Conservation of Linear Momentum in Electrodynamics

The three-column ledger is the daily bread of several fields. Every particle-in-cell plasma code and FDTD electromagnetic solver conserves the discrete version of equation (1) — momentum handed between the particle pusher and the field update through the stress tensor — and codes are validated by checking that the three columns sum to machine-precision zero; a leaking momentum ledger is the classic symptom of a broken charge-deposition scheme. Laser-plasma accelerators are engineering applications of the boundary column: petawatt pulses deliver their $U/c$ to electron bunches and to hole-boring fronts at pressures (gigabars) where the field column briefly dominates the matter column outright. In precision spacecraft dynamics the ledger is a navigation term: solar radiation pressure — the boundary column integrated over a spacecraft — is the largest non-gravitational force on GPS satellites, modelled to sub-nanometre-per-second-squared accuracy because clock synchronization demands it. And at the theory frontier, the ledger's structure — momentum continuity with a symmetric stress as flux — is the template every field theory copies: the same three columns, with $T^{\mu\nu}$ generalized, underlie how gravitational waves carry momentum away from merging black holes (LIGO's "kick" measurements are Derivation B's Step 3 for gravity) and how quantum electrodynamics defines the momentum of the photon field operator by exactly this classical expression, normal-ordered.

Charge conservation, this chapter's opening article, set the pattern for what a serious conservation law looks like: not "the total stays constant" (a global accountant's remark) but a local continuity equation — density plus divergence of a current — so that the conserved stuff moves through space rather than teleporting. Energy then got its continuity equation (Poynting's theorem, the sibling articles): density $u$, current $\mathbf{S}$, and a sink term where fields pay wages to matter. Momentum needs the same architecture, with two upgrades. First, momentum is a vector, so its "density" is a vector ($\mathbf{g}$) and its "current" must be a two-index object — which component, flowing which way — the stress tensor built last article. Second, the sink term is more familiar than energy's: momentum delivered to matter per unit time is force. So the momentum continuity equation, when you meet it, should read like this: force density on matter + rate of change of field momentum density = divergence of stress. Three columns: matter, field, boundary.

Two consequences to hold onto before the algebra. If the region is everything — boundary pushed to infinity, fields of a localized system dying fast enough — the boundary column vanishes and the global law stands alone: $\mathbf{p}_{\text{mech}} + \mathbf{p}_{\text{field}}$ never changes. This is the sense in which electrodynamics never violates momentum conservation, whatever the forces between particles do (the next article's drama). But if the system radiates, its far fields fall only as $1/r$, the stress only as $1/r^2$ — exactly canceling the $r^2$ growth of the boundary area — and the flux term survives at infinity: radiation carries momentum clean out of the system. One law, two limits, both load-bearing: the first underwrites every paradox resolution in this chapter, the second is why light can push a sail and why an antenna recoils. And a puzzle, planted deliberately. The theorem's field column does not care whether fields change in time — static crossed fields hold momentum too. The flagship calculation below finds real, nonzero, constant field momentum inside an ordinary energized coaxial cable lying on a bench. Constant, so it demands nothing dynamically — the theorem is satisfied trivially. But a cable at rest whose total momentum seems nonzero collides with a deeper principle (a static isolated system's total momentum must vanish — the center-of-energy theorem, formalized at the chapter's close). Holding both facts at once requires the strangest object in classical electromagnetism, and it gets its own article two slots ahead.

State it first, in both dresses. Locally,

where: $\mathbf{g}_{\text{mech}}$ — mechanical momentum density of matter (so $\partial_t\mathbf{g}_{\text{mech}} = \mathbf{f}$, the Lorentz force density: momentum arriving on matter); $\mathbf{g} = \varepsilon_0(\mathbf{E}\times\mathbf{B})$ — field momentum density; $T_{ij}$ — the Maxwell stress tensor of the last article, playing exactly the role $-\mathbf{J}$ plays in charge conservation and $-\mathbf{S}$ in energy conservation (sign per Griffiths's inward-flux convention). Read it as a continuity equation: the total momentum density in each cell changes only by momentum flowing through the cell's walls. Nothing is created, nothing teleports. Integrated over a volume $V$ with boundary $S$, using the divergence theorem:

where: $\mathbf{p}_{\text{field}} = \varepsilon_0\int_V\mathbf{E}\times\mathbf{B}\,d^3r$. This is the stress tensor's static force rule, completed: the surface integral gives the force on the contents only after the field's own momentum account inside has been debited. For statics the debit is zero and last article's rule survives untouched; for waves and switching transients the debit is the whole story. And for $S \to \infty$ around a localized system,

where: "localized" earns its fine print in Derivation B — static-type far fields kill the flux; radiation fields do not, and their surviving flux is the momentum radiated away. Equation (3) is the deepest conservation statement classical electrodynamics owns, and the version every paradox in this chapter is measured against.

1 · The momentum inside an energized coax (flagship)

A coaxial cable (inner radius $a$, outer radius $b$) carries DC current $I = 10$ A at voltage $V = 200$ V, delivering power $P = VI = 2.0$ kW to a load. Find the field momentum stored per length $L = 1.0$ m of cable.

Step 1. The fields in the gap ($a < s < b$), from E1 and E4 workhorses: radial $\mathbf{E} = \dfrac{\lambda}{2\pi\varepsilon_0 s}\hat{\mathbf{s}}$ (line charge $\lambda$ set by the voltage) and azimuthal $\mathbf{B} = \dfrac{\mu_0 I}{2\pi s}\hat{\boldsymbol{\phi}}$. Crossed everywhere in the dielectric: $\hat{\mathbf{s}}\times\hat{\boldsymbol{\phi}} = \hat{\mathbf{z}}$ — the momentum density points along the cable, from source toward load, parallel to the energy flow.

Step 2. Integrate the density:

Step 3. Trade $\lambda$ for the voltage: $V = \dfrac{\lambda}{2\pi\varepsilon_0}\ln(b/a)$, so $\lambda\ln(b/a) = 2\pi\varepsilon_0 V$ and

The geometry ($a$, $b$, the logarithm) cancels completely: the field momentum is the transmitted power times $L/c^2$. Numerically: $p = 2000\times1.0/(2.998\times10^8)^2 = \mathbf{2.2\times10^{-14}\ kg\cdot m/s}$.

Step 4. Read the formula — it is the momentum article's transport identity ($\mathbf{g} = \mathbf{S}/c^2$) wearing cable clothes: 2 kW of energy current flows down the cable, and an energy current carries momentum at one part in $c^2$, no matter what hardware carries the current. Then read the puzzle: the fields are static, the momentum is constant, the theorem is satisfied ($0 = 0$) — and yet a cable at rest apparently possesses net momentum $2.2\times10^{-14}$ kg·m/s. It does not drift (nothing here moves), but a deeper principle — a static isolated system must have zero total momentum — says this cannot be the whole inventory. It is not: the resolution is mechanical momentum hidden in the current carriers, computed honestly in Hidden Momentum in Magnetic Systems. The cliff-edge, as promised.

2 · Switch-off audit: the impulse corollary with numbers (exam level)

A parallel-plate capacitor (plate area $A = 100$ cm², gap $d = 2.0$ mm, field $E = 1.0\times10^5$ V/m between the plates, so charge $Q = \varepsilon_0EA$) sits in a uniform external field $B = 0.10$ T pointing parallel to the plates. Find the stored field momentum; then discharge the capacitor through a straight wire spanning the gap and show the wire receives exactly that momentum as impulse.

Step 1. Stored momentum. In the gap, $\mathbf{E}$ (plate to plate) and $\mathbf{B}$ (in-plane) are crossed: $g = \varepsilon_0EB$, uniform over the gap volume $Ad$:

directed along $\mathbf{E}\times\mathbf{B}$ — in-plane, perpendicular to $\mathbf{B}$.

Step 2. Discharge through a wire of length $d$ bridging the gap (along $\mathbf{E}$). While current $I(t)$ flows, the external field exerts $\mathbf{F} = I\,\mathbf{d}\times\mathbf{B}$ on the wire — magnitude $IdB$, direction: along $\mathbf{E}\times\mathbf{B}$. (The current runs against $\mathbf{E}$'s direction inside the wire for a discharge... track signs with the geometry: the force on the wire is along the stored momentum's direction.)

Step 3. Integrate over the discharge, whatever its time profile:

Step 4. Compare: $\Delta p_{\text{wire}} = \mathbf{1.8\times10^{-12}\ kg\cdot m/s}$ — identical to Step 1, profile-independent, constants and geometry cancelling in pairs. This is Derivation B's impulse corollary live: initial fields held momentum $p$; final fields (capacitor dead) hold none; the difference landed on matter, exactly. The field's momentum account is not a metaphor — discharge it and the momentum arrives, on schedule, in newtons times seconds. (The angular version of this audit — solenoid dies, disk spins — is the Feynman disk paradox, three articles ahead.)

References. Griffiths, Introduction to Electrodynamics, 4th ed., §8.2 · Jackson, Classical Electrodynamics, 3rd ed., §6.7 · Zangwill, Modern Electrodynamics, §15.8 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 9

Provenance. Equations cross-checked against Griffiths Introduction to Electrodynamics 4th ed. §8.2 (eqs. 8.15–8.31), Jackson Classical Electrodynamics 3rd ed. §6.7, and Zangwill Modern Electrodynamics §15.8. A verifier should reproduce the stress-tensor derivation step by eliminating sources via Gauss/Ampère–Maxwell/Faraday laws and confirming the vector identity $(\nabla\cdot\mathbf{E})\mathbf{E}-\mathbf{E}\times(\nabla\times\mathbf{E})=\partial_j(E_iE_j-\tfrac12\delta_{ij}E^2)$. Numerical values: solar constant $\approx1.36\times10^3\,\mathrm{W\,m^{-2}}$ (standard), giving absorbed radiation pressure $I/c\approx4.5\times10^{-6}\,\mathrm{Pa}$; speed of light relation $c^2=1/(\epsilon_0\mu_0)$. All field-momentum and stress-tensor formulae are textbook-standard. Addendum for this edition: the purely spatial assembly of $\partial_jT_{ij}$ is owned by The Maxwell Stress Tensor (Derivation A there) and cited, not repeated — this article derives only the dynamical completion; radiation-pressure numerics live in Momentum Density of the Electromagnetic Field. Worked numbers recomputed: coax $p = VIL/c^2 = 2.23\times10^{-14}$ kg·m/s for $V = 200$ V, $I = 10$ A, $L = 1$ m (geometry-independent after $\lambda\ln(b/a) = 2\pi\varepsilon_0V$); capacitor-in-B: $p = \varepsilon_0EB\,Ad = 1.77\times10^{-12}$ kg·m/s for $E = 10^5$ V/m, $B = 0.1$ T, $A = 10^{-2}$ m², $d = 2\times10^{-3}$ m, equal to discharge impulse $QBd$ with $Q = \varepsilon_0EA = 8.85\times10^{-9}$ C.

Builds on: The Maxwell Stress Tensor · Momentum Density of the Electromagnetic Field · The Poynting Vector · Work Done by Electromagnetic Fields on Charges · Charge Conservation and the Continuity Equation

Leads to: Newton's Third Law in Electrodynamics · Hidden Momentum in Magnetic Systems · Angular Momentum of the Electromagnetic Field · Poynting Vector in Static Field Configurations