Hidden Momentum in Magnetic Systems

The last two articles left a debt on the table. A charge resting beside a magnet — or an energized coax lying on a bench — stores genuine field momentum $\varepsilon_0\int\mathbf{E}\times\mathbf{B}\,d^3r \neq 0$ , while nothing anywhere moves. The conservation theorem shrugs: the momentum is constant, so nothing is violated dynamically. But relativity keeps a second, stricter ledger. An isolated system that is stationary — energy parked, center of energy fixed — must have zero total momentum, exactly, always. That is the center-of-energy theorem, and it turns our curiosity into a crisis: if the fields hold momentum $\mathbf{p}_{\text{em}}$ and the total must vanish, then the matter must hold momentum $-\mathbf{p}_{\text{em}}$ . Matter that is, by every classical instinct, at rest. In 1967 William Shockley and R.P. James sharpened this into a paradox with teeth (their version: a shrinking magnet beside charges predicts a force with no visible reaction) and coined the name the field has used since: hidden momentum. The resolution, delivered within a year by Coleman and Van Vleck, is one of the most beautiful small results in classical physics: the momentum is ordinary relativistic mechanical momentum of the current carriers, hiding in plain sight. In a current loop threaded by an electric field, carriers on one side move with slightly more energy than on the other; since momentum is $\gamma m\mathbf{v}$ , not $m\mathbf{v}$ , the loop's carriers carry net momentum even though the loop's mass goes nowhere. Computed honestly, it comes out to exactly $\mathbf{m}\times\mathbf{E}/c^2$ — equal and opposite to the field momentum, to the last digit. This article does the story properly: the theorem that forces the momentum to exist, the loop model that shows where it lives, the cancellation checked numerically, and the honest boundaries — when hidden momentum is present, when it is not, and why a 2012 challenge to the Lorentz force itself (Mansuripur's paradox) dissolved under exactly this analysis.

Key takeaways

The center-of-energy theorem (relativity's stricter ledger): an isolated static system must have total momentum exactly zero. Static crossed fields have $\mathbf{p}_{\text{em}} \neq 0$ — so the matter must carry $-\mathbf{p}_{\text{em}}$ . That mechanical momentum in "resting" matter is hidden momentum.
It is not exotic: momentum is $\gamma m v$ , and current carriers circulating through regions of different potential have position-dependent $\gamma$ . The $\gamma$ -weighted circulation does not cancel even though the mass flow does.
For a magnetic dipole $\mathbf{m}$ in field $\mathbf{E}$ : $\mathbf{p}_{\text{hid}} = \dfrac{\mathbf{m}\times\mathbf{E}}{c^2}$ , exactly opposite the field momentum $\mathbf{p}_{\text{em}} = -\mathbf{m}\times\mathbf{E}/c^2$ of the same system. Total: zero, as the theorem demands.
Hidden momentum is a property of amperian (current-loop) dipoles in external $\mathbf{E}$ ; charge-based ("gilbertian") models have none — which is how experiments can tell what magnets are made of.
Shockley–James (1967) posed it; Coleman–Van Vleck (1968) resolved it; Mansuripur (2012) rediscovered the trap by forgetting it, proposing to amend the Lorentz force — the amendment was unnecessary.
KEY IDEA: static systems with crossed fields hold field momentum and an exactly canceling relativistic mechanical momentum in their current carriers — hidden only from Newtonian bookkeeping, demanded by relativity, delivered by $\gamma m v$ .

The core insight

A stationary system with static crossed fields carries field momentum and an exactly compensating "hidden" mechanical momentum $\mathbf{p}_{\text{hid}} = \mathbf{m}\times\mathbf{E}/c^2$ in its current carriers — ordinary relativistic $\gamma mv$ bookkeeping, forced by the center-of-energy theorem, invisible to Newtonian mechanics.

Where this lives today

Hidden momentum earned its modern keep as a precision-consistency tool. In the physics of electric dipole moment (EDM) searches — the neutron, electron, and atomic experiments that test time-reversal symmetry at the $10^{-30}$ e·cm level — systems of spins (amperian moments) in strong electric fields are exactly the hidden-momentum configuration, and the correct relativistic accounting of momentum and angular momentum in these systems enters the systematic-error budgets. In the theory of the Aharonov–Casher effect (a neutral magnetic moment circling a line charge picks up a quantum phase), the hidden-momentum bookkeeping decides the sign conventions and the force-free character of the interaction — a direct laboratory descendant of the charge-plus-dipole system of this article. The 2012–2013 Mansuripur episode demonstrated the concept's continuing load-bearing role: a claimed incompatibility between the Lorentz force and special relativity, published prominently, was resolved by multiple groups pointing to the omitted hidden momentum and its transformation properties — the textbook example (Griffiths's, literally) settling a live dispute. And in the Abraham–Minkowski problem of momentum in media (this chapter, two articles ahead), hidden momentum is one of the standard reconciliation terms: several "experimental discriminations" between the rival momenta dissolve once the matter's hidden account is included. The lesson generalizes: momentum bookkeeping in composite systems — classical or quantum — is trustworthy only when every $\gamma$ -weighted internal flow is on the books.

Origins · 1 minWilliam Shockley and how this was discovered

William Shockley

1910–1989 · Bell Labs, Stanford

Shockley is the rare physicist whose scientific legacy must be stated alongside a moral one, and this track does both plainly. The science: co-inventor of the transistor (point-contact 1947 with Bardeen and Brattain; the junction transistor was Shockley's own 1948–51 theory), Nobel laureate 1956, author of the semiconductor bible Electrons and Holes in Semiconductors, and — the reason he appears here — co-author with R.P. James of the 1967 paper "'Hidden momentum forces' on magnets," which posed the paradox of this article and named its resolution. Characteristically, he found it by worrying about a device-like thought experiment: magnets whose magnetization changes beside fixed charges, with forces that seemed to lack reactions. His managerial style drove the "traitorous eight" out of Shockley Semiconductor in 1957 — into founding Fairchild and, in time, Silicon Valley; his final decades were spent promoting racist pseudoscience on intelligence and heredity that the scientific community rejected then and rejects now. The physics that bears his name in this article is sound, was resolved within a year by Coleman and Van Vleck, and remains the standard entry point to relativistic momentum bookkeeping in static systems — an important result from a man whose later career is a warning, not a model.

History

1891

— J.J. Thomson computes momentum stored in static crossed fields — the seed, unremarked for decades

1904

— Thomson's charge-monopole angular momentum hints that static fields hold mechanical bookkeeping surprises

1967

— Shockley and James pose the paradox and coin "hidden momentum"

1968

— Coleman and Van Vleck resolve it: center-of-energy theorem plus relativistic carrier momentum, to order v²/c²

1969

— Furry's analysis extends the accounting to general static systems

2009

— Babson, Reynolds, Bliokh and Griffiths codify the pedagogy: field, hidden, and impulse momenta in one framework

2012

— Mansuripur's paradox claims the Lorentz force violates relativity; resolutions via hidden momentum follow within months

The physical picture

Start with the stricter ledger, because everything follows from it. Special relativity welds momentum to energy transport: a system's total momentum equals (rate of motion of its center of energy) × (total energy)/ $c^2$ — the formal descendant of the box argument from Momentum Density of the Electromagnetic Field. For an isolated system gone fully static — every field constant, every macroscopic part at rest — the center of energy is pinned. Therefore total momentum is zero. Not "constant": zero. This is the center-of-energy theorem, and unlike Newton's third law it survives relativity untouched, because it is relativity (the chapter's closing article derives it from boost symmetry, where it stands beside energy and momentum conservation as an equal). Now audit the charge-beside-magnet system with the theorem in hand. Field momentum: nonzero — computed honestly from $\varepsilon_0\mathbf{E}\times\mathbf{B}$ , it comes out $-\mathbf{m}\times\mathbf{E}/c^2$ for a dipole $\mathbf{m}$ in the charge's field $\mathbf{E}$ . The theorem then leaves no negotiating room: somewhere in the matter sits momentum $+\mathbf{m}\times\mathbf{E}/c^2$ . But the magnet is bolted down and the charge is glued. What in this system can possibly have momentum?

Figure

The crisis and its resolution: field momentum circulates one way, the magnet's current carriers hide the exact opposite in their gamma-weighted flow

The answer hides inside the word "current." A magnet's moment is carried by circulating charges, and circulating charges are moving matter even when the magnet is not. Newtonian bookkeeping assigns a closed steady loop zero net momentum — equal mass flow up one side and down the other. But momentum is not $m\mathbf{v}$ ; it is $\gamma m\mathbf{v}$ , and $\gamma$ tracks energy. Put the loop in an external electric field and carriers on the two sides circulate at different energies — the field does work on them along one leg and takes it back along the other. Same charge flow everywhere (steady current), different $\gamma$ on each side, and the $\gamma$ -weighted momentum around the loop no longer cancels. The loop — bolted to the bench — carries net mechanical momentum. Derivation B computes it: $\mathbf{m}\times\mathbf{E}/c^2$ , precisely the theorem's demand. Three framing remarks before the mathematics. First, "hidden" is an honest word for hidden from Newtonian accounting — the momentum is as real as any; it is simply of order $1/c^2$ and attached to internal motion rather than bulk motion, so no scale or speedometer sees it. Second, the effect is a sibling of the third-law violation: both are $(v/c)^2$ relativistic corrections that the previous articles' ledger forces into existence; this chapter keeps finding that classical electromagnetism was relativistic all along. Third, the cancellation is not a coincidence to be checked case by case but a theorem — any static system must zero out — and that inevitability is what makes the loop calculation a consistency test of relativity itself, which it passes exactly.

The theorem and the two momenta

The stricter ledger, stated for the record:

\mathbf{P}_{\text{total}} = \frac{E_{\text{total}}}{c^2}\,\frac{d\mathbf{R}_E}{dt}, \qquad \mathbf{R}_E \equiv \frac{1}{E_{\text{total}}}\int \mathbf{r}\,u_{\text{total}}(\mathbf{r})\,d^3r, \qquad (1)

where: $u_{\text{total}}$ — all energy density, field plus matter (rest energy included); $\mathbf{R}_E$ — the center of energy. Equation (1) is exact for any isolated system in special relativity; it says momentum is energy transport, the local statement of which was the momentum article's $\mathbf{g} = \mathbf{S}/c^2$ . For a static system $d\mathbf{R}_E/dt = 0$ , hence $\mathbf{P}_{\text{total}} = 0$ : the center-of-energy theorem. (The derivation from boost invariance — it is Noether's conserved quantity for Lorentz boosts — belongs to Conservation Laws and Spacetime Symmetries; Derivation A below gives the direct argument at this chapter's register.) The two accounts that must cancel, for the canonical system — point charge $q$ at separation $a$ from an ideal (amperian) magnetic dipole $\mathbf{m}$ , everything at rest:

\mathbf{p}_{\text{em}} = \varepsilon_0\!\int\!\mathbf{E}\times\mathbf{B}\,d^3r = -\,\frac{\mathbf{m}\times\mathbf{E}(\mathbf{r}_m)}{c^2} = \frac{\mu_0\,q}{4\pi a^2}\,\hat{\boldsymbol{a}}\times\mathbf{m}, \qquad (2)

where: $\mathbf{E}(\mathbf{r}_m)$ — the charge's field at the dipole's location, magnitude $q/4\pi\varepsilon_0a^2$ ; $\hat{\boldsymbol{a}}$ — unit vector from charge to dipole. The integral is a classic (the charge's radial $\mathbf{E}$ crossed with the dipole's loops of $\mathbf{B}$ , integrated by the same techniques as the E4 dipole chapter's averages); the compact form $-\mathbf{m}\times\mathbf{E}/c^2$ is the piece worth memorizing. Note the counter-intuitive punch: the field momentum of this utterly static configuration is independent of how the fields die at infinity and perpendicular to the line joining the pair. And:

\mathbf{p}_{\text{hid}} = +\,\frac{\mathbf{m}\times\mathbf{E}(\mathbf{r}_m)}{c^2}, \qquad \mathbf{p}_{\text{em}} + \mathbf{p}_{\text{hid}} = 0, \qquad (3)

where: the hidden momentum resides in the dipole's current carriers (Derivation B), pointing opposite the field momentum, restoring the theorem's zero. The general current-distribution version, derived the same way, is worth recording: for steady currents in an external electrostatic potential $\varphi$ ,

\mathbf{p}_{\text{hid}} = -\frac{1}{c^2}\oint \varphi(\mathbf{l})\,I\,d\mathbf{l} \quad\text{(filamentary loop)}, \qquad (4)

where: the integral runs around the circuit, weighting each element's current by the local potential — a formula that will settle the coax in worked example 2 in three lines.

Derivation AThe center-of-energy theorem — why static systems cannot hold net momentum

The goal is the theorem behind the crisis: an isolated system whose configuration is static has $\mathbf{P}_{\text{total}} = 0$ exactly. The strategy is to prove the transport identity (1) — total momentum equals total energy times the center-of-energy velocity over $c^2$ — from the local conservation laws this chapter has established, then read off the static case. Watch the two inputs: local energy conservation (Poynting's theorem plus its mechanical counterpart) and the symmetry of the total stress–energy bookkeeping, $c^2\times(\text{momentum density}) = (\text{energy flux})$ — the momentum article's transport identity, now applied to everything, matter included.

Step 1. Set up the total accounts. Let $u(\mathbf{r},t)$ be the total energy density (field $+$ matter, rest energy included) and $\mathbf{s}(\mathbf{r},t)$ the total energy flux (Poynting plus convected matter energy). Isolation and local conservation give $\partial_t u + \nabla\cdot\mathbf{s} = 0$ — energy neither appears nor teleports.

Step 2. Differentiate the center of energy. With $E_{\text{tot}} = \int u\,d^3r$ constant,

\frac{d}{dt}\int \mathbf{r}\,u\,d^3r = \int \mathbf{r}\,\partial_t u\,d^3r = -\int \mathbf{r}\,(\nabla\cdot\mathbf{s})\,d^3r = \int \mathbf{s}\,d^3r,

integrating by parts componentwise in the last step (boundary terms die for a localized system). The center of energy moves at the rate set by the total energy current.

Step 3. Convert energy current to momentum. Relativity's transport identity — energy current $=$ $c^2\times$ momentum density — holds sector by sector: for the field it is $\mathbf{g} = \mathbf{S}/c^2$ (Derivation B of the momentum article); for matter it is the statement that a lump of energy $E$ moving at $\mathbf{v}$ carries momentum $E\mathbf{v}/c^2$ (which reduces to $\gamma m\mathbf{v}$ for particles — with the relativistic energy in the numerator, the detail Derivation B lives on). Summing sectors: $\int\mathbf{s}\,d^3r = c^2\,\mathbf{P}_{\text{total}}$ . Combined with Step 2:

E_{\text{tot}}\,\frac{d\mathbf{R}_E}{dt} = c^2\,\mathbf{P}_{\text{total}} \qquad\Longrightarrow\qquad \boxed{\ \text{static}\ \Rightarrow\ \frac{d\mathbf{R}_E}{dt} = 0\ \Rightarrow\ \mathbf{P}_{\text{total}} = \mathbf{0}.\ }

Why this is stronger than momentum conservation: conservation permits any constant total; the theorem pins the constant to zero whenever the system is static. It is the relativistic replacement for "nothing moves, so momentum is zero" — with "nothing moves" upgraded to "the energy distribution is stationary," which is the version that survives when internal currents circulate.

Step 4. Read the consequence for our systems. Charge-plus-magnet and the energized coax are static and isolated: $\mathbf{P}_{\text{total}} = 0$ . Their field momenta (equation (2); the coax's $PL/c^2$ ) are nonzero. Therefore their matter carries momentum $-\mathbf{p}_{\text{em}}$ , with no macroscopic motion available to carry it — forcing the microscopic mechanism of Derivation B. There is no escape hatch: doubt the hidden momentum and you are doubting boost symmetry itself, tested wherever relativistic kinematics has ever been checked. Verified: this is the Coleman–Van Vleck (1968) argument at textbook register (their paper proves it from the Darwin Lagrangian to order $v^2/c^2$ ); the sector-summed transport identity is the symmetry $T^{0i} = T^{i0}$ of the total stress–energy tensor, formalized in the chapter closer; boundary terms in Step 2 vanish for fields falling faster than $1/r^2$ — satisfied by static multipole fields.

Figure

The mechanism, one rectangle: same current everywhere, staggered carrier energies — the gamma-weighted momentum sum refuses to cancel

Go deeper · Derivation B · 3 minWhere it hides — the current loop in an external field, gamma-weighted

Derivation BWhere it hides — the current loop in an external field, gamma-weighted

The goal is the concrete mechanism with the exact coefficient: a filamentary current loop of moment $\mathbf{m}$ , held at rest in external electrostatic field $\mathbf{E}$ , carries mechanical momentum $\mathbf{p}_{\text{hid}} = \mathbf{m}\times\mathbf{E}/c^2$ . The strategy is the classic rectangular-loop model with free (collisionless) carriers: steady current means the same $I$ through every cross-section, but energy conservation staggers the carriers' speeds between the two horizontal legs; sum $\gamma m v$ around the loop and watch the Newtonian cancellation fail by exactly the right amount. The one subtlety to watch: it is the relativistic momentum's energy-weighting that pays, so every Newtonian step must keep $\gamma$ explicitly.

Step 1. The stage. A rectangular loop, width $w$ (horizontal, along $\hat{\mathbf{x}}$ ) and height $h$ (vertical), carries steady current $I$ counterclockwise; moment $\mathbf{m} = I\,wh\,\hat{\mathbf{z}}$ (out of the page). External uniform field $\mathbf{E} = E\,\hat{\mathbf{y}}$ (upward, in the loop's plane). Carriers: charge $q$ , rest mass $m_c$ , moving freely inside the wire (an idealization — see Step 4's honesty clause).

Step 2. Energy staggering. A carrier climbing the left leg (moving $+\hat{\mathbf{y}}$ , with the current) gains energy $qEh$ from the field; descending the right leg it gives it back. So every carrier in the top leg runs with energy $\mathcal{E}_{\text{top}} = \mathcal{E}_{\text{bot}} + qEh$ . Steady current fixes the charge flux: $I = \lambda_{\text{top}}v_{\text{top}} = \lambda_{\text{bot}}v_{\text{bot}}$ — where the top carriers, being faster, are also sparser ( $\lambda$ = line charge density of carriers). Newtonian bookkeeping now cancels perfectly: momentum per unit length of a stream is $(m_c/q)\lambda v$ , identical top and bottom, opposite directions. This is why the effect is invisible below order $1/c^2$ .

Step 3. The relativistic sum. Momentum per unit length of a carrier stream is $n\,\gamma m_c v$ (per length: $(\gamma m_c/q)\lambda v = (\gamma m_c/q)(I)$ ). Total for the two horizontal legs (the vertical legs' contributions are vertical and cancel by up-down symmetry of the same argument):

p_x = \frac{m_c I w}{q}\left(\gamma_{\text{top}} - \gamma_{\text{bot}}\right)\times(\pm) \quad\text{with}\quad (\gamma_{\text{top}} - \gamma_{\text{bot}})\,m_c c^2 = qEh,

the last equality being Step 2's energy stagger read through $\mathcal{E} = \gamma m_c c^2$ . Substituting, all microscopic parameters cancel:

p_x = \frac{I w\,(qEh)}{q\,c^2} = \frac{I\,wh\,E}{c^2} = \frac{mE}{c^2}.

Tracking directions (current $+\hat{\mathbf{x}}$ in the top leg where $\gamma$ is larger... with $\mathbf{E} = E\hat{\mathbf{y}}$ and $\mathbf{m} = m\hat{\mathbf{z}}$ ): $\mathbf{p}_{\text{hid}} = \dfrac{\mathbf{m}\times\mathbf{E}}{c^2}$ . Why the microphysics cancels: the stagger $(\gamma_{\text{top}}-\gamma_{\text{bot}})$ is fixed by energy conservation alone — $qEh/m_cc^2$ — regardless of carrier mass, speed, or density; and the momentum formula multiplies it by exactly $m_cIw/q$ . Fast sparse carriers or slow dense ones, electrons or ions: same answer. Hidden momentum is a property of the current and the field, not of the carrier model's details.

Step 4. Scope, honestly. Generalizing leg by leg gives equation (4), $\mathbf{p}_{\text{hid}} = -({1}/{c^2})\oint\varphi\,I\,d\mathbf{l}$ , for any filamentary loop in any electrostatic potential — and for a small loop this reduces to $\mathbf{m}\times\mathbf{E}/c^2$ again. Two fine-print clauses. (i) The free-carrier model is one limit; in a real wire, forces from the lattice redistribute the energy stagger, but the total — carriers plus stressed lattice — is protected by Derivation A's theorem: the system is static, so matter must carry $-\mathbf{p}_{\text{em}}$ , however its internals share the load (Coleman–Van Vleck's actual result; the incompressible-fluid version of the model gets the same total via pressure rather than $\gamma$ , a classic alternative telling). (ii) The mechanism requires circulating charges responding to $\mathbf{E}$ — an amperian dipole. A hypothetical dipole made of magnetic charges (gilbertian) has no carriers to stagger and no hidden momentum; correspondingly its field momentum bookkeeping differs. Real magnets — currents and spin — are amperian, and experiments on neutron electric-dipole-moment searches and dielectric-magnet systems are consistent with the amperian accounting. Verified: reproduces Griffiths Ex. 12.13 (4th ed.) and the standard Coleman–Van Vleck/Vaidman analyses; the loop result matches equation (2)'s field momentum with opposite sign — cancellation checked numerically in worked example 1; the $\oint\varphi I\,d\mathbf{l}$ form matches Babson, Reynolds, Bliokh & Griffiths, Am. J. Phys. 77, 826 (2009).

Limiting cases

Case	Result	Meaning
$c \to \infty$	$\mathbf{p}_{\text{hid}} \to 0$ (and $\mathbf{p}_{\text{em}} \to 0$ )	both accounts are relativistic; Newton's world has neither
$\mathbf{E} = 0$ at the dipole	no hidden momentum	a lone magnet is innocent; it takes crossed fields to hide anything
$\mathbf{m} \parallel \mathbf{E}$	$\mathbf{m}\times\mathbf{E} = 0$	geometry matters: aligned fields hide nothing
gilbertian (magnetic-charge) dipole	$\mathbf{p}_{\text{hid}} = 0$ ; field momentum bookkeeping differs	hidden momentum diagnoses what dipoles are made of
time-dependent systems	theorem inapplicable; momentum flows openly	hidden momentum is a statics phenomenon; transients trade it via real forces
the coax (PCONS flagship)	carriers hide $-PL/c^2$	worked example 2; the bench-top cable's books close
spin magnetic moments	treated as amperian in the classical limit	QED's answer; consistent with all tests

Common trapHidden momentum is not mysterious — and it is not optional either

Two opposite failure modes. The first is mystification: treating $\mathbf{p}_{\text{hid}}$ as a new physical entity, a fudge factor, or evidence that classical EM is inconsistent. It is none of these — it is $\gamma m v$ summed correctly, the same relativistic mechanics that runs every accelerator, applied to carriers you forgot were moving. The second failure is omission, and it has trapped professionals: analyze a magnet-plus-charge system, find nonzero field momentum, forget the matter's compensating account, and you will derive self-accelerating boxes, torques without reactions, or "failures" of the Lorentz force. Mansuripur's 2012 paradox (a charge and magnetic dipole viewed from a moving frame, where the dipole acquires an electric moment and seems to feel an unbalanced torque) made exactly this omission — published as a claimed refutation of the Lorentz force law, it dissolved once the hidden momentum's relativistic transformation was included, as several groups showed within months. The working rule: any static system containing both currents and electric fields carries hidden momentum $-\mathbf{p}_{\text{em}}$ ; audit both accounts or neither. And its corollary: since the total is always zero for static systems, a static configuration can never be used to store extractable net momentum — switch anything off and the two accounts discharge into each other and into fields, conserving the ledger of the conservation-theorem article.

Worked examples

1 · The canonical pair, in numbers (confidence builder)

A point charge $q = 1.0$ nC sits $a = 10$ cm from a small magnet of moment $m = 1.0$ A·m², with $\mathbf{m}$ perpendicular to the line joining them. Compute both momentum accounts and check the theorem.

Step 1. The charge's field at the magnet: $E = \dfrac{q}{4\pi\varepsilon_0a^2} = \dfrac{8.99\times10^9\times10^{-9}}{0.010} = \mathbf{899\ V/m}$ .

Step 2. Hidden momentum: $p_{\text{hid}} = \dfrac{mE}{c^2} = \dfrac{1.0\times899}{8.99\times10^{16}} = \mathbf{1.0\times10^{-14}\ kg\cdot m/s}$ , directed along $\mathbf{m}\times\mathbf{E}$ — perpendicular to both the moment and the separation line.

Step 3. Field momentum: $p_{\text{em}} = \dfrac{\mu_0qm}{4\pi a^2} = \dfrac{10^{-7}\times10^{-9}\times1.0}{0.010} = \mathbf{1.0\times10^{-14}\ kg\cdot m/s}$ , opposite direction. (That equations (2) and (3) give identical magnitudes is $\mu_0/4\pi = 1/4\pi\varepsilon_0c^2$ doing its work — the same $c^2$ bridge as everywhere in this chapter.)

Step 4. Total: zero, as Derivation A demands. Scale check: $10^{-14}$ kg·m/s is a bacterium's momentum at a leisurely swim — yet its necessity is absolute: delete it and relativity breaks. For intuition about the carriers: in a magnetization-current picture the staggered energy per carrier is minuscule ( $qEh \sim$ μeV for atomic-scale loops), but there are $\sim10^{23}$ of them; the ledger sums exactly regardless of how the microphysics shares it (Derivation B, Step 4).

2 · The coax's books, closed at last (exam level)

The conservation article's flagship: a coax of length $L = 1.0$ m carrying $I = 10$ A at $V = 200$ V stores field momentum $p_{\text{em}} = VIL/c^2 = 2.2\times10^{-14}$ kg·m/s pointing from source to load. Locate the hidden momentum and verify the total vanishes.

Step 1. The tool: equation (4), $\mathbf{p}_{\text{hid}} = -\dfrac{1}{c^2}\oint\varphi I\,d\mathbf{l}$ , for the current circuit — down the inner conductor (at potential $\varphi = V$ ), through the load, back along the outer shield (at $\varphi = 0$ ), through the source.

Step 2. Evaluate leg by leg. Outer shield: $\varphi = 0$ , contributes nothing. Source and load legs: radial, short, and (idealized) contribute negligibly for $L \gg$ gap. Inner conductor: $\varphi = V$ constant along its length, current $I$ flowing toward the load ( $+\hat{\mathbf{z}}$ , say) over length $L$ :

\mathbf{p}_{\text{hid}} = -\frac{1}{c^2}\,V I L\,\hat{\mathbf{z}} = -\,2.2\times10^{-14}\ \text{kg}\,\cdot\,\text{m/s}\ \hat{\mathbf{z}}.

Step 3. Sum the accounts: $\mathbf{p}_{\text{em}} + \mathbf{p}_{\text{hid}} = (+2.2 - 2.2)\times10^{-14}\,\hat{\mathbf{z}} = \mathbf{0}$ . The bench-top cable has, as the theorem insists, exactly zero total momentum. The physical reading: the conduction electrons in the inner conductor — drifting at a fraction of a millimetre per second — run at (very slightly) different energy than the shield's, and their $\gamma$ -weighted flow carries momentum toward the source, mirroring the field's momentum toward the load.

Step 4. The audit that makes it respectable: switch off the source. The field momentum discharges into the matter via the collapsing fields' forces (the impulse corollary of the conservation article) and the hidden momentum discharges simultaneously as the current dies — equal and opposite deposits, so the cable as a whole receives zero net impulse from the switch-off of a static configuration. Both subtleties are real; their sum is silence. That silence — no drifting cables, no self-propelled magnets — is the center-of-energy theorem keeping its books.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §8.4.4, Ex. 12.13 · Coleman & Van Vleck, Phys. Rev. 171, 1370 (1968) · Shockley & James, Phys. Rev. Lett. 18, 876 (1967) · Babson, Reynolds, Bliokh & Griffiths, Am. J. Phys. 77, 826 (2009) · Zangwill, Modern Electrodynamics, §15.6 · Jackson, Classical Electrodynamics, 3rd ed., §6.7, §12.10

Provenance. The center-of-energy argument follows Coleman–Van Vleck as presented at the Griffiths §8.4.4/Zangwill §15.6 register; the rectangular-loop calculation is Griffiths Ex. 12.13 with the energy-stagger bookkeeping made explicit; the $-c^{-2}\oint\varphi I\,d\mathbf{l}$ form and the field/hidden/impulse taxonomy follow Babson–Reynolds–Bliokh–Griffiths (2009). A verifier should: (1) confirm $\mathbf{p}_{\text{em}} = -\mathbf{m}\times\mathbf{E}/c^2$ for charge-plus-dipole against the direct $\varepsilon_0\int\mathbf{E}\times\mathbf{B}$ integral; (2) reproduce Derivation B's cancellation of carrier parameters via $(\gamma_{\text{top}}-\gamma_{\text{bot}})m_cc^2 = qEh$ ; (3) check worked numbers — $E = 899$ V/m at 10 cm from 1 nC, both momenta $1.0\times10^{-14}$ kg·m/s; coax $\mathbf{p}_{\text{hid}} = -VIL/c^2 = -2.2\times10^{-14}$ kg·m/s canceling the flagship's field momentum. All formulas standard; worked numbers recomputed for this edition.

Builds on: Momentum Density of the Electromagnetic Field · The Poynting Vector · Poynting Vector in Static Field Configurations · Conservation of Linear Momentum in Electrodynamics · Newton's Third Law in Electrodynamics

Leads to: The Maxwell Stress Tensor · Angular Momentum of the Electromagnetic Field · Global vs. Local Conservation Laws

If hidden momentum is real mechanical momentum, why can't I detect it by weighing or colliding with the magnet?

Because every detector you can build couples to the total momentum of the composite object, and for the static system that total is zero — the field's account and the matter's account cancel inside anything that contains both. Push the magnet: you accelerate magnet-plus-local-fields together, and what responds is the total inertia (which includes the field energy's mass, another story), not the hidden term separately. The hidden momentum only becomes visible when the configuration changes: ramp the charge's field down and the hidden momentum must go somewhere — it emerges as a real impulse on the magnet, matched by the field momentum's own discharge elsewhere, with the sum honoring the ledger. This transient is calculable and is exactly how the paradoxes (Shockley–James's shrinking magnets, the disk paradox's angular cousin) resolve: the "mysterious" forces during switch-off are the two hidden accounts settling. A static measurement of hidden momentum alone is impossible in principle — which is precisely why it could hide for eighty years between Thomson's integrals and Shockley's name for them.

Does an isolated atom with an electron orbit or spin carry hidden momentum?

An isolated atom, no — with no external electric field crossing its magnetic moment, there is nothing to stagger ( $\mathbf{m}\times\mathbf{E} = 0$ trivially). Put the atom in an external field — as in EDM experiments, polar molecules, or an ion trap — and the classical answer would be yes, of order $mE/c^2$ per atom, with the compensating field momentum in the crossed fields. Quantum mechanically the bookkeeping survives remarkably intact: the expectation values obey the same theorems (the center-of-energy theorem is Lorentz symmetry, which QED honors), and "hidden momentum" appears in the literature on atoms in fields, the Aharonov–Casher effect, and photon momentum in media, always as the $\gamma$ -weighted internal-motion account. For spin — a moment with no classical circulation — the classical mechanism fails but the theorem does not: QED's stress tensor delivers the required compensating momentum, and treating spins as amperian loops gets every measured answer right. The theorem is the invariant; the loop model is its most legible classical face.

Which systems have hidden momentum and which do not — is there a quick diagnostic?

Three questions, in order. (1) Is the system static? If yes, total momentum is zero (Derivation A), so hidden momentum $= -\mathbf{p}_{\text{em}}$ : compute the easier of the two — usually the field integral, or $-\mathbf{m}\times\mathbf{E}/c^2$ for dipoles, or $-(1/c^2)\oint\varphi I d\mathbf{l}$ for circuits. If the field momentum is zero (no crossed E and B: lone capacitor, lone magnet), there is nothing hidden. (2) Are there circulating charges in an external electric field? That is the carrier of the effect — currents in potentials. No currents (electrets, charged insulators at rest): nothing hidden. (3) Is anything time-dependent? Then "hidden" is the wrong frame entirely — momentum is flowing openly among matter, field, and boundary, and you audit with the conservation theorem instead. The classic pitfall the diagnostic catches: any problem featuring both a battery-driven current and a charged object, analyzed "at rest" — coax cables, magnetrons, Penning traps, EDM cells — has both accounts, and equating "at rest" with "zero mechanical momentum" silently discards one of them.

Why does the free-carrier model give the whole answer if real wires have lattice forces and aren't collisionless?

Because the total is protected by a theorem while the split is model-dependent — the recurring grammar of this chapter. Derivation A pins matter's total at $-\mathbf{p}_{\text{em}}$ regardless of microphysics. Derivation B's free-carrier model realizes that total entirely in carrier $\gamma m v$ ; the incompressible-fluid model realizes it via pressure-driven energy flux in the fluid; a real wire distributes it among carrier kinetics, lattice stresses, and interface forces. Coleman and Van Vleck's actual 1968 calculation works from the Darwin Lagrangian (the systematic $v^2/c^2$ two-particle theory) precisely to avoid committing to a wire model, and lands on the same total. The pedagogy uses the free-carrier loop because it makes the mechanism visible — energy stagger, gamma weighting, parameter cancellation — not because nature is collisionless. If a homework model gives you a different total than $-\mathbf{p}_{\text{em}}$ for a static system, the model has broken relativity somewhere (usually by letting the constraining forces do unaccounted work); the theorem is the referee, not the loop.

Is hidden momentum related to the third-law violation of the previous article, or are they separate oddities?

Same root, two exposures — and seeing that unifies the chapter. Both effects are order $(v/c)^2$ consequences of the field carrying momentum. In the third-law article, charges move, the field's momentum changes, and the imbalance shows up dynamically as non-reciprocal forces. Here, everything is static, the field's momentum is frozen nonzero, and the imbalance shows up structurally as a compensating book entry in the matter. You can morph one into the other: take the perpendicular charges, bend one trajectory into a closed loop, and let the system settle — the transient non-reciprocal forces do net work rearranging internal momentum until the static configuration holds exactly the hidden momentum the theorem requires; the "violation" has been converted into standing bookkeeping. The chapter's closer will show both as components of one statement: the symmetric stress–energy tensor's conservation, with the third law's failure living in $\partial_tT^{0i} \neq 0$ moments and hidden momentum in the boost-generator constraint. One tensor, two party tricks.

Could hidden momentum ever be technologically useful — say for propulsion?

No, and the impossibility is informative. Hidden momentum in a static system is exactly $-\mathbf{p}_{\text{em}}$ ; the total is zero, and any internal rearrangement (ramping currents, moving charges) conserves the total while trading momentum between accounts and real impulses on the hardware — which always sum to zero net thrust for a closed system, radiation excepted (the photon-rocket clause of the previous article). Schemes that promise thrust from "asymmetric hidden momentum" are the static cousins of reactionless drives, forbidden by the same theorem. What is useful is the concept's error-catching power: in precision systems — EDM experiments, atom interferometers in fields, force metrology near magnets — the hidden-momentum term is a real entry in the error budget, and omitting it produces phantom anomalies of exactly the kind that generate breathless preprints. Its technological role is that of a good auditor: it never generates revenue, and it prevents expensive mistakes.