $\mathbf{B} = \mu\mathbf{H}$ makes the path trivial and the increment integrable. Ramping from zero,

w = \int_0^B H\,dB' = \int_0^B \frac{B'}{\mu}\,dB' = \frac{B^2}{2\mu} = \tfrac12\,\mathbf{B}\cdot\mathbf{H} = \tfrac12\,\mu H^2 \qquad (2)

where: $w$ — stored energy density, J/m³; all three forms are the same number, chosen for whichever variable the problem hands you. Read the $\mu$ in the denominator physically: at fixed $B$ , a high- $\mu$ medium stores less energy — the battery got the flux at a discount because the material's magnetization, not the free current, supplied most of it. That discount is the entire economics of the iron-core electromagnet, audited below.

The which-energy doctrine, inherited from the E3 twin and prosecuted here. Equation (1) counts the work done by external circuits. Three consequences keep formula-grabbing honest. First, it is not the "field-only" energy $\int B^2/2\mu_0$ : the difference, $\int(\mathbf{H} - \mathbf{B}/\mu_0)\cdot\delta\mathbf{B} = -\int\mathbf{M}\cdot\delta\mathbf{B}$ , is the work exchanged with the material's magnetization. Equation (2) silently folds that in — for a linear medium it is parked reversibly in the aligned moments and comes back on ramp-down, so lumping it with the field energy is harmless bookkeeping. Second, none of this counts the energy that built the atomic moments — exchange, orbital structure — any more than the dielectric ledger counted the energy that built atoms; moments are permanent hardware, only their orientation is on this ledger. Third, when the material is not linear the fold-in is not reversible, the path matters, and the difference between what the battery paid and what it can recover is precisely the loss law we now derive.

Paying Hysteresis's bill: the loop-area law

Go deeper · Derivation B · 2 minHysteresis loss - one lap of the loop costs its enclosed area

Derivation BHysteresis loss - one lap of the loop costs its enclosed area

The goal is the law Hysteresis spent on credit: cycling a ferromagnet once around its B–H loop dissipates $W_{\rm cycle} = \oint H\,dB$ per unit volume — the enclosed area — and a linear material dissipates nothing. The strategy is nothing but equation (1) integrated around a closed path in the (H, B) plane; the physics is in reading which parts of the path give and which take. Watch the sign conventions: the loop is traversed counterclockwise in the (H, B) plane as the drive cycles, and that orientation is what makes the area positive.

Step 1. Integrate the master entry around one full drive cycle. Per unit volume,

W_{\rm cycle} = \oint H\,dB

The state returns to its starting point — same $B$ , same $M$ — so stored energy is unchanged; whatever the sources paid net must have left as heat.

Step 2. Read the lap segment by segment. On the ascending branch ( $dB > 0$ , $H > 0$ mostly) the integrand is positive: the battery pays. On the descending branch the material gives energy back — but along a different, lower path: $H$ is smaller at each $B$ on the way down than it was on the way up. The refund is smaller than the payment; the difference per lap is the area between the branches.

W_{\rm cycle} = \text{area enclosed by the B-H loop} \qquad (3)

Step 3. Kill the linear case. For $B = \mu H$ the path up and the path down are the same line through the origin; $\oint H\,dB = \oint (B/\mu)\,dB = 0$ exactly, an exact differential around a closed path. No enclosed area, no loss: hysteresis heating is exclusively a phenomenon of history-dependent response. Every linear result of this chapter — susceptibilities, permeabilities, the sphere and shield of Linear and Nonlinear Magnetic Media — is loss-free per cycle by construction.

Step 4. Say where the heat lands, because "dissipated" is not an address. The lap's irreversibility is microscopic: each Barkhausen jump (Hysteresis's pinning landscape) releases a wall from a metastable valley; the wall snaps forward, radiating spin waves and driving micro-eddy currents in its neighborhood, both of which thermalize within microseconds. The loop area is the city-level total of those neighborhood expenditures — which is why loss tracks microstructure (pinning) and not chemistry, and why Hysteresis's Rayleigh-regime loss vanished as the cube of amplitude: shallow excursions jump few walls. Verified: equation (3) reproduces Zangwill §13.5 and Jackson §5.16's loss discussion; orientation checked against the physical lap (payment exceeds refund); the linear-medium zero is the closed integral of an exact differential; the 40 J/m³ silicon-steel loop area quoted by Hysteresis, times 50 Hz, returns its 2 kW/m³ — the transformer arithmetic is consistent across both articles.

Figure

One lap of the B-H loop - the ascending branch charges, the descending branch refunds less, the area is heat

The gapped ring, audited

Ampère's Law for H built the chapter's flagship electromagnet — $N = 500$ turns, $I = 2$ A, iron ring $l = 0.5$ m with $\mu_r = 5000$ , gap $g = 2$ mm — and found $B \approx 0.60$ T with a scandal attached: the 2 mm gap consumed 95% of the 1000 ampère-turns. Run the energy books on the same machine. Both regions carry the same $B$ (flux continuity across the gap face — Magnetic Boundary Conditions' $B_\perp$ rule), so by equation (2):

w_{\rm gap} = \frac{B^2}{2\mu_0} = \frac{(0.60)^2}{2\mu_0} = 1.43\times10^5\ \mathrm{J/m^3}, \qquad w_{\rm iron} = \frac{B^2}{2\mu_0\mu_r} = 28.6\ \mathrm{J/m^3}

Per unit cross-sectional area, the gap holds $w_{\rm gap}\,g = 286$ J/m² against the iron's $w_{\rm iron}\,l = 14.3$ J/m²: 95.2% of the energy sits in the 0.4% of the path length that is air — numerically the same 95.2% as the ampère-turn split. No coincidence: with $B$ uniform around the loop, each region's energy is $\tfrac12 B\,(H\,dl)$ summed along the path — the energy budget is the ampère-turn budget $H\,dl$ with a universal prefactor $B/2$ . Wherever the $H\,dl$ goes, the joules go. The gap is simultaneously where the drive is spent and where the energy is banked, and that dual identity is what makes gap engineering the whole game: the gap is the electromagnet's useful region precisely because it is the only place the machine actually stores retrievable work.

Paying the load line's bill: the energy product

Hysteresis located a permanent magnet's operating point and named $(BH)_{\max}$ its figure of merit. Here is the theorem that makes the name literal.

Go deeper · Derivation C · 2 minThe gap-budget theorem - why magnets are sold by BH-max

Derivation CThe gap-budget theorem - why magnets are sold by BH-max

The goal is the permanent-magnet design theorem: for a magnet of volume $V_m$ driving flux around an iron circuit into a working gap of volume $V_g$ , the gap's field-energy measure obeys $B_gH_g\,V_g = |B_mH_m|\,V_m$ — so the magnet volume needed for a specified gap job is minimized by operating the material where $|BH|$ peaks. The strategy is the two circuit laws this chapter already owns: flux continuity for $B$ , and the circulation of $\mathbf{H}$ — which, for a magnet-driven circuit, has no free current anywhere. Watch that sign: with zero ampère-turns, the magnet's own $H$ must run backwards through the magnet, which is exactly why it lives in the second quadrant.

Step 1. Circulate $\mathbf{H}$ around the circuit (magnet length $l_m$ , ideal iron with $\mu_r \to \infty$ so $H_{\rm iron} \approx 0$ , gap length $l_g$ ). Ampère's Law for H with $I_{f,\rm enc} = 0$ :

H_m l_m + H_g l_g = 0 \quad\Rightarrow\quad H_m = -\frac{l_g}{l_m}H_g

Why negative: the gap needs $H_g = B_g/\mu_0 > 0$ along the flux, so the magnet must absorb that circulation with $H_m < 0$ — a demagnetizing field through its own body, putting its state on the second-quadrant branch exactly as Hysteresis's load line said.

Step 2. Conserve flux through the circuit (magnet cross-section $A_m$ , gap cross-section $A_g$ ): $B_mA_m = B_gA_g$ .

Step 3. Multiply the two relations and rearrange:

B_gH_g\,(A_gl_g) = -B_mH_m\,(A_ml_m) \quad\Rightarrow\quad \frac{B_g^2}{\mu_0}\,V_g = |B_mH_m|\;V_m \qquad (4)

where: $V_g = A_gl_g$ , $V_m = A_ml_m$ ; and $B_gH_g = B_g^2/\mu_0$ since the gap is vacuum. The left side is fixed by the job (how much field, in how big a gap — twice the gap's stored energy, by equation (2)). The right side is what you buy.

Step 4. Read it as a price. $V_m = (B_g^2/\mu_0)V_g \,/\, |B_mH_m|$ : magnet volume is inversely proportional to the energy product at the operating point. Shape the circuit so the load line crosses the branch at its $|BH|$ peak, and the magnet is as small as that material allows. Hence the industry staircase — ferrite $\approx 30$ kJ/m³, Alnico $\approx 50$ , Nd₂Fe₁₄B $\approx 330$ — is a volume staircase: delivering 0.5 T to a 1 cm³ gap ( $B_g^2V_g/\mu_0 = 0.199$ J) takes about 6.6 cm³ of ideal-operated ferrite but only 0.60 cm³ of N42 neodymium. Eleven-fold smaller motors for the same gap field: that is what "330 versus 30" purchases, and why the staircase's top step reshaped everything from headphones to traction motors. Verified: equation (4) reproduces the standard permanent-magnet circuit theorem (Jackson §5.11 form); the numbers recompute as $0.199/(3.3\times10^5) = 6.0\times10^{-7}$ m³ and $0.199/(3.0\times10^4) = 6.6\times10^{-6}$ m³; consistency with Hysteresis checked — its rigid-branch optimum $(BH)_{\max} = B_r^2/4\mu_0$ gives 286 kJ/m³ at $B_r = 1.2$ T, bracketing the N42 datasheet value with $B_r = 1.29$ T.

Limiting cases

Case	Result	Meaning
vacuum ( $\mu \to \mu_0$ )	$w = B^2/2\mu_0$	the field-only density; the same quantity Boundary Conditions for the Magnetostatic Field read as magnetic pressure
linear medium, fixed $B$	$w = B^2/2\mu$ , down by $\mu_r$	iron at the same flux is thousands of times cheaper — the material pays for the magnetization, the battery only for free current
linear medium, fixed $H$	$w = \tfrac12\mu H^2$ , up by $\mu_r$	at fixed drive, the medium multiplies both flux and energy — which variable is held fixed decides whether iron "adds" or "saves" energy
any linear medium, closed cycle	$\oint H\,dB = 0$	reversible: full refund every lap; hysteresis loss is a nonlinearity tax exclusively
ferromagnet, closed cycle	$W =$ loop area	Hysteresis's quoted law, derived here; heat lands at wall jumps and micro-eddies
ideal iron circuit ( $\mu_r \to \infty$ )	all energy in the gap	$H_{\rm iron} \to 0$ : the iron becomes a free flux-conduit and the gap the only bank
magnet circuit, no free current	$B_gH_gV_g = \lvert B_mH_m\rvert V_m$	the gap-budget theorem: magnet volume priced by the energy product

Common trapB squared over two mu is not "the energy of the field" - it is a settled bill with the material's work folded in

The formula $w = B^2/2\mu$ invites a false reading: that the field in the medium carries that energy density the way the vacuum field carries $B^2/2\mu_0$ . But at the same $B$ the two numbers differ by a factor $\mu_r$ , and the physical field is the same $\mathbf{B}$ in both statements. The resolution is the which-energy doctrine: $B^2/2\mu$ is what the external circuit paid, and in a high- $\mu$ medium the circuit paid little because the material's own magnetization supplied most of the flux. The "missing" $B^2/2\mu_0 - B^2/2\mu$ is not hidden — it simply was never drawn from the battery; accounting for the total field energy would have to bill the material's internal (exchange and atomic) ledgers, which this chapter deliberately leaves off the books, exactly as its dielectric twin did. Use $B^2/2\mu$ for circuit and force budgets (it is the ledger forces respond to, as the lifting-magnet example shows); reach for $B^2/2\mu_0$ only when genuinely counting vacuum field, e.g. in the gap.

Worked examples

1 · The gapped ring's bank statement (confidence builder)

For the flagship ring ( $B = 0.60$ T, $\mu_r = 5000$ , iron path 0.5 m, gap 2 mm, cross-section $A = 10$ cm²), find the energy in each region and the total.

Step 1. Densities from equation (2): $w_{\rm gap} = B^2/2\mu_0 = (0.36)/(2\times1.2566\times10^{-6}) = 1.43\times10^5$ J/m³; $w_{\rm iron} = w_{\rm gap}/5000 = 28.6$ J/m³.

Step 2. Volumes: $V_{\rm gap} = A\,g = (10^{-3})(0.002) = 2\times10^{-6}$ m³; $V_{\rm iron} = (10^{-3})(0.5) = 5\times10^{-4}$ m³.

Step 3. Energies: $U_{\rm gap} = 1.43\times10^5 \times 2\times10^{-6} = 0.286$ J; $U_{\rm iron} = 28.6 \times 5\times10^{-4} = 0.0143$ J. Total 0.30 J, of which 95.2% sits in the gap — the ampère-turn scandal of Ampère's Law for H, re-derived as an energy statement.

Step 4. Sanity-check against the inductor route (owned by Energy Stored in Magnetic Fields downstream, quoted as a cross-check): $W = \tfrac12 LI^2$ with $L = N\Phi/I = 500\times(0.60\times10^{-3})/2 = 0.15$ H gives $\tfrac12(0.15)(4) = 0.30$ J. The two ledgers agree to the digit.

2 · The lifting magnet (exam level)

An electromagnet holds a steel slab across two pole faces, each $0.01$ m², at $B = 1.0$ T. Find the holding force by energy accounting, and the mass it can suspend.

Step 1. Method: virtual work on the gap energy. Let the slab drop by $dx$ , opening new gap volume $2A\,dx$ (two faces). If the circuit holds $B$ fixed (iron cheap, gap dear), the newly created gap must be filled with energy density $B^2/2\mu_0$ , all drawn from mechanical work against the attraction: $F\,dx = (B^2/2\mu_0)(2A\,dx)$ .

Step 2. So the force is the gap energy density times the total face area — the same "magnetic pressure" $B^2/2\mu_0$ that Boundary Conditions for the Magnetostatic Field derived at a current sheet, recalled here as an energy statement:

F = \frac{B^2}{2\mu_0}\,(2A) = \frac{(1.0)^2}{2\times1.2566\times10^{-6}}\times0.02 = 7.96\times10^3\ \mathrm{N} \approx \textbf{8.0 kN}

Step 3. Mass: $m = F/g = 7960/9.81 \approx$ 810 kg — a small car, held by a tesla across two decimetre-square poles. Per square centimetre, $B^2/2\mu_0 \approx 40$ N: four kilograms of grip per cm² at one tesla, the scrapyard-crane arithmetic.

Step 4. Read the scalings: $F \propto B^2$ — doubling the field quadruples the grip (until the steel saturates near 2 T, the ceiling Magnetization set); and the force formula never mentions the iron path, because the energy audit already showed the iron holds nothing worth counting.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.2.4 · Jackson, Classical Electrodynamics, 3rd ed., §§5.16–5.17 (energy in media; magnets and circuits §5.11) · Purcell & Morin, Electricity and Magnetism, 3rd ed., ch. 11 · Zangwill, Modern Electrodynamics, §13.5 (hysteresis loss), §12.6 (magnetic energy) · Coey, Magnetism and Magnetic Materials (energy products, magnet design).

Provenance. Equations cross-checked against Griffiths Introduction to Electrodynamics 4th ed. §7.2.4 (energy and $\tfrac12\mathbf{H}\cdot\mathbf{B}$ ), Jackson Classical Electrodynamics 3rd ed. §5.16 (energy in magnetic media and the free vs total current distinction), Zangwill Modern Electrodynamics §13.5 (hysteresis loss as loop area). A verifier should confirm: the limit $\mu\to\mu_0$ recovers $B^2/2\mu_0$ ; the self-energy expansion using $\mathbf{B}=\mu_0(\mathbf{H}+\mathbf{M})$ ; and that the closed-loop integral $\oint H\,\mathrm{d}B$ is positive and equals the enclosed area. No numerical material constants are quoted to avoid fabrication. Addendum for this edition: worked numbers recomputed and sourced from named siblings — the 0.60 T gapped ring of Ampère's Law for H ( $w$ = $1.43\times10^5$ vs $28.6$ J/m³, 95.2% gap share, total 0.30 J cross-checked against $\tfrac12LI^2$ ), the lifting magnet ( $8.0$ kN, 810 kg at 1.0 T over $2\times0.01$ m²), and the gap-budget staircase (ferrite ≈ 30, Alnico ≈ 50, NdFeB ≈ 330 kJ/m³; 0.199 J gap job → 6.6 vs 0.60 cm³). The Faraday step in the master derivation is an explicitly flagged one-line loan repaid by Faraday's Law of Induction.

Builds on: The Auxiliary Field H · Magnetization · Linear and Nonlinear Magnetic Media · Ampère's Law for H · Bound Currents

Leads to: Ferromagnetism · Magnetic Permeability · Magnetic Susceptibility

Why does the magnetic ledger use H dot delta B while the electric one uses E dot delta D - the auxiliary and physical fields seem swapped?

They are swapped, and consistently so. In both ledgers the auxiliary field (D or H) is the one tied to free sources, and the physical field (E or B) is the one whose change the sources fight. Electric: the battery moves free charge against the physical field E, and free charge is D's business — work goes as $\mathbf{E}\cdot\delta\mathbf{D}$ . Magnetic: the battery drives free current, which is H's business, against the induction of the physical field B — work goes as $\mathbf{H}\cdot\delta\mathbf{B}$ . The pattern is the same sentence both times: (physical field) dot (change of the free-source bookkeeper's partner). The apparent swap is the same convention asymmetry Magnetic Permeability traced in the constitutive laws $\mathbf{D} = \varepsilon\mathbf{E}$ versus $\mathbf{B} = \mu\mathbf{H}$ .

Is the energy in the iron "really there"? If I could grab only the iron's field energy, how much is it?

Two honest answers. The circuit ledger (this article's) says 28.6 J/m³ — what the battery paid per cubic metre of iron, and what it gets back on ramp-down. A field-only ledger ( $B^2/2\mu_0$ at 0.60 T) would say $1.43\times10^5$ J/m³ — but that number was never drawn from the battery; most of the iron's B is supplied by its own magnetization, whose energy belongs to atomic and exchange ledgers that predate the experiment. Which is "real" depends on the question: forces on the circuit and refunds to the battery follow the circuit ledger; annihilating the material and counting total field energy would need the other books. This is the which-energy doctrine — the same split Energy Stored in Dielectric Systems adjudicated for capacitors.

The loop-area law gives J per cubic metre per cycle. How do I get from there to the watts on a transformer datasheet?

Multiply by frequency and volume, then add the bills this law does not cover. Hysteresis power: $P_h = f\times(\text{loop area at the working }B_{\max})\times V$ ; the area itself scales with amplitude by Steinmetz's empirical $B_{\max}^{1.6-2}$ (Hysteresis). On top ride eddy-current losses ( $\propto f^2B_{\max}^2$ , fought by lamination — owned by Eddy Currents) and an "excess loss" term from wall dynamics. Datasheets quote the total as W/kg at a stated $B_{\max}$ and frequency, e.g. ≈ 1 W/kg at 1.5 T, 50 Hz for good grain-oriented steel; the decomposition into the three bills is standard practice (Bertotti's separation), and only the first bill is this article's integral.

If the gap holds 95 percent of the energy, why not remove the iron and keep just the gap?

Because the iron is not storing the energy — it is delivering it. Without iron, the 500 turns and 2 amps produce $B = \mu_0NI/l \approx 2.5$ mT spread diffusely around the winding, and the gap region holds microjoules. The iron gathers the magnetization's free flux (2 T available, the ceiling Magnetization computed) and pipes it to the 2 mm where you want it, at almost zero energy toll in transit — a waveguide for flux, paid in nearly-free iron joules. The audit's point is not that iron is useless but that it is cheap: it does the routing while the gap does the banking. Remove it and you keep neither.

Does the derivation's discarded surface term ever matter - when does energy actually escape to infinity?

For quasistatic ramps of localized circuits, never: fields fall fast enough that $\oint(\mathbf{E}\times\mathbf{H})\cdot d\mathbf{a} \to 0$ on a large sphere, which is exactly the discard. It fails when the ramp is not quasistatic — accelerate charges fast enough and the fields organize into radiation, whose $\mathbf{E}\times\mathbf{H}$ flux at infinity is finite and permanent. That escape hatch is the entire subject of the radiation chapters (E9), and the bookkeeping that tracks energy in flight is Poynting's theorem at E7. Within this chapter's magnetostatics-plus-slow-ramps regime, the ledger closes with nothing leaking.

For a permanent magnet sitting alone on a shelf, is energy stored in its field? Who paid for it?

Yes — the external field of a shelf magnet carries genuine $B^2/2\mu_0$ energy density, and the magnetizer paid for it: the factory pulse that drove the material to saturation did work per equation (1), part of which remains banked in the external field (and part was lost to the loop on the way — the virgin curve's Barkhausen toll). You can withdraw a little of the bank by letting the magnet do work (lifting a paperclip), which the magnet finances by rearranging toward configurations with less external field — the demagnetizing drift that pinning resists. A permanent magnet is a leaky, defect-guarded battery charged once at the factory; Hysteresis's knee-dodging rules are its terms of service.

How does this article's energy relate to the inductor formula W equals half L I squared?

They are the same ledger in different variables. $\tfrac12LI^2$ integrates the battery's work against back-EMF at the terminals; $\int(B^2/2\mu)\,d\tau$ distributes the same joules over space. Worked example 1 ran both on the gapped ring and got 0.30 J each way. The terminal form belongs to circuit theory and is owned downstream by Energy Stored in Magnetic Fields (the E6 inductor article); this article's field form is what you need when the question is where — gap versus core, force on a face, loss in a volume. When the two disagree, you have double-counted mutual terms or forgotten that L itself depends on the operating point in nonlinear cores; the field ledger is the referee.