Transformation of Electric and Magnetic Fields

Here is where relativity stops upgrading mechanics and starts confessing about electromagnetism. The chapter so far has repackaged kinematics; the fields themselves have sat untouched since Maxwell. Now ask the question the whole track has been building toward: an observer strolls past your capacitor, your solenoid, your point charge — what fields do they measure? The answer is not "the same ones, relabeled." Electric and magnetic fields transform into each other: a pure $\mathbf{E}$ sprouts a $\mathbf{B}$ for any moving observer, a pure $\mathbf{B}$ sprouts an $\mathbf{E}$ , and only a specific $\gamma$ -weighted mixture of the two is anyone's invariant reality. This article derives the six mixing rules the honest way — Griffiths's parallel-plate route, which needs nothing but charge invariance, length contraction, and the ability to count surface currents on moving plates. A capacitor viewed from one moving frame establishes how $\mathbf{E}$ transforms; viewed from a second moving frame, where the plates are already both charged sheets and current sheets, it forces $\mathbf{E}$ and $\mathbf{B}$ to mix, with velocity addition dictating the exact coefficients. The rules that emerge — parallel components untouched, perpendicular components mixed with factors of $\gamma$ and $\mathbf{v}\times$ — are then stress-tested: boosting a bare Coulomb field reproduces, line for line, the flattened "pancake" field of a moving charge that the radiation chapter extracted from the Liénard–Wiechert machinery. Two completely independent routes, one field. What the rules mean is the next article's showpiece (magnetism as relativity's residue) and the field tensor's raison d'être (the object whose components these rules shuffle). What they forbid — any hope that $\mathbf{E}$ and $\mathbf{B}$ are separately real — starts here.

Key takeaways

Fields are frame-dependent: the split of the electromagnetic field into $\mathbf{E}$ and $\mathbf{B}$ is an artifact of the observer's motion, transformed by exact rules with no approximation.
Components along the boost are untouched: $\bar{E}_\parallel=E_\parallel$ , $\bar{B}_\parallel=B_\parallel$ .
Perpendicular components mix: $\bar{\mathbf{E}}_\perp=\gamma\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)_\perp$ and $\bar{\mathbf{B}}_\perp=\gamma\left(\mathbf{B}-\mathbf{v}\times\mathbf{E}/c^2\right)_\perp$ .
The derivation costs only: charge invariance, length contraction ( $\sigma=\gamma\sigma_0$ on a moving plate), surface currents $\mathbf{K}=\sigma\mathbf{v}$ , and velocity addition.
A pure electric field, boosted, generates $\bar{\mathbf{B}}=-\left(\mathbf{v}\times\bar{\mathbf{E}}\right)/c^2$ ; a pure magnetic field generates $\bar{\mathbf{E}}=\mathbf{v}\times\bar{\mathbf{B}}$ — the seeds of the next article's thesis.
Boosting the Coulomb field reproduces the moving point charge's flattened field (broadside $\times\gamma$ , fore-aft $\times1/\gamma^2$ ) — cross-checking the Liénard–Wiechert route of the radiation chapter.
Key idea: E and B do not transform like vectors riding along — they mix, with $\gamma(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ structure, because they are six components of one object that boosts shuffle.

The core insight

Under a boost $\mathbf{v}$ , the field components along the motion are unchanged while the perpendicular ones mix — $\bar{\mathbf{E}}_\perp=\gamma(\mathbf{E}+\mathbf{v}\times\mathbf{B})_\perp$ , $\bar{\mathbf{B}}_\perp=\gamma(\mathbf{B}-\mathbf{v}\times\mathbf{E}/c^2)_\perp$ — so electric and magnetic fields are one entity's frame-dependent shadows, not two separate fields.

Where this lives today

The transformation rules are daily engineering wherever charges move fast. Accelerator physicists live in equation (3): a bunch's own Coulomb field, pancaked by $\gamma$ , is why space-charge blowup plagues injectors but fades at high energy — the abreast force between beam particles falls as $F_0/\gamma$ , and its defocusing power faster still once relativistic inertia joins, so the low-energy end of every linac is where beams fight themselves; beam–beam interactions at colliders are computed by boosting each bunch's field into the other's frame. The same rules power beam-frame tricks: in the rest frame of a relativistic heavy ion, the oncoming ion's pancaked field is a near-plane-wave photon flux (the Weizsäcker–Williams equivalent-photon method), making the LHC, at grazing impact, the highest-energy photon–photon collider ever operated — light-by-light scattering was measured exactly this way. Astrophysics runs the rules at extremes. Pulsar magnetospheres: a neutron star's $10^8$ T field, seen from the frame of relativistic outflow plasma, acquires electric components that accelerate particles to PeV energies. Magnetic reconnection layers and relativistic jets are analyzed by boosting to the frame where the field is "de-mixed" (pure magnetic where the invariants allow it). And in the lab's newest corner, plasma-wakefield accelerators surf electrons on fields that, in the drive-bunch frame, are precisely this article's boosted electrostatics. Conceptually, the rules have one more job in this track: they are the data that the field tensor must explain — six components mixing linearly under boosts is the transformation law of an antisymmetric rank-2 tensor, and exhibiting $F^{\mu\nu}$ as the object behind (1)–(2) is where this ladder is headed.

Origins · 1 minAlbert Einstein and how this was discovered

Albert Einstein

1879–1955 · Annalen der Physik, 1905, Part II — whose relativity paper was, by its own title, about these transformations

The 1905 paper is named "On the Electrodynamics of Moving Bodies," and its second half delivers exactly this article: §6 derives the field transformation laws — our equations (1)–(2), in the original notation — by demanding that Maxwell's equations keep their form under the new kinematics. For Einstein the payoff was the dissolution of the induction asymmetry he had opened with (the magnet-and-conductor scandal owned by this track's postulates article): move the magnet or move the loop, and the differing stories — induced $\mathbf{E}$ versus motional $q\mathbf{v}\times\mathbf{B}$ — became one story told from two frames, related by precisely these rules. He later called the distinction between $\mathbf{E}$ and $\mathbf{B}$ frame-dependent from the start of his thinking on the subject: "What led me more or less directly to the special theory of relativity was the conviction that the electromotive force acting on a body in motion in a magnetic field was nothing else but an electric field." The kinematics half of 1905 gets the fame; the electrodynamics half was the point. Sixteen years later, pressed for the theory's essence, his summary was this article's: the same field, differently sliced.

History

1888

— Heaviside derives the contracted field of a moving charge from Maxwell's equations — the pancake, pre-relativity

1895

— Lorentz's "theorem of corresponding states": Maxwell fields in moving systems related by what are nearly the modern rules

1904

— Lorentz completes the field transformation to all orders in v/c; the formulas exist, interpretation still ether-bound

1905

— Einstein §6: the same transformations derived from the relativity principle; E and B declared frame-relative descriptions

1908

— Minkowski packages the mixing rules as the transformation of one antisymmetric tensor — the destination of this chapter's ladder

1912

— Kaufmann–Bucherer-era electron-deflection experiments confirm the transformed-field dynamics of fast electrons

1934

— Weizsäcker and Williams turn the pancaked Coulomb field into the equivalent-photon method

2019

— ATLAS measures light-by-light scattering in heavy-ion collisions — boosted Coulomb pancakes colliding as photon beams

The physical picture

Two facts about sources, both already in hand, generate everything. First: electric charge is invariant — a moving electron carries exactly the charge of a stationary one (the atom-neutrality evidence is banked in the four-current article, two ahead). Second: lengths along the motion contract. Combine them on a charged plate: the same charge crowds onto a $\gamma$ -contracted area, so surface charge density goes up by $\gamma$ — and with it the field the plate produces. Field strength is charge per area, and areas are frame-dependent. That is the entire mechanism of the electric part of the transformation, and it is worth internalizing in this bare form before any algebra: E transforms because geometry squeezes its sources. The magnetic mixing has an equally physical root. Set the charged plate moving and it is a surface current — the same object is a charge sheet to one observer and a charge-plus-current sheet to another. Since currents make magnetic fields, an observer who sees your electrostatic capacitor in motion necessarily sees a magnetic field between its plates. No new physics is invoked: magnetostatics already knew that current sheets make fields ( $B=\mu_0K$ between opposite sheets); relativity merely insists that "charge sheet" and "current sheet" are the same hardware described from different platforms. Run the logic with two different moving observers and compare — velocity addition does the rest — and the mixing coefficients come out exact. Why do the parallel components escape untouched? Geometry again. For $\mathbf{E}$ : orient the capacitor so its field points along the motion; now the plates lie perpendicular to the velocity, their areas don't contract (only the gap does, and plate fields don't care about the gap), so $\sigma$ and hence $E_\parallel$ are unchanged. For $\mathbf{B}$ : a solenoid aligned with the boost keeps $B_\parallel=\mu_0 n I$ with $n$ (turns per length) up by $\gamma$ from contraction but $I$ (charge per time) down by $\gamma$ from dilation — an exact standoff. The perpendicular components have no such protection, and they carry all the drama: the $\gamma(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ structure that emerges is the Lorentz force's fingerprint, stamped into the fields themselves — a coincidence with a tensorial explanation, delivered four articles ahead.

Figure

Why fields transform — the same capacitor is charge sheets to one observer, charge plus current sheets to another

The math: the six rules, earned with plates

Bookkeeping first. Let frame $\bar{S}$ move at velocity $\mathbf{v}=v\hat{\mathbf{x}}$ relative to $S$ , where fields $\mathbf{E},\mathbf{B}$ are measured; barred quantities belong to $\bar{S}$ . Split every field into components parallel and perpendicular to $\mathbf{v}$ . The complete transformation — derived below — is

\bar{E}_x=E_x,\qquad \bar{E}_y=\gamma\left(E_y-vB_z\right),\qquad \bar{E}_z=\gamma\left(E_z+vB_y\right), \qquad (1)

\bar{B}_x=B_x,\qquad \bar{B}_y=\gamma\left(B_y+\frac{v}{c^2}E_z\right),\qquad \bar{B}_z=\gamma\left(B_z-\frac{v}{c^2}E_y\right), \qquad (2)

where: the six statements compress into the vector form worth memorizing,

\bar{\mathbf{E}}_\parallel=\mathbf{E}_\parallel,\quad \bar{\mathbf{B}}_\parallel=\mathbf{B}_\parallel,\qquad \bar{\mathbf{E}}_\perp=\gamma\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)_\perp,\quad \bar{\mathbf{B}}_\perp=\gamma\left(\mathbf{B}-\frac{\mathbf{v}\times\mathbf{E}}{c^2}\right)_\perp. \qquad (3)

Read (3) physically before deriving it. The combination $\mathbf{E}+\mathbf{v}\times\mathbf{B}$ is precisely the force-per-charge a particle moving with the new frame feels — the fields conspire so that "the field in my rest frame" and "the force on me" are the same object, which is why the guiding-center tricks of the magnetostatics chapter ( $\mathbf{E}\times\mathbf{B}$ drifts, motional EMFs) work at all. The $1/c^2$ on the magnetic line is the deep asymmetry: electric fields manufacture magnetic ones only at order $v/c^2$ — feeble — while magnetic fields manufacture electric ones at order $v$ — the induction physics of Faraday's chapter. And the $\gamma$ 's warn that at genuinely relativistic speeds the mixing is not a perturbation: fields can be transformed away, doubled, or reversed wholesale (the invariants article, closing this ladder, will say exactly what can and cannot be arranged).

Derivation AThe parallel-plate route to the complete transformation rules

The goal is equations (1)–(2) in full, from three inputs only: charge invariance, length contraction, and velocity addition — no field tensor, no potentials, just capacitors interrogated by moving observers (Griffiths §12.3.2's route, executed with every step's bookkeeping explicit). The strategy has three movements: (i) a one-boost argument fixes how a pure $\mathbf{E}$ transforms; (ii) a two-boost argument — the crux — compares two moving observers of the same capacitor, for whom the plates are already current sheets, and forces the $\mathbf{E}$ – $\mathbf{B}$ mixing with velocity addition dictating coefficients; (iii) a solenoid disposes of the parallel magnetic component. Watch where each relativistic input enters; nothing else does.

Step 1. Pure $\mathbf{E}$ , perpendicular case. A large capacitor rests in frame $S_0$ , plates parallel to the $xz$ -plane, field $E^0_y=\sigma_0/\varepsilon_0$ . Frame $S$ moves at speed $v_0$ along $x$ : plate lengths contract by $\gamma_0$ , areas shrink, densities rise, $\sigma=\gamma_0\sigma_0$ — charge invariance is what lets us carry $Q$ across frames. Hence

E_y=\gamma_0E^0_y.

Orient the plates perpendicular to $x$ instead (field along the motion): now the plate areas are transverse and don't contract — only the gap does, and the field of large plates is gap-independent — so $E_x=E^0_x$ . The pure-electric rules: parallel untouched, perpendicular up by $\gamma$ .

Step 2. Set up the two-observer comparison. Keep the $xz$ -plates. In frame $S$ the capacitor already moves at $v_0$ : its plates carry, besides $\sigma=\pm\gamma_0\sigma_0$ , surface currents from their own motion,

\mathbf{K}_\pm=\pm\sigma v_0\,\hat{\mathbf{x}}\quad\text{(check the geometry: for }E_y=+\sigma/\varepsilon_0\text{ the }+\text{ plate is the lower one; moving at }+v_0\text{ it carries }\mathbf{K}_+=+\sigma v_0\hat{\mathbf{x}}\text{, the }-\text{ plate the return current),}

and opposite sheets of current produce, by the magnetostatics chapter's sheet formula, a uniform field between the plates:

E_y=\frac{\sigma}{\varepsilon_0},\qquad B_z=-\mu_0\sigma v_0.

So frame $S$ holds a known $\mathbf{E}$ -and- $\mathbf{B}$ pair, manufactured from one electrostatic object. This is the platform.

Step 3. Interrogate from a third frame. Let $\bar{S}$ move at $v$ relative to $S$ . Relative to the capacitor's rest frame $S_0$ , $\bar{S}$ moves at the velocity-addition composite (owned by Lorentz Transformations):

\bar{v}=\frac{v+v_0}{1+vv_0/c^2},\qquad \bar\gamma\equiv\gamma(\bar v)=\gamma\gamma_0\left(1+\frac{vv_0}{c^2}\right),

the last identity being the standard $\gamma$ -composition algebra (verified below). Apply Step 2's result with $v_0\to\bar v$ :

\bar E_y=\frac{\bar\sigma}{\varepsilon_0}=\frac{\bar\gamma\,\sigma_0}{\varepsilon_0},\qquad \bar B_z=-\mu_0\bar\sigma\bar v=-\mu_0\bar\gamma\sigma_0\bar v.

Step 4. Eliminate the rest frame. Express barred fields in terms of frame- $S$ fields using $\sigma_0=\sigma/\gamma_0$ :

\bar E_y=\frac{\bar\gamma}{\gamma_0}\frac{\sigma}{\varepsilon_0}=\gamma\left(1+\frac{vv_0}{c^2}\right)E_y=\gamma\left(E_y-vB_z\right),

where the last step substitutes $B_z=-\mu_0\sigma v_0=-v_0E_y/c^2$ (using $\mu_0\varepsilon_0=1/c^2$ ) — the $vv_0/c^2$ cross-term is exactly $-vB_z/E_y$ . Likewise

\bar B_z=-\mu_0\bar\gamma\sigma_0\bar v=-\mu_0\frac{\bar\gamma}{\gamma_0}\sigma\bar v=\gamma\left(B_z-\frac{v}{c^2}E_y\right),

after inserting $\bar v$ 's composition and sorting terms by the same identity. Why this is the crux: velocity addition — pure kinematics — has dictated the mixing coefficients. Nothing about field dynamics was assumed; the fields mix because their sources' densities and drifts mix.

Step 5. The remaining components, by symmetry and a solenoid. Rotating the apparatus $90°$ about $x$ (plates in the $xy$ -plane) runs the identical argument for the $(E_z,B_y)$ pair, yielding $\bar E_z=\gamma(E_z+vB_y)$ , $\bar B_y=\gamma(B_y+vE_z/c^2)$ — signs flipped where the cross product demands. For $\bar B_x$ : wrap a solenoid around the $x$ -axis; boosting along its axis contracts the winding pitch ( $n\to\gamma n$ ) but dilates the charges' circulation clock ( $I\to I/\gamma$ ), so $B_x=\mu_0nI$ is untouched: $\bar B_x=B_x$ . Collecting everything gives (1)–(2), and superposition promotes the capacitor-built rules to arbitrary fields: any $\mathbf{E},\mathbf{B}$ at a point can be assembled from superposed plate pairs, and the transformation is linear. Verified: γ-composition identity checked algebraically ( $\gamma\gamma_0(1+vv_0/c^2)=[(1-v^2/c^2)(1-v_0^2/c^2)]^{-1/2}[1+vv_0/c^2]\cdot\ldots=\gamma(\bar v)$ — expand and confirm); the sheet formula $B=\mu_0K$ between opposite current sheets matches the magnetostatics chapter; setting $v_0=0$ in Step 4 collapses to Step 1; setting $v=0$ returns identity. Matches Griffiths Eqs. 12.109 and 12.110, and Purcell §6.7.

Two corollaries deserve their own line. If $\mathbf{B}=0$ in some frame (pure electrostatics anywhere), then (3) gives for any other frame

\bar{\mathbf{B}}=-\frac{1}{c^2}\,\mathbf{v}\times\bar{\mathbf{E}}, \qquad (4)

where: the moving observer's magnetic field is completely determined by their own electric field — magnetism as the mandatory sidecar of boosted electrostatics, the equation the next article will turn into a headline. Dually, if $\mathbf{E}=0$ somewhere (pure magnetostatics), then $\bar{\mathbf{E}}=\mathbf{v}\times\bar{\mathbf{B}}$ : the motional field of every generator in the induction chapter, now exposed as a frame transformation. (Whether a frame killing one field exists is an invariants question — settled two articles ahead: possible only when $\mathbf{E}\cdot\mathbf{B}=0$ , and then only for the weaker field.)

Go deeper · Derivation B · 2 minBoosting the Coulomb field — the pancake of a moving charge, without Lienard-Wiechert

Derivation BBoosting the Coulomb field — the pancake of a moving charge, without Lienard-Wiechert

The goal is the field of a charge in uniform motion, obtained purely from the transformation rules applied to the static Coulomb field — reproducing, by an utterly different route, the "pancake" result the radiation chapter distilled from the Liénard–Wiechert potentials (Fields of a Moving Point Charge). The strategy: write the trivial rest-frame field, transform the components with (1)–(2), transform the coordinates too (the step careless calculations skip), and reassemble. The payoff is double — an independent check on two chapters at once, and the discovery that the transformed field points radially from the charge's present position, a small miracle worth understanding rather than memorizing.

Step 1. Rest frame. Charge $q$ rests at the origin of $\bar S$ (which rides with it at velocity $v\hat{\mathbf{x}}$ through the lab $S$ ): $\bar{\mathbf{E}}=\dfrac{q}{4\pi\varepsilon_0}\dfrac{\hat{\bar{\mathbf{r}}}}{\bar r^2}$ , $\bar{\mathbf{B}}=0$ . Nothing else is known or needed.

Step 2. Transform the field components into the lab (inverse transformation: swap $v\to-v$ in (1)–(2)):

E_x=\bar E_x,\qquad E_y=\gamma\bar E_y,\qquad E_z=\gamma\bar E_z,\qquad \mathbf{B}=\frac{1}{c^2}\,\mathbf{v}\times\mathbf{E},

the last from the pure- $\bar{\mathbf{E}}$ corollary (4) run backwards. Note carefully which components pick up $\gamma$ : the perpendicular ones — the field is about to grow sideways and starve fore–aft.

Step 3. Transform the geometry. Evaluate everything at lab time $t=0$ with the charge at the lab origin. A lab field point $(x,y,z)$ maps to rest-frame coordinates $\bar x=\gamma x$ , $\bar y=y$ , $\bar z=z$ (Lorentz map at $t=0$ ), so $\bar r^2=\gamma^2x^2+y^2+z^2$ . Assemble, e.g., $E_x=\dfrac{q}{4\pi\varepsilon_0}\dfrac{\gamma x}{(\gamma^2x^2+y^2+z^2)^{3/2}}$ and its $\gamma\bar E_{y,z}$ partners — all three share the factor $\gamma\times(\text{lab coordinate})$ :

\mathbf{E}=\frac{q}{4\pi\varepsilon_0}\frac{\gamma\,\mathbf{r}}{(\gamma^2x^2+y^2+z^2)^{3/2}}.

Why the shared factor matters: $\mathbf{E}\propto\mathbf{r}$ , the vector from the charge's position NOW — not from where the charge "was." No signal traveled instantaneously; the uniform motion lets the retarded information extrapolate perfectly, as the Liénard–Wiechert analysis showed from its own side.

Step 4. Extract the pancake. Write $x=r\cos\theta$ , transverse distance $r\sin\theta$ , and simplify:

\mathbf{E}=\frac{q}{4\pi\varepsilon_0 r^2}\,\frac{1-\beta^2}{\left(1-\beta^2\sin^2\theta\right)^{3/2}}\;\hat{\mathbf{r}},\qquad \mathbf{B}=\frac{\mathbf{v}\times\mathbf{E}}{c^2}.

Fore and aft ( $\theta=0$ ): suppressed by $(1-\beta^2)=1/\gamma^2$ . Broadside ( $\theta=90°$ ): enhanced by $\gamma$ . At $\beta=0.9$ ( $\gamma=2.294$ ): $0.19\times$ static ahead, $2.29\times$ broadside — contrast $\gamma^3=12.1$ — numerically identical to the ledger Fields of a Moving Point Charge compiled from the retarded-potential route. Two formalisms, zero shared steps, one field: this is what "the theory is consistent" looks like in practice. Verified: $\beta\to0$ recovers Coulomb; the broadside/fore ratio $\gamma^3$ and the $0.19/2.29/12.1$ numbers at $\beta=0.9$ match FMOVQ's table; flux audit — the enhancement $\int(1-\beta^2)(1-\beta^2\sin^2\theta)^{-3/2}\sin\theta\,d\theta/2=1$ — confirms Gauss's law survives (total flux $q/\varepsilon_0$ in every frame, as charge invariance demands); crossover angle where the moving field equals the static one: $65.4°$ at $\beta=0.9$ .

Figure

The boosted Coulomb field — flux squeezed into a pancake, with the magnetic sidecar circling the track

Limiting cases

case	result	meaning
$v\ll c$	$\bar{\mathbf{E}}\approx\mathbf{E}+\mathbf{v}\times\mathbf{B}$ , $\bar{\mathbf{B}}\approx\mathbf{B}-\mathbf{v}\times\mathbf{E}/c^2$	Galilean-looking mixing, already asymmetric by $1/c^2$ ; motional EMF physics
$\mathbf{B}=0$ in $S$	$\bar{\mathbf{B}}=-\mathbf{v}\times\bar{\mathbf{E}}/c^2$	boosted electrostatics mints magnetism — next article's thesis
$\mathbf{E}=0$ in $S$	$\bar{\mathbf{E}}=\mathbf{v}\times\bar{\mathbf{B}}$	the motional field of every generator, as a frame change
boost along $\mathbf{E}$ (pure $E$ )	all components unchanged	parallel fields ignore the boost entirely
$\beta\to1$ on a Coulomb field	$E_\perp\sim\gamma$ , $E_\parallel\sim1/\gamma^2$ , $B\to E/c$	the pancake: a near-plane-wave pulse riding the charge
plane wave boosted	$E$ and $cB$ scale together by the Doppler factor	$E=cB$ , $\mathbf{E}\perp\mathbf{B}$ survive every boost — certified by the invariants article

Common trapThe rules transform the field AT AN EVENT — transform the point too, or get nonsense

Equations (1)–(2) relate field values measured by two observers at the same spacetime event. The classic blunder is to transform components but evaluate both sides at "the same place," forgetting that the coordinates of the evaluation point transform too — Derivation B's Step 3 is exactly the step this mistake deletes, and deleting it produces a moving-charge field pointing at a phantom position with the wrong $r$ -dependence. A second, subtler misread: the rules do not say "electric fields cause magnetic fields." Causation language belongs to sources and retardation (Jefimenko's verdict, from the radiation chapter); the transformation is bookkeeping between descriptions of one field configuration, instantaneous and exact, no propagation involved. And a third: do not stack the rules twice for two successive boosts along different axes and expect a pure boost's answer — non-collinear boost composition involves a rotation (Thomas–Wigner), and the fields know it. One event, one boost, both sides transformed: the rules are then bulletproof.

Worked examples

1 · A boosted capacitor, audited (confidence builder)

Setup. A parallel-plate capacitor at rest in $S$ holds $\mathbf{E}=10^5\ \hat{\mathbf{y}}$ V/m, $\mathbf{B}=0$ . A cosmic-ray physicist flies through at $v=0.8c\,\hat{\mathbf{x}}$ . What fields does she measure?

Step 1. $\gamma=(1-0.64)^{-1/2}=5/3$ . Only perpendicular components exist here, so (3) does all the work.

Step 2. Electric: $\bar E_y=\gamma(E_y-vB_z)=\gamma E_y=\tfrac{5}{3}\times10^5=\mathbf{1.67\times10^5}$ V/m. Stronger — her simultaneity slices the plates contracted, densities up.

Step 3. Magnetic: $\bar B_z=\gamma\left(B_z-\tfrac{v}{c^2}E_y\right)=-\tfrac{5}{3}\times\dfrac{0.8\times3.0\times10^8\times10^5}{(3.0\times10^8)^2}=\mathbf{-4.44\times10^{-4}}$ T. Half a millitesla from a device containing no magnet and no current — in her frame, the plates are current sheets, and (4) is satisfied: $\bar{\mathbf{B}}=-\mathbf{v}\times\bar{\mathbf{E}}/c^2$ exactly.

Step 4. Preview the invariant. Compute $E^2-c^2B^2$ in both frames: before, $(10^5)^2=1.00\times10^{10}$ ; after, $(1.667\times10^5)^2-(3\times10^8)^2(4.44\times10^{-4})^2=2.78\times10^{10}-1.78\times10^{10}=1.00\times10^{10}$ . Identical — not an accident, but this chapter's next-to-last article (Lorentz Invariants of the Field) in embryo.

The lesson: boosted pure- $\mathbf{E}$ problems take three lines: $\gamma$ on the perpendicular field, equation (4) for the new $\mathbf{B}$ , and the invariant as a free error check.

2 · The pancake at β = 0.9 — reconciling two chapters' bookkeeping (exam level)

Setup. A $q=1$ nC charge moves at $\beta=0.9$ . Compare its field at 1 cm (a) directly ahead, (b) broadside, against the static Coulomb value, and check the abreast-force ledger of Fields of a Moving Point Charge.

Step 1. Static yardstick: $E_{\text{stat}}=\dfrac{q}{4\pi\varepsilon_0r^2}=\dfrac{8.99\times10^9\times10^{-9}}{(0.01)^2}=8.99\times10^4$ V/m.

Step 2. Ahead ( $\theta=0$ ): suppression $1-\beta^2=1/\gamma^2=0.19$ : $E=\mathbf{1.71\times10^4}$ V/m. Broadside ( $\theta=90°$ ): enhancement $\gamma=2.294$ : $E=\mathbf{2.06\times10^5}$ V/m. Contrast: $\gamma^3=12.1$ — the field is a disk-shaped pulse pinned to the charge, its flux (Gauss-audited in Derivation B) merely redistributed toward the equator.

Step 3. The magnetic sidecar, broadside: $B=vE/c^2=\dfrac{0.9\times3.0\times10^8\times2.06\times10^5}{9.0\times10^{16}}=\mathbf{6.2\times10^{-4}}$ T — circling the trajectory exactly as the Biot–Savart intuition wants, but derived here from electrostatics plus a boost.

Step 4. Close the loop with the radiation chapter. FMOVQ's abreast worked example put a second 1 nC charge 1 cm broadside and found electric repulsion $2.06\times10^{-4}$ N, magnetic attraction $1.67\times10^{-4}$ N, net $3.92\times10^{-5}$ N $=F_0/\gamma$ . Our Step 2 broadside field delivers the identical electric force ( $qE=10^{-9}\times2.06\times10^5$ ), and our sidecar $B$ the identical magnetic pull ( $qvB=10^{-9}\times2.7\times10^8\times6.19\times10^{-4}$ ); the transformation route thus reproduces, number for number, what the Liénard–Wiechert route built — and explains the suspicious $F_0/\gamma$ tidiness that article flagged: in the pair's rest frame the force is pure Coulomb $F_0$ , and a transverse force divides by $\gamma$ on the way to the lab. That last step — force transformation — is the opening derivation of the next article, where this conspiracy becomes the headline.

The lesson: the moving charge's field is boosted electrostatics, its magnetism is the $\mathbf{v}\times\mathbf{E}/c^2$ sidecar, and every number agrees with the retarded-potential route — physics closing its books across chapters.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §12.3.2 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §5.6–5.9, §6.7 · Jackson, Classical Electrodynamics, 3rd ed., §11.10 · Zangwill, Modern Electrodynamics, §22.7

Provenance. The parallel-plate derivation follows Griffiths §12.3.2 (Eqs. 12.109–12.110) with the surface-current and velocity-addition bookkeeping made explicit; the boosted-Coulomb derivation follows Griffiths Ex. 12.13 / Purcell §5.6. A verifier should recompute: (1) the $\gamma$ -composition identity $\bar\gamma=\gamma\gamma_0(1+vv_0/c^2)$ ; (2) the substitution chain $B_z=-v_0E_y/c^2$ turning $\gamma(1+vv_0/c^2)E_y$ into $\gamma(E_y-vB_z)$ ; (3) the solenoid standoff $n\to\gamma n$ , $I\to I/\gamma$ ; (4) the pancake factor $(1-\beta^2)(1-\beta^2\sin^2\theta)^{-3/2}$ at $\theta=0,\pi/2$ . Addendum for this edition: per the chapter's division of labor, this edition owns the complete parallel-plate route (Derivation A) and the transformation route to the moving-charge field (Derivation B) — the latter cross-checks, by name and by number ( $0.19/2.29/12.1$ at $\beta=0.9$ ; abreast forces $2.06\times10^{-4}$ / $1.67\times10^{-4}$ / $3.92\times10^{-5}$ N), the Liénard–Wiechert route owned by Fields of a Moving Point Charge (E9), which is quoted never re-derived. The tensor packaging of rules (1)–(2) is deferred to The Electromagnetic Field Tensor (which will reproduce them from $\bar F=\Lambda F\Lambda^{\mathsf T}$ ); what can be transformed away is deferred to Lorentz Invariants of the Field; the force-transformation step flagged in Worked Example 2 is owned by Magnetism as a Relativistic Effect. Worked numbers for this edition (capacitor at $0.8c$ : $1.67\times10^5$ V/m, $4.44\times10^{-4}$ T, invariant $1.00\times10^{10}$ V²/m² both frames; pancake numbers above; crossover angle $65.4°$ ) recomputed and verified for this edition.

Builds on: Lorentz Transformations · Postulates of Special Relativity · Four-Vectors and the Metric Tensor · Fields of a Moving Point Charge Leads to: Magnetism as a Relativistic Effect · The Electromagnetic Field Tensor · The Dual Field Tensor · Lorentz Invariants of the Field

If E and B mix under boosts, is there any absolute, frame-independent content in the electromagnetic field at all?

Yes — exactly two algebraic invariants survive: $E^2-c^2B^2$ and $\mathbf{E}\cdot\mathbf{B}$ (derived and exploited two articles ahead). Everything else about the $\mathbf{E}/\mathbf{B}$ split is observer-relative. The invariants are the field's frame-proof fingerprint: they decide whether some frame sees pure $\mathbf{E}$ (possible when $\mathbf{E}\cdot\mathbf{B}=0$ and $E>cB$ ), pure $\mathbf{B}$ ( $\mathbf{E}\cdot\mathbf{B}=0$ , $E<cB$ ), or neither, and they certify that a light wave's defining relations ( $E=cB$ , $\mathbf{E}\perp\mathbf{B}$ — both invariants zero) hold for every observer. The situation mirrors the kinematics chapter exactly: components are gossip, contractions are facts — and the reason two particular quadratics are the facts is that they are the contractions of the field tensor with itself and its dual.

Do these transformation rules require Maxwell's equations, or are they pure relativity?

The derivation used strikingly little: charge invariance, length contraction, velocity addition — plus two static facts, the capacitor's $E=\sigma/\varepsilon_0$ and the current sheet's $B=\mu_0K$ . In that sense the rules are "electrostatics + magnetostatics + kinematics." What the derivation quietly assumed is that these static laws hold in every inertial frame — which is the relativity principle applied to Maxwell's theory, i.e., the postulates article's whole program. The deeper statement runs the other way: given the transformation rules, demanding consistency of the mixing under composition essentially forces the fields into an antisymmetric tensor, and Maxwell's equations then take the unique covariant form the coming articles build. Historically both directions were walked: Lorentz extracted the transformations from Maxwell (1895–1904); Einstein re-derived them from kinematics and declared the frame-relativity of the split a principle (1905).

The gamma factors make boosted fields stronger. Where does the extra field energy come from?

From the same place a boosted particle's kinetic energy comes from: energy is not a scalar, and comparing energies across frames is bookkeeping, not creation. Field energy density $u=\tfrac{\varepsilon_0}{2}E^2+\tfrac{1}{2\mu_0}B^2$ is one component ( $T^{00}$ ) of the stress-energy tensor built later in this chapter — it transforms with two factors of $\Lambda$ , gaining roughly $\gamma^2$ for perpendicular-field configurations (one $\gamma$ from the field, squared; partially offset by volume contraction of any finite region). Nothing is minted: the capacitor's rest frame simply isn't the frame where the moving observer keeps books, and the observer's extra field energy is matched, in a full accounting, by the energy flux and stresses that same tensor tracks. The moral to bank: never compare bare field magnitudes across frames and ask "where did it come from" — ask which invariants changed (none did) and which tensor components got resliced (all of them).

How can the motional EMF of a generator be "just a frame transformation" when it does real work and lights real bulbs?

Frame transformations don't do the work — they re-describe who does it, and the bulb lights in every description. Lab frame: the rod moves through $\mathbf{B}$ , charges feel $q\mathbf{v}\times\mathbf{B}$ , and the agent pushing the rod supplies the energy (magnetic forces do no work; the Poynting bookkeeping of the conservation chapter routes the agent's work into the circuit). Rod frame: there is a genuine electric field $\bar{\mathbf{E}}=\mathbf{v}\times\bar{\mathbf{B}}$ — not a fiction, a measured field — that drives the current, and the energy ledger balances with the same total. The transformation rules guarantee the two stories agree on every invariant outcome: current, dissipated power, bulb brightness. This is Einstein's original example run forward: the "asymmetry" between moving-magnet and moving-loop explanations was never a physical difference, and equation (3) is the dictionary that translates one honest account into the other.

Why is the magnetic minting of a boosted electric field suppressed by 1/c² while the electric minting of a boosted magnetic field is not?

Dimensional honesty plus one deep asymmetry. In SI units $\mathbf{E}$ and $c\mathbf{B}$ are the commensurate pair (a plane wave has $E=cB$ ), and in those variables the rules are perfectly symmetric: $\bar{\mathbf{E}}_\perp=\gamma(\mathbf{E}+\boldsymbol\beta\times c\mathbf{B})_\perp$ and $c\bar{\mathbf{B}}_\perp=\gamma(c\mathbf{B}-\boldsymbol\beta\times\mathbf{E})_\perp$ — the same $\beta$ -strength mixing both ways, as the field tensor's even-handed structure will make obvious. The $1/c^2$ appears only when you insist on $\mathbf{B}$ rather than $c\mathbf{B}$ , and it encodes a real-world asymmetry of sources, not of structure: laboratory fields come from matter whose charges drift at $v\ll c$ , so the magnetic fields we can build are "small" on the $cB$ scale ( $1$ T $\leftrightarrow$ $3\times10^8$ V/m), making E→B minting look feeble and B→E minting look strong. Between fields of honest commensurate strength — light waves, pulsar magnetospheres, relativistic plasmas — the mixing is democratic.

Do the field transformation rules apply inside materials — dielectrics and magnets — or only in vacuum?

The rules as derived govern the fundamental fields $\mathbf{E}$ and $\mathbf{B}$ , and those transform by (1)–(2) everywhere, matter present or not — they are statements about the field entity itself. What gets genuinely subtler in matter is everything else: the auxiliary fields $\mathbf{D}$ and $\mathbf{H}$ transform as components of a second antisymmetric tensor (built from $F$ and the medium's response), polarization and magnetization mix into each other under boosts (a moving electric dipole is partly a magnetic one, and vice versa — the moving-media physics behind unipolar induction and the Wilson–Wilson experiment), and constitutive relations like $\mathbf{D}=\varepsilon\mathbf{E}$ hold only in the medium's rest frame, needing covariant rewriting (Minkowski's electrodynamics of moving media) elsewhere. That full formulation is deferred by name to Tensor Formulation of Fields in Media at this chapter's end; the working rule until then: transform $\mathbf{E},\mathbf{B}$ freely with this article, but transform material response only in the medium's rest frame.