Newton's Third Law in Electrodynamics

Every so often physics owes you an uncomfortable calculation, performed without anesthetic. Here is this chapter's. Take two point charges moving at right angles to each other — one heading east, one heading north, both approaching the same corner. Compute the full electromagnetic force each exerts on the other, Coulomb plus magnetic. The electric parts balance. The magnetic parts do not: charge 1 pushes charge 2 west while charge 2 pushes charge 1 south. Equal magnitudes, perpendicular directions. Action does not equal minus reaction. Newton's third law — the axiom your first mechanics course treated as bedrock, the guarantor of momentum conservation itself — fails, in plain quasistatic electrodynamics, for the simplest imaginable system. This article refuses to soften that result. The violation is real, not an artifact of approximation: we compute it explicitly, trace exactly why two centuries of magnetostatics never noticed (closed steady circuits obey the third law as a theorem — the loophole is precise), and identify the true casualty. It is not momentum conservation. It is the picture of forces as private transactions between pairs of particles, settled instantaneously at a distance. The rescue was assembled over the last three articles: the field carries momentum, and the previous article's theorem guarantees that mechanical-plus-field momentum is exactly conserved, always. What the perpendicular charges lose as a pair, their shared field gains, instant by instant. The third law survives in the only form nature ever promised: not "every force has a counter-force" but "momentum has three columns and the ledger closes." Newton's axiom was the $c \to \infty$ shadow of that deeper law — and this is the article where the shadow and the law part company.

Key takeaways

For isolated moving point charges, the magnetic interaction forces are generally not equal and opposite: in the perpendicular-approach geometry, $\mathbf{F}_{12}$ and $\mathbf{F}_{21}$ have equal magnitudes but perpendicular directions. Computed, not hand-waved.
The violation is order $(v/c)^2$ relative to the Coulomb force — invisible to slow mechanics, structural at accelerator speeds.
Magnetostatics hid it for a reason: for closed steady circuits the loop-integrated forces obey $\mathbf{F}_{12} = -\mathbf{F}_{21}$ exactly (the gradient term dies on closed loops). Only isolated charges expose the failure.
The deep cause is retardation: forces mediated at finite speed cannot be settled pairwise-instantaneously. Newton's third law is an instantaneous-action-at-a-distance fossil.
Nothing is violated globally: the field's momentum completes the ledger, and $\mathbf{p}_{\text{mech}} + \mathbf{p}_{\text{field}}$ is exactly conserved (previous article's theorem — recalled, not re-proved).
KEY IDEA: the third law fails for moving charges because momentum flows through the field with finite speed; what survives — strengthened — is momentum conservation with the field as a full participant.

The core insight

Isolated moving charges exert electromagnetic forces on each other that are not equal and opposite — Newton's third law fails — yet total momentum is exactly conserved, because the electromagnetic field itself carries the balance.

Where this lives today

The failure of pairwise force balance is working physics wherever charges move fast and freely. Beam–beam interactions at particle colliders are computed field-through-particle, never force-pair by force-pair: the electromagnetic wake one bunch leaves is a field configuration that kicks the oncoming bunch with no meaningful "reaction on the first bunch now" — the momentum ledger is kept per bunch plus field, exactly this chapter's three columns. Plasma physics institutionalized the lesson decades ago: particle-in-cell codes move charges in fields and update fields from charges, and pairwise Newton pairs simply never appear in the algorithm — momentum conservation is verified globally, particles plus field, each timestep. The conceptual legacy runs deeper. The third law's failure was an early crack in mechanical worldview through which field theory fully entered: if conservation laws are to be exact, fields must be first-class owners of momentum and energy — the very premise on which quantum field theory later built (particles themselves becoming excitations of fields). And the modern effective-field-theory view completes the arc: whenever physicists today write "instantaneous potential" models (molecular dynamics, gravitational few-body codes), they knowingly work in the $c \to \infty$ corner where the third law is a valid theorem — with post-Newtonian and Darwin corrections, order $(v/c)^2$ , as the systematic bill for the fiction. The perpendicular-charge audit of this article is, in that language, the leading Darwin term made visible.

Origins · 1 minIsaac Newton and how this was discovered

Isaac Newton

1643–1727 · Woolsthorpe, Cambridge, London

It is fair, in an article about breaking his third law, to let Newton state what he actually claimed. Law III of the Principia (1687): to every action there is always opposed an equal reaction — distilled from collision experiments (Wren, Wallis, Huygens on colliding pendulums) and load-bearing for his system in exactly one way: it made momentum conservation automatic for any assembly of bodies, letting him treat planets, tides, and cannonballs as systems without inventorying their internals. Newton was carefully uneasy about the instantaneous action-at-a-distance his gravity required — "that one body may act upon another at a distance through a vacuum... is to me so great an absurdity that I believe no man who has... a competent faculty of thinking can ever fall into it" — and hoped a mediating agent would be found. Two centuries later Faraday and Maxwell supplied one, and it promptly did what mediators must: it took time to carry influence, held momentum in transit, and retired the pairwise law while preserving the thing Newton built it to guarantee. The third law failed; the conservation law it protected was strengthened. It is hard to imagine Newton — who prized the conserved quantity, distrusted the action at a distance, and demanded the mediator — regarding this chapter as anything but a settling of his own accounts.

History

1687

— Principia states Law III; momentum conservation for systems follows

1820

s — Ampère insists force between current elements must be central and reciprocal; Grassmann's (1845) element force is neither — the dispute is undecidable because closed circuits hide the difference

1893

— J.J. Thomson locates momentum in the field around moving charges

1900

— Poincaré: without field momentum, radiation recoil breaks the reaction principle and centers of mass self-accelerate

1905

— Special relativity makes simultaneous action-reaction between distant bodies undefinable in principle

1912

— Von Laue completes the relativistic momentum bookkeeping for extended electromagnetic systems

1960

s — Particle simulation codes adopt field-mediated momentum accounting; pairwise force symmetry quietly exits computational physics

The physical picture

Newton's third law was never really an independent axiom about forces; it was momentum conservation wearing a particular costume. If interactions are private, instantaneous transactions between pairs — I push you, you push me, right now, along the line between us — then $\mathbf{F}_{12} = -\mathbf{F}_{21}$ is the statement that our momentum exchange nets to zero at every instant. The costume fits any world where influence travels infinitely fast. In Newton's world it did: his gravity acted at a distance with no delay, and the third law was its bookkeeping. Electrodynamics broke the costume's central seam. Influence now travels at $c$ , through a field that takes time to update: when charge 1 swerves, charge 2 does not learn of it for $r/c$ seconds — and during that interval, whatever momentum charge 1 gave up is in transit, held by neither particle. Pairwise instant bookkeeping cannot survive this. The only question is where the failure first shows up in the equations, and the answer is: at order $(v/c)^2$ , in the magnetic force — because magnetism, as the E4 chapter kept hinting and the E10 chapter will prove, is the first relativistic correction to Coulomb's law. The perpendicular-charge geometry below is simply the cleanest stage for the seam to tear.

Figure

The audit that fails: two charges crossing at right angles push on each other with equal magnitudes in perpendicular directions

Before the calculation, calibrate your suspicion correctly. This is not a paradox of bad approximations — the quasistatic fields used below are the legitimate leading behavior, and pushing to exact retarded fields makes the imbalance worse, not better (the extreme case being radiation, where a single accelerated charge sends momentum to infinity with no partner anywhere). Nor is it a sign that something is wrong with the Lorentz force. It is a discovery about what kind of law the third law always was: a theorem of instantaneous mechanics, valid exactly in the $c \to \infty$ limit where the field's momentum column (suppressed by $1/c^2$ — the momentum article's ledger) vanishes identically. Electrodynamics keeps the theorem's parent — momentum conservation, now with three columns — and retires the theorem. One more honest note about why you never met this in E4. Everything magnetostatics ever computed involved closed steady circuits, and for those, Derivation B proves the third law holds exactly — the violating terms integrate to zero around any closed loop. The static chapters weren't lying; they were protected by their own geometry. The protection lapses the moment a charge travels alone.

The audit, quantitatively

The tools are all in hand from E4. A point charge in uniform (or slowly changing) motion carries, to leading order in $v/c$ , its Coulomb field and the magnetic field of the Biot–Savart point-charge form:

\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\,\frac{q\,\mathbf{v}\times\hat{\boldsymbol{r}}}{r^2}, \qquad (1)

where: $\hat{\boldsymbol{r}}$ , $r$ — unit vector and distance from the charge's present position to the field point; valid for $v \ll c$ and away from the acceleration's radiation zone (the exact fields are E9/E10 business; corrections enter at relative order $(v/c)^2$ , which will not change the verdict — they deepen it). Note the geometry that drives everything below: the field vanishes along the line of motion ( $\mathbf{v}\times\hat{\boldsymbol{r}} = 0$ fore and aft) and is strongest broadside. The interaction forces to audit are the full Lorentz pair:

\mathbf{F}_{2\leftarrow 1} = q_2\left[\mathbf{E}_1(\mathbf{r}_2) + \mathbf{v}_2\times\mathbf{B}_1(\mathbf{r}_2)\right], \qquad \mathbf{F}_{1\leftarrow 2} = q_1\left[\mathbf{E}_2(\mathbf{r}_1) + \mathbf{v}_1\times\mathbf{B}_2(\mathbf{r}_1)\right]. \qquad (2)

where: each charge responds to the other's fields evaluated at its own position. The electric parts, at this order, are central and equal-opposite — they pass the audit and we set them aside. The magnetic parts are the defendants.

Derivation AThe violation, computed — perpendicular charges push in perpendicular directions

The goal is the overview's claim with every vector shown: for two equal charges approaching a corner at right angles, the magnetic forces $\mathbf{F}_{2\leftarrow1}$ and $\mathbf{F}_{1\leftarrow2}$ are equal in magnitude and perpendicular in direction, so their sum — the net electromagnetic force on the isolated pair — is nonzero. The strategy is direct evaluation of (1) and (2) in the symmetric geometry, chosen so every cross product is a one-liner and the punchline is not hidden in algebra. Watch where each right-hand rule fires; the violation is pure geometry.

Step 1. The stage. Put $q_1$ at $(-d, 0, 0)$ moving with $\mathbf{v}_1 = v\,\hat{\mathbf{x}}$ , and $q_2$ at $(0, -d, 0)$ moving with $\mathbf{v}_2 = v\,\hat{\mathbf{y}}$ — both heading for the origin, separation $\sqrt{2}\,d$ . Unit vector from 1 to 2: $\hat{\boldsymbol{r}}_{12} = (\hat{\mathbf{x}} - \hat{\mathbf{y}})/\sqrt{2}$ .

Step 2. Field of 1 at 2. From (1): $\mathbf{v}_1\times\hat{\boldsymbol{r}}_{12} = v\,\hat{\mathbf{x}}\times(\hat{\mathbf{x}}-\hat{\mathbf{y}})/\sqrt{2} = -(v/\sqrt{2})\,\hat{\mathbf{z}}$ , so

\mathbf{B}_1(\mathbf{r}_2) = -\frac{\mu_0 q v}{8\sqrt{2}\,\pi d^2}\,\hat{\mathbf{z}}

— into the page at $q_2$ 's location. Why into the page: $q_1$ moves east; broadside-south of its track, the right-hand rule wraps its field lines clockwise seen from ahead, giving $-\hat{\mathbf{z}}$ below the track.

Step 3. Force on 2. $\mathbf{F}_{2\leftarrow1}^{\text{mag}} = q\,\mathbf{v}_2\times\mathbf{B}_1 = qv\,\hat{\mathbf{y}}\times\left(-\tfrac{\mu_0 qv}{8\sqrt{2}\pi d^2}\hat{\mathbf{z}}\right) = -\dfrac{\mu_0 q^2v^2}{8\sqrt{2}\,\pi d^2}\,\hat{\mathbf{x}}$ — due west.

Step 4. Repeat with the labels swapped. $\hat{\boldsymbol{r}}_{21} = -\hat{\boldsymbol{r}}_{12}$ ; $\mathbf{v}_2\times\hat{\boldsymbol{r}}_{21} = v\,\hat{\mathbf{y}}\times(-\hat{\mathbf{x}}+\hat{\mathbf{y}})/\sqrt{2} = +(v/\sqrt{2})\,\hat{\mathbf{z}}$ , so $\mathbf{B}_2(\mathbf{r}_1) = +\dfrac{\mu_0qv}{8\sqrt{2}\pi d^2}\hat{\mathbf{z}}$ (out of the page at $q_1$ ), and

\mathbf{F}_{1\leftarrow2}^{\text{mag}} = qv\,\hat{\mathbf{x}}\times\mathbf{B}_2(\mathbf{r}_1) = -\frac{\mu_0q^2v^2}{8\sqrt{2}\,\pi d^2}\,\hat{\mathbf{y}}

— due south. Equal magnitude to Step 3's force. Perpendicular to it.

Step 5. The verdict, in three lines. The electric forces cancel pairwise; the magnetic forces sum to

\mathbf{F}_{\text{net on pair}} = -\frac{\mu_0 q^2 v^2}{8\sqrt{2}\,\pi d^2}\,(\hat{\mathbf{x}} + \hat{\mathbf{y}}) \neq \mathbf{0}.

An isolated two-particle system experiences a net force: pairwise mechanical momentum is not conserved. Relative to the Coulomb force $q^2/4\pi\varepsilon_0(2d^2)$ , the imbalance is of magnitude $(v/c)^2/\sqrt{2}$ — invisible for drift velocities, order-unity for relativistic beams. And the asymmetric limiting geometry is starker still: put one charge directly on the other's line of motion, and (1) gives zero field there while the perpendicular partner's field does not vanish — one charge feels a magnetic force, the other feels none at all. Verified: geometry cross-checked numerically (electrons, $v = 0.5c$ , $d = 0.5\ \mu$ m: each magnetic force $8.16\times10^{-17}$ N, sum $1.15\times10^{-16}$ N along $-(\hat{\mathbf{x}}+\hat{\mathbf{y}})/\sqrt{2}$ , electric forces $4.61\times10^{-16}$ N canceling exactly); matches Griffiths §8.2.1's qualitative audit; ratio to Coulomb $= \mu_0\varepsilon_0v^2/\sqrt{2} = (v/c)^2/\sqrt{2} = 0.18$ at $v = 0.5c$ , checked.

Figure

The loophole that protected magnetostatics: current elements violate the third law, but closing the loops integrates the offense away

Go deeper · Derivation B · 2 minWhy magnetostatics never noticed — the closed-loop protection theorem

Derivation BWhy magnetostatics never noticed — the closed-loop protection theorem

The goal is to prove the claim that shielded two chapters of magnetostatics: for two closed loops carrying steady currents, the total magnetic forces satisfy $\mathbf{F}_{12} = -\mathbf{F}_{21}$ exactly — the third law holds — even though the force between individual current elements violates it. The strategy: start from the Lorentz (Grassmann) force between elements, split it with the BAC–CAB identity into a symmetric central term plus an offender, and show the offender is a perfect gradient whose closed-loop integral dies. Watch how the loophole closes only for complete circuits — the same term that dies on a loop survives for an isolated moving charge, which is Derivation A's whole story.

Step 1. The element–element force. Loop 1 (current $I_1$ , element $d\mathbf{l}_1$ at $\mathbf{r}_1$ ) acts on loop 2's element via Biot–Savart plus Lorentz:

d\mathbf{F}_{2\leftarrow1} = \frac{\mu_0 I_1I_2}{4\pi}\,\frac{d\mathbf{l}_2\times(d\mathbf{l}_1\times\hat{\boldsymbol{r}}_{12})}{r^2},

with $\hat{\boldsymbol{r}}_{12}$ , $r$ from element 1 to element 2. Element-wise this is not antisymmetric under $1\leftrightarrow2$ — swap the labels and the double cross product changes shape, not just sign. Current elements, like isolated charges, violate the third law.

Step 2. Split with BAC–CAB: $d\mathbf{l}_2\times(d\mathbf{l}_1\times\hat{\boldsymbol{r}}) = d\mathbf{l}_1\,(d\mathbf{l}_2\cdot\hat{\boldsymbol{r}}) - \hat{\boldsymbol{r}}\,(d\mathbf{l}_1\cdot d\mathbf{l}_2)$ . The second term is central (along $\hat{\boldsymbol{r}}$ ) and manifestly antisymmetric: swapping labels flips $\hat{\boldsymbol{r}}_{12} \to -\hat{\boldsymbol{r}}_{12}$ while $d\mathbf{l}_1\cdot d\mathbf{l}_2$ is symmetric. If only this term existed, the third law would hold element-wise. The first term is the offender.

Step 3. Kill the offender on closed loops. Its contribution to the total force is

\frac{\mu_0I_1I_2}{4\pi}\oint_1 d\mathbf{l}_1\oint_2\frac{d\mathbf{l}_2\cdot\hat{\boldsymbol{r}}_{12}}{r^2}.

But $\hat{\boldsymbol{r}}_{12}/r^2 = -\nabla_2(1/r)$ , so the inner integral is $-\oint_2 d\mathbf{l}_2\cdot\nabla_2(1/r) = 0$ — the line integral of a gradient around a closed path. Why this is the precise loophole: the offender is an exact differential in the "receiving" coordinate; only complete circuits integrate it away. An isolated moving charge is an open current element with nowhere to close, and keeps the offending force in full.

Step 4. What survives is symmetric. With the offender gone,

\mathbf{F}_{2\leftarrow1} = -\frac{\mu_0I_1I_2}{4\pi}\oint_1\oint_2\frac{(d\mathbf{l}_1\cdot d\mathbf{l}_2)\,\hat{\boldsymbol{r}}_{12}}{r^2} = -\,\mathbf{F}_{1\leftarrow2},

central and antisymmetric — Newton's third law as a theorem for closed steady circuits. Every force magnetostatics ever computed (parallel wires, coil pairs, motor torque reactions) lived under this theorem's protection, which is why the E4 chapter's article on forces between currents could verify action–reaction with a clear conscience. The protection is not approximate — it is exact for steady loops — but it is conditional, and Derivation A is what happens the moment the condition fails. Verified: reproduces the standard Neumann-symmetrization argument (Griffiths Problem 5.49 lineage; Jackson §5.2's remark on the Ampère–Grassmann equivalence for closed circuits); the surviving double integral is Neumann-form symmetric, consistent with $F = -\partial W/\partial x$ from the mutual-inductance energy (E6); the gradient identity $\hat{\boldsymbol{r}}/r^2 = -\nabla(1/r)$ is the E0 workhorse.

The rescue — recalled, not re-proved

Where does the perpendicular pair's missing momentum go? Into the only other participant on stage. The previous article proved — from the same Maxwell equations that produced the forces above — that

\frac{d}{dt}\left(\mathbf{p}_1 + \mathbf{p}_2 + \varepsilon_0\!\int\mathbf{E}\times\mathbf{B}\,d^3r\right) = \oint_{S_\infty}\overleftrightarrow{T}\cdot d\mathbf{a} \to 0, \qquad (3)

where: the surface term dies for this non-radiating quasistatic pair (the fall-off audit of Conservation of Linear Momentum in Electrodynamics, Derivation B there). So the net southwest force on the particles is exactly balanced by momentum accumulating in their overlapping fields: $d\mathbf{p}_{\text{field}}/dt = +\dfrac{\mu_0q^2v^2}{8\sqrt{2}\pi d^2}(\hat{\mathbf{x}}+\hat{\mathbf{y}})$ , northeast, at this instant. No new derivation is needed and none is given — the theorem was built precisely so that every such audit closes by citation. What this article adds is the reading: the third law's failure and the field's momentum are not two curiosities but one fact seen from two sides. Forces between charges are not private transactions; they are each particle's local settlement with the field, and the field's own account is what makes the books balance. The relativistic postscript sharpens it. Retardation makes any pairwise-instantaneous force law untenable in principle: "equal and opposite right now" is not even well-defined when simultaneity is frame-dependent (the E10 chapter's territory). The third law's true successor is the local continuity law — momentum flowing through the field at finite speed, the stress tensor as its current — which needs no notion of "now" shared between distant particles. Newton's axiom is recovered exactly where it was discovered: slow charges, closed circuits, $c$ effectively infinite.

Limiting cases

Case	Result	Meaning
$v \ll c$	violation $\sim (v/c)^2$ of Coulomb	Newtonian mechanics safe to that order; two centuries of innocence explained
closed steady circuits	$\mathbf{F}_{12} = -\mathbf{F}_{21}$ exactly	Derivation B's theorem: magnetostatics was protected, not lucky
collinear motion (one charge on the other's track)	one magnetic force zero, the other not	maximal asymmetry; same ledger closes it
parallel same-speed charges	magnetic forces equal-opposite	special geometries can pass the audit; do not generalize from them
$c \to \infty$	$\mathbf{B} \to 0$ , field momentum $\to 0$	third law restored identically; it is the instantaneous limit's theorem
radiation (accelerated charge)	momentum leaves with no partner at all	the violation's extreme case; boundary column collects (previous article)
relativity	"equal-opposite now" loses meaning	continuity law replaces pairwise bookkeeping outright

Common trapDo not patch the third law — retire it

Three tempting repairs, all wrong. "Include the reaction of the field on itself and the third law holds pairwise somehow" — no: the field is not a third particle exerting forces; it is a continuum carrying momentum, and forcing its bookkeeping into action–reaction pairs produces nonsense (whose reaction is radiation escaping to infinity?). "The quasistatic fields were the approximation; exact retarded fields restore the balance" — backwards: exact fields make pairwise imbalance generic, and radiation makes it maximal. "Weak law broken, strong law kept" — the traditional textbook split (weak: forces equal-opposite; strong: also central) invites the false memory that some pairwise law survives; for isolated moving charges both fail. The correct posture: pairwise force symmetry was always a stand-in for momentum conservation, valid when interaction momentum-in-transit is negligible. Electrodynamics promotes the real law — three-column continuity — and demotes the stand-in to a limit theorem (exact again only for $c \to \infty$ or closed steady loops). Exams love to ask "does electromagnetism violate Newton's third law?" The full-credit answer is yes, pairwise — and total momentum is conserved anyway, because the field carries the difference.

Worked examples

1 · The crossing electron beams, in numbers (confidence builder)

Two electrons cross at right angles, each with $v = 0.5c$ , in the symmetric geometry of Derivation A with $d = 0.5\ \mu$ m (separation $r = \sqrt{2}\,d = 0.71\ \mu$ m). Audit all four interaction forces.

Step 1. Electric pair. $F_E = \dfrac{e^2}{4\pi\varepsilon_0 r^2} = \dfrac{(1.60\times10^{-19})^2\times8.99\times10^9}{(7.07\times10^{-7})^2} = \mathbf{4.6\times10^{-16}\ N}$ on each, directed along the separation line, exactly opposite: this pair passes.

Step 2. Magnetic pair. From Derivation A, each magnitude is $F_B = \dfrac{\mu_0e^2v^2}{8\sqrt{2}\pi d^2}$ . Numerically: $\mu_0e^2v^2 = 1.257\times10^{-6}\times(1.60\times10^{-19})^2\times(1.50\times10^8)^2$ , and dividing by $8\sqrt{2}\pi d^2 = 8\sqrt{2}\pi\times(5\times10^{-7})^2$ : $F_B = \mathbf{8.2\times10^{-17}\ N}$ — electron 2 pushed along $-\hat{\mathbf{x}}$ , electron 1 along $-\hat{\mathbf{y}}$ .

Step 3. The imbalance. $|\mathbf{F}_1 + \mathbf{F}_2| = \sqrt{2}\,F_B = \mathbf{1.2\times10^{-16}\ N}$ , pointing southwest — a net force on an isolated pair, 25% of the Coulomb magnitude ( $F_B/F_E = (v/c)^2/\sqrt{2} = 0.18$ each; the net is $\sqrt{2}$ of one of them). At this speed the effect is not a refinement; it is a quarter of the interaction.

Step 4. The ledger line. By equation (3), the pair's fields are gaining momentum northeast at $1.2\times10^{-16}$ N. If the electrons were slower by ten — $v = 0.05c$ , beam-physics territory — every magnetic entry drops by 100: the $(v/c)^2$ dial that kept the nineteenth century asleep.

2 · The maximal asymmetry: one pushes, the other ignores (exam level)

Electron A is at the origin moving along $+\hat{\mathbf{y}}$ with $v = 0.5c$ ; electron B sits at $(-1.0\ \mu\text{m}, 0, 0)$ moving along $+\hat{\mathbf{x}}$ — directly toward A, along the line joining them. Compute both magnetic forces and the net, and state where the momentum is going.

Step 1. Force on A from B. B moves along the separation line, so at A's location $\mathbf{v}_B\times\hat{\boldsymbol{r}} = 0$ : B produces no magnetic field at A whatsoever. $\mathbf{F}_{A\leftarrow B}^{\text{mag}} = 0$ . (Fore-and-aft blindness of equation (1) — a moving charge's magnetic field vanishes on its own track.)

Step 2. Force on B from A. A moves broadside to the line: $\mathbf{v}_A\times\hat{\boldsymbol{r}}_{AB} = v\,\hat{\mathbf{y}}\times(-\hat{\mathbf{x}}) = +v\,\hat{\mathbf{z}}$ , giving $\mathbf{B}_A(\mathbf{r}_B) = \dfrac{\mu_0ev}{4\pi r^2}\hat{\mathbf{z}}$ (with the electron's negative charge flipping it to $-\hat{\mathbf{z}}$ ), and

F_{B\leftarrow A}^{\text{mag}} = \frac{\mu_0e^2v^2}{4\pi r^2} = \frac{1.257\times10^{-6}\times(1.60\times10^{-19})^2\times(1.5\times10^8)^2}{4\pi\times(10^{-6})^2} = \mathbf{5.8\times10^{-17}\ N},

directed along $-\hat{\mathbf{y}}$ (work the two sign flips: $\mathbf{v}_B\times\mathbf{B}_A$ with both charges negative).

Step 3. Net on the pair: $5.8\times10^{-17}$ N with no reaction at all — the starkest form of the failure. One particle is being pushed by a partner it cannot push back on magnetically.

Step 4. The audit closes anyway: the pair's field momentum changes at exactly $+5.8\times10^{-17}$ N in $+\hat{\mathbf{y}}$ (equation (3)); as the geometry evolves, that momentum flows partly back to the particles, and if either accelerates enough to radiate, partly to infinity. Nothing drifts unaccounted; only the pairwise fiction died. The one-sentence exam summary: magnetic interactions between isolated charges need not be reciprocal even in magnitude; conservation is enforced by the field, not by the pair.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §8.2.1 · Jackson, Classical Electrodynamics, 3rd ed., §6.7, §12.10 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §5.7–5.8 · Zangwill, Modern Electrodynamics, §15.5 · Feynman Lectures, Vol. II, §26-2, §27-6

Provenance. The perpendicular-charge audit follows Griffiths §8.2.1's geometry with the quasistatic point-charge fields (Biot–Savart form); the closed-loop symmetrization is the standard Neumann-form argument (Griffiths Problem 5.49; Jackson §5.2). The rescue statement cites the momentum conservation theorem of the preceding article rather than re-deriving it. A verifier should: (1) reproduce Derivation A's vectors and confirm the net pair force $-\mu_0q^2v^2(\hat{\mathbf{x}}+\hat{\mathbf{y}})/8\sqrt{2}\pi d^2$ ; (2) confirm the gradient-term vanishing $\oint d\mathbf{l}\cdot\nabla(1/r) = 0$ in Derivation B; (3) check the numerical audits (electrons, $v = 0.5c$ : $F_E = 4.6\times10^{-16}$ N at $r = 0.71\ \mu$ m; $F_B = 8.2\times10^{-17}$ N each; head-on case $5.8\times10^{-17}$ N at 1 μm with zero reaction) against the stated formulas. All formulas textbook-standard; worked numbers recomputed for this edition.

Builds on: Work Done by Electromagnetic Fields on Charges · The Poynting Vector · Charge Conservation and the Continuity Equation

Leads to: Momentum Density of the Electromagnetic Field · The Maxwell Stress Tensor · Conservation of Linear Momentum in Electrodynamics · Hidden Momentum in Magnetic Systems

Doesn't the electric force also violate the third law once retardation is included?

Yes — the quasistatic audit assigns the violation to magnetism only because quasistatics is the leading order. Work with the exact retarded fields and the electric forces between accelerating charges fail pairwise balance too: each charge responds to where the other was, and "equal and opposite now" fails in every sector. The magnetic violation is merely the first term of the failure in a $(v/c)$ expansion — which is one more way of seeing that magnetism is relativity's first correction to electrostatics (the E10 chapter's opening theme). The honest hierarchy: at order $(v/c)^0$ , Coulomb central forces, third law exact; at $(v/c)^2$ , magnetic and Darwin corrections, pairwise balance fails but a conserved total still exists with the quasistatic field momentum; at radiation order, momentum leaves the system entirely and only the full three-column law survives. Each rung keeps conservation and loses a little more of the pairwise fiction.

If the field carries the missing momentum, can I think of the field as a third body and restore the third law that way?

As a mnemonic, fine; as physics, it fails in an instructive way. A "third body" would be a localized object with a position, exchanging forces pairwise with each charge. The field is a continuum: its momentum is distributed (density $\varepsilon_0\mathbf{E}\times\mathbf{B}$ everywhere), its exchanges with matter are local (Lorentz force at each charge's position), and there is no meaningful "force of charge 1 on the field as a whole, reciprocated." Try to build the pairwise picture and radiation breaks it immediately — momentum departs to infinity, and a third law demands a partner that does not exist. What actually replaces pairwise bookkeeping is continuity: momentum flows from charge 1 into the field locally, propagates through the field at $c$ (stress tensor as current), and lands on charge 2 later. Newton's law was a statement about transactions; the truth is a statement about transport. Keep the transport picture and every case — quasistatic, radiative, relativistic — is one law.

How did Ampère and his contemporaries miss this — didn't they measure forces between currents all the time?

They not only missed it, they institutionalized the protection theorem without knowing it. Ampère insisted on philosophical grounds that element–element forces be central and reciprocal, and constructed his force law accordingly; Grassmann's 1845 alternative (the modern $Id\mathbf{l}\times\mathbf{B}$ form) is neither central nor reciprocal element-wise. The nineteenth century could not settle the dispute experimentally — because every experiment used closed circuits, and Derivation B shows the two laws differ by a term that integrates to zero around any closed loop. Identical predictions for every measurable case; a debate about invisible bookkeeping. The resolution had to wait for isolated charges (cathode rays, then particle beams) where the Grassmann/Lorentz form wins outright — and where its third-law violation, no longer integrable away, surfaces as this article's subject. It is a clean historical lesson in how a symmetry can be an artifact of the apparatus: the third law held in every lab for a century because every lab soldered its circuits shut.

Does the violation mean I could build a reactionless drive from crossed charges or currents?

No — and the impossibility is exactly the conservation theorem. The perpendicular pair feels a net force southwest, but the total (pair plus field) momentum is constant: as the system evolves, the field momentum changes compensate, and any closed cycle of the machinery returns every column to its starting balance with zero net impulse (non-radiating case). What a "reactionless" EM drive can legally do is radiate: push momentum out through the boundary column as directed radiation — a photon rocket — with thrust $P/c$ , about 3.3 μN per megawatt: legal, conserved, and heroically inefficient. Claims of thrust from closed cavities with no radiated momentum (the periodic "EM drive" news cycle) would require the three-column ledger to fail, which is why the physics community's prior against them is effectively absolute — momentum conservation in electrodynamics is not an assumption but a theorem of Maxwell plus Lorentz, tested wherever charges and light interact.

In the head-on example, charge B feels nothing magnetically. Doesn't that break even the ledger — what supplies B's reaction?

Nothing needs to "supply a reaction" — that demand is the retired law talking. The ledger only requires that total momentum be constant: A gains momentum southward from B's field; therefore the field account loses momentum southward at the same rate (its $\int\mathbf{E}\times\mathbf{B}$ is evolving as the charges move — remember both charges' fields overlap, and it is the cross terms that hold interaction momentum, exactly as in the stress-tensor article's warning). B's own motion, meanwhile, is unaffected magnetically at this instant — perfectly permissible, since no principle ties A's gain to B's loss specifically. If it still feels wrong, run time forward: the geometry changes, B stops being fore-aft aligned, forces reappear on it, and integrated over the whole encounter each particle's impulse plus the field's net change sums to zero. Instant-by-instant pairing was Newton's costume; the conservation law never wore it.

Is momentum conservation itself ever violated in electrodynamics — even in exotic cases?

Within classical electrodynamics with the standard stress tensor: never — it is an identity of the equations, as the previous article's derivation makes plain, holding for radiation, relativistic motion, media (with care about which momentum you call "field" — the Abraham–Minkowski article), and arbitrary time dependence. The places where the accounting gets subtle are instructive, not threatening: self-force (an accelerating charge's field acts back on it — radiation reaction, E11 — where the classical point-charge model strains but conservation holds for any finite structure); "static systems with net field momentum" (resolved by hidden mechanical momentum, next article, with the center-of-energy theorem enforcing zero total); and quantum electrodynamics, where the conservation law is promoted to an operator identity tied to translation symmetry — Noether's theorem, the chapter's closing article — and verified in every scattering experiment ever done. If a calculation appears to violate momentum conservation, the modern posture is the correct one: the calculation has mislaid a column, not nature.