Work Done by Electromagnetic Fields on Charges

The conservation-law chapter has its template; now it needs its first transaction. Every energy story in electromagnetism — the resistor warming, the beam accelerating, the battery draining, the antenna broadcasting — funnels through a single scalar field, and this article is its biography: $\mathbf{J}\cdot\mathbf{E}$ , the rate per unit volume at which electromagnetic fields do work on charged matter. It is the term that will sit on the right-hand side of Poynting's theorem three articles from now, the one entry in the field's energy ledger that names a partner. Its anatomy is almost embarrassingly simple, and the simplicity is a theorem: of the full Lorentz force $q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ , only the electric piece can change a particle's speed — the magnetic force is forever sideways, a steering wheel with no engine, as the magnetostatics chapter proved once and for all. Power flows between field and matter through $\mathbf{E}$ alone, and summing over carriers turns $q\mathbf{v}\cdot\mathbf{E}$ into $\mathbf{J}\cdot\mathbf{E}$ . Everything else is bookkeeping of where that power goes: into heat when collisions randomize it (Joule's law, recovered here from the microscopic side), into clean kinetic energy when there are no collisions (every particle accelerator), or out of matter and into the field when chemistry or machinery pushes charge against the field — the negative- $\mathbf{J}\cdot\mathbf{E}$ regions that make batteries and generators sources rather than sinks. Learn to read the sign and fate of $\mathbf{J}\cdot\mathbf{E}$ point by point and circuit diagrams become energy maps: chemical power entering the field inside the battery, flowing silently through space (the shock of the coming Poynting articles), and surrendering to matter in the resistor. This article builds that reading.

Key takeaways

The power the Lorentz force delivers to a charge is $\mathbf{F}\cdot\mathbf{v}=q\,\mathbf{v}\cdot\mathbf{E}$ : the magnetic term dies because $\mathbf{v}\times\mathbf{B}\perp\mathbf{v}$ — magnetic fields steer, never pump (theorem recalled from the Lorentz-force article).
Summed over carriers, fields transfer energy to matter at rate $\mathbf{J}\cdot\mathbf{E}$ per unit volume — the universal field–matter coupling term, and the source term of the chapter's energy ledger.
Its fate splits: with collisions it becomes heat ( $\mathbf{J}\cdot\mathbf{E}=J^2/\sigma$ in Ohmic media — Joule's law); without, it becomes coherent kinetic energy (accelerators); the split is derivable, not assumed.
Its sign flips in sources: inside a battery or generator, current is driven against $\mathbf{E}$ , so $\mathbf{J}\cdot\mathbf{E}<0$ — matter pumps energy into the field.
Total mechanical power in a volume: $\mathrm{d}W/\mathrm{d}t=\int\mathbf{J}\cdot\mathbf{E}\,\mathrm{d}\tau$ ; for a uniform wire this integrates to exactly $I^2R$ .
KEY IDEA: $\mathbf{J}\cdot\mathbf{E}$ is the hinge of electromagnetic energy — the one term through which fields and matter settle every energy transaction, with its sign naming the direction.

The core insight

The rate of work done by the field on charge, per unit volume, is $\mathbf{J}\cdot\mathbf{E}$ ; the magnetic force never contributes because it is always perpendicular to the velocity.

Where this lives today

Power electronics is applied $\mathbf{J}\cdot\mathbf{E}$ management at the cubic-micron scale. A modern CPU dissipates on the order of $100$ W in a square centimetre of silicon — local power densities in the interconnects and channels reach $10^{15}$ W/m³, a billion times our copper wire — and "dark silicon," throttling, and the entire heat-sink industry exist because Derivation B's steady-drift branch converts every switching event's field work into lattice heat. At the other extreme, superconducting magnets and cables run the $\mathbf{J}\cdot\mathbf{E}=0$ row of the limiting-cases table commercially: persistent currents in MRI solenoids circulate for years without payment. In fusion research the two-fates split is a design constraint with a famous ceiling. Tokamaks begin by Ohmic heating — driving mega-amps through the plasma and collecting $J^2/\sigma$ — but plasma resistivity falls as $T^{-3/2}$ (Spitzer), so the method self-extinguishes around a few keV: the hotter the plasma, the less the field's payment can be laundered into heat. Every auxiliary-heating system (neutral beams, RF at cyclotron resonances) is an engineered workaround, choosing collisionless, resonant versions of $\mathbf{J}\cdot\mathbf{E}$ that pay particular particle populations directly. And space physics reads the sign of $\mathbf{J}\cdot\mathbf{E}$ as a diagnostic: NASA's MMS spacecraft measure it point-wise in the magnetosphere to locate where magnetic reconnection converts field energy to particle energy — literally mapping the payroll of a plasma the size of a planet.

The physical picture

Put a charge on an electric field line and it is pushed along its motion: the force has a component parallel to $\mathbf{v}$ , and kinetic energy flows in. Now add a magnetic field. The charge's path curves — sometimes spectacularly — but the magnetic force plays the role of the string in a whirling-ball problem: always perpendicular to the motion, changing direction relentlessly and speed never. That is not an approximation but the geometry of the cross product, and the magnetostatics chapter (The Lorentz Force Law, recalled by name) elevated it to a theorem and then survived its most treacherous test — the "but magnets lift cars!" paradox, resolved there with a crank-arm audit showing the magnetic force redirects work done by something else. This article inherits both the theorem and the paradox's resolution, and spends them at the continuum level. The continuum is where energy bookkeeping actually happens, because real matter carries current as a density. Each species of carrier — electrons in copper, ions and electrons in a plasma — receives power $q\,\mathbf{v}\cdot\mathbf{E}$ per particle, and the species' contributions assemble into one scalar field, $\mathbf{J}\cdot\mathbf{E}$ watts per cubic metre. This is the field theory's payroll: positive where the field pays matter, negative where matter pays the field, zero where they merely coexist. In a flashlight circuit the payroll map is vivid — strongly negative in the chemical interior of the battery (charge pushed uphill against the electrostatic field), nearly zero along the fat copper, intensely positive in the filament — and the map raises the question that will detonate two articles from now: if energy leaves the battery's volume and arrives in the filament's, by what route does it travel? The answer (through the fields, in the space around the wires) belongs to the Poynting articles by name; this article's job is to nail the endpoints of the journey.

Figure

E pumps, B steers — and the circuit's payroll map: where fields pay matter and where matter pays the field

One promissory note, issued now and cashed by the whole rest of the chapter: when $\mathbf{J}\cdot\mathbf{E}$ is written into the field's own ledger, it will enter with a minus sign — $\partial_t u+\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}$ — the transfer clause of the template two articles back, paired exactly with the $+\mathbf{J}\cdot\mathbf{E}$ that credits matter here. This article is, in the template's language, the definition and audit of the chapter's one transfer term.

The mathematics of the hinge

Derivation AFrom the Lorentz force to J dot E - the continuum assembly

We derive the field-to-matter power density for an arbitrary distribution of moving charge, with the honesty that real media carry several species at once. Strategy: start from a single carrier's power (where the magnetic kill is inherited as a theorem), sum over a species, then over species — and watch what the assembly does and does not assume, because the multi-species step is where naive treatments quietly cheat.

Step 1. One carrier of charge $q$ , velocity $\mathbf{v}$ : the delivered power is $\mathbf{F}\cdot\mathbf{v} = q(\mathbf{E}+\mathbf{v}\times\mathbf{B})\cdot\mathbf{v} = q\,\mathbf{v}\cdot\mathbf{E}$ , the magnetic term vanishing by the scalar-triple-product identity $\mathbf{v}\cdot(\mathbf{v}\times\mathbf{B})=0$ — the no-work theorem of The Lorentz Force Law, recalled with its crank-arm paradox resolution rather than re-litigated.

Step 2. A species $s$ with number density $n_s$ , charge $q_s$ , mean velocity $\mathbf{v}_s$ : power density $n_s q_s \langle\mathbf{v}\rangle_s\cdot\mathbf{E} = \mathbf{J}_s\cdot\mathbf{E}$ , where $\mathbf{J}_s=n_sq_s\mathbf{v}_s$ is the species' current density (the carrier-summing assembly owned by Electric Current and Current Density, by name). Why the average suffices: the power is linear in $\mathbf{v}$ , so thermal spread around the mean contributes nothing to the average power — a linearity gift that the energy bookkeeping of Step 4 will not enjoy.

Step 3. Sum species: total power density $\sum_s\mathbf{J}_s\cdot\mathbf{E} = \mathbf{J}\cdot\mathbf{E}$ , since fields act on all charge democratically. Integrated over any volume:

\frac{\mathrm{d}W}{\mathrm{d}t}=\int_V \mathbf{J}\cdot\mathbf{E}\;\mathrm{d}\tau,

the total rate at which electromagnetic fields do work on everything charged inside $V$ . This is the exact, assumption-light statement: no Ohm's law, no steady state, no material model — just Lorentz plus additivity.

Step 4. Now the honesty about whose energy grows. Step 3 credits each species' energy separately, and in media with more than one carrier the split matters: in a Hall-effect geometry or a two-fluid plasma, $\mathbf{J}\cdot\mathbf{E}$ can pay electrons while ions coast, driving the species apart in temperature — fusion devices measure exactly this. The single scalar $\mathbf{J}\cdot\mathbf{E}$ is the total invoice; its division among carriers is transport physics, not field theory.

Step 5. Also note what was not included: the power delivered by whatever non-electromagnetic forces act (chemical gradients in a battery, the lattice forces of a spinning armature, gravity in a falling charged drop). Those appear in matter's ledger as separate income lines; $\mathbf{J}\cdot\mathbf{E}$ is only the electromagnetic line — the one that will reappear, sign flipped, in the field's own books. Quick checks: dimensions $[\mathbf{J}][\mathbf{E}]=(\mathrm{A/m^2})(\mathrm{V/m})=\mathrm{W/m^3}$ ✓; for a single point charge the volume integral with $\mathbf{J}=q\mathbf{v}\,\delta^3(\mathbf{r}-\mathbf{r}_0)$ collapses back to $q\mathbf{v}\cdot\mathbf{E}$ ✓; and the uniform-wire integration in Worked Example 1 recovers $I^2R$ exactly, the classical calibration (Griffiths §8.1.1).

Go deeper · Derivation B · 2 minThe two fates of J dot E - coherent push versus heat

Derivation BThe two fates of J dot E - coherent push versus heat

The invoice $\mathbf{J}\cdot\mathbf{E}$ says nothing about what matter does with the payment. We derive the split — bulk kinetic energy versus thermal energy — for a single carrier species, because the split is the difference between an accelerator and a toaster, and because it shows Joule's law emerging from mechanics rather than being postulated. Strategy: write the species' momentum equation with a collision drag, form the bulk-kinetic-energy rate, and subtract.

Step 1. Carriers of mass $m$ , density $n$ , drift velocity $\mathbf{v}$ , colliding with a background (lattice, neutrals) at rate $1/\tau$ . Momentum balance per carrier (the Drude picture of Ohm's Law and Conductivity, recalled by name):

m\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}=q\mathbf{E}+q\,\mathbf{v}\times\mathbf{B}-\frac{m\mathbf{v}}{\tau}.

Step 2. Dot with $n\mathbf{v}$ to get the bulk (coherent) kinetic-energy density rate; the magnetic term dies as always:

\frac{\mathrm{d}}{\mathrm{d}t}\left(\tfrac12 n m v^2\right)=\mathbf{J}\cdot\mathbf{E}-\frac{n m v^2}{\tau}.

where the last term is the rate collisions bleed coherent motion into disordered motion — heat, delivered to the lattice. The field's full payment $\mathbf{J}\cdot\mathbf{E}$ splits, on the spot, into acceleration and dissipation.

Step 3. Fate one — no collisions ( $\tau\to\infty$ ): every watt becomes bulk kinetic energy. This is the accelerator regime: beams in vacuum, electrons in a cathode-ray gap, the collisionless plasmas of space. $\mathbf{J}\cdot\mathbf{E}$ is then reversible — decelerate the charges and they pay the field back in full (the generator half of the cycle, and eventually the radiation story of E9, by name).

Step 4. Fate two — steady drift ( $\mathrm{d}\mathbf{v}/\mathrm{d}t=0$ ): acceleration by the field exactly balances collisional drag, the left side vanishes, and all of $\mathbf{J}\cdot\mathbf{E}$ flows to heat. Solving the balance gives $\mathbf{J}=(nq^2\tau/m)\mathbf{E}=\sigma\mathbf{E}$ — Ohm's law with its Drude conductivity — and the payment becomes

\mathbf{J}\cdot\mathbf{E}=\sigma E^{2}=\frac{J^{2}}{\sigma},

Joule's law as a derived identity: heating is what the field's work becomes when collisions launder it. The bulk kinetic inventory in this regime is comically small (drift energy of copper's electrons at 10 A/mm² is ~ $10^{-17}$ J/m³) — the field pays continuously, and collisions confiscate the payment within a femtosecond-scale $\tau$ .

Step 5. The general case interpolates: switch on a field and the current ramps over $\sim\tau$ (inductive-like inertia of the carriers), then settles into pure dissipation; switch off and the stored drift energy — those meagre attojoules — is the only refund. This is why "resistors dissipate $J^2/\sigma$ " is instantaneous and honest at power frequencies, and why plasma physicists, whose $\tau$ can be long, must track both fates explicitly. As a check: Step 4 reproduces the Drude conductivity and Joule's law exactly as Ohm's Law and Conductivity derived them (recalled, consistent); dimensional audit of every term gives W/m³; and the collisionless limit's reversibility matches the accelerator energy-gain bookkeeping of Worked Example 2.

Figure

The two fates of the field's payment — banked as coherent motion, or laundered into heat by collisions

Go deeper · Derivation C · 2 minSources - where J dot E goes negative and matter pays the field

Derivation CSources - where J dot E goes negative and matter pays the field

Every circuit needs at least one region where the energy flows the other way. We audit the interior of a source — battery, generator, thermocouple — and show that $\mathbf{J}\cdot\mathbf{E}<0$ there is not an anomaly but the defining transaction, with the EMF as its integrated price. Strategy: use the source model of Electromotive Force (recalled by name): inside a source, a non-electrostatic agency drives charge with an effective force per unit charge $\mathbf{f}_s$ , against the electrostatic field.

Step 1. Inside an open-circuit battery, equilibrium of the carriers demands $\mathbf{f}_s=-\mathbf{E}$ : chemistry pushes exactly as hard as the built-up charge pulls back, and the terminal voltage integrates $\int\mathbf{f}_s\cdot\mathrm{d}\boldsymbol{\ell}=\mathcal{E}$ , the EMF. No current, no energy flow — a loaded spring.

Step 2. Close the circuit: current $\mathbf{J}$ now flows in the direction of $\mathbf{f}_s$ — from $-$ to $+$ inside the source — which is against the interior electrostatic field $\mathbf{E}$ . Hence, pointwise in the source's volume, $\mathbf{J}\cdot\mathbf{E}<0$ : the electromagnetic field is doing negative work on the charge, i.e. the charge (pushed by chemistry) is doing positive work on the field's account. Energy enters the electromagnetic ledger here.

Step 3. Integrate over the source volume, with the ideal-source idealization ( $\mathbf{f}_s=-\mathbf{E}$ maintained under load):

\int_{\rm source}\mathbf{J}\cdot\mathbf{E}\,\mathrm{d}\tau=-\int_{\rm source}\mathbf{J}\cdot\mathbf{f}_s\,\mathrm{d}\tau=-\,\mathcal{E}I,

the field gains $\mathcal{E}I$ watts — precisely the chemical power the cell converts. A real cell with internal resistance $r$ keeps back $I^2r$ of heat: the net export is $\mathcal{E}I-I^2r$ , terminal volts times current, as circuit theory always claimed. Here the claim has an address: the conversion happens in the electrode interfaces, watt by watt, cubic millimetre by cubic millimetre.

Step 4. The same audit runs every source. A generator's moving armature: the "chemistry" is the $\mathbf{v}\times\mathbf{B}$ pump of Motional EMF, and the crank-arm theorem (recalled by name) shows the mechanical crank supplies what the field receives — magnetic force still doing zero net work, merely brokering. A solar cell: photon energy ejects carriers against the junction field. In each, the source region is where some other ledger buys into the electromagnetic one, and $\mathbf{J}\cdot\mathbf{E}<0$ is the receipt. As a check: the integrated $-\mathcal{E}I$ is consistent with standard source bookkeeping; signs check against the discharge direction (current from − to + internally); and the generator case reproduces the exact power balance of the rail flagship in Motional EMF (0.18 W in = 0.18 W out, recalled).

Limiting cases

case	$\mathbf{J}\cdot\mathbf{E}$	reading
electrostatics, no current	$0$	fields and charges coexist, no energy moves
ideal conductor / superconductor	$0$	current flows with $\mathbf{E}=0$ : transport without payment
Ohmic steady state	$J^2/\sigma>0$	all payment laundered to heat — Joule's law
collisionless beam	$\mathrm{d}(\text{KE})/\mathrm{d}t$	all payment banked as kinetic energy, reversible
source interior	$-\mathcal{E}I/V<0$	matter (chemistry, cranks, photons) pays the field
pure magnetic force	$0$ exactly	steering only — the inherited no-work theorem

Common trapMagnetic fields do no work — and yet the motor lifts the load

The most persistent apparent contradiction in electromagnetism, inherited here at the continuum level. $\mathbf{J}\cdot\mathbf{E}$ is the whole electromagnetic payment — nothing like " $\mathbf{J}\cdot(\mathbf{v}\times\mathbf{B})$ " ever appears — yet motors, magnets, and eddy brakes visibly trade energy. The resolution, established in The Lorentz Force Law and Motional EMF and recalled here: in every such machine the magnetic force redirects energy while induced electric fields and internal mechanical stresses deliver it. In a motor, the working energy enters via the back-EMF's $\mathbf{J}\cdot\mathbf{E}$ in the windings; the magnetic field is the broker who closes every deal and takes no fee. Writing a magnetic work term into an energy budget is a sign error against a theorem.

Worked examples

1 · One metre of warm copper, audited twice (confidence builder)

A copper wire of cross-section $A=1.0$ mm² carries $I=10$ A. Find the power the field delivers per metre, via $\mathbf{J}\cdot\mathbf{E}$ and via $I^2R$ , and locate where it goes.

Step 1. Current density $J=I/A=10/10^{-6}=1.0\times10^{7}$ A/m². Field required (Ohm): $E=J/\sigma = 10^{7}/5.96\times10^{7} = \mathbf{0.17\ V/m}$ — the modest interior field that a metre of wire needs 0.17 V to sustain.

Step 2. Payment density: $\mathbf{J}\cdot\mathbf{E} = 10^{7}\times0.17 = \mathbf{1.7\times10^{6}\ W/m^3}$ . Per metre of wire: $\times A = 1.7\times10^{6}\times10^{-6} = \mathbf{1.7\ W/m}$ .

Step 3. Circuit route: $R/L = 1/\sigma A = 1/(5.96\times10^{7}\times10^{-6}) = 0.017$ Ω/m, so $I^2R = 100\times0.017 = \mathbf{1.7\ W/m}$ — identical, as Derivation A's calibration promised. The lumped formula is the volume integral wearing circuit clothes.

Step 4. Fate audit (Derivation B): steady drift, so all 1.7 W/m is heat — the wire runs warm, and at this loading (10 A/mm² is aggressive but legal for short runs) the temperature rise is what limits ampacity tables. The coherent drift energy in that metre is $\tfrac12 nmv^2A\approx10^{-17}$ J — switch off the battery and that is the field's total refund; everything else has already left as lattice heat.

2 · An accelerator pays a beam (exam level)

A linac section sustains an axial field $E=30$ MV/m over $L=1.0$ m, threaded by a beam of average current $I=1.0$ mA. Audit the energy flow both per-volume and per-particle.

Step 1. Global invoice: the beam is a current $I$ through potential drop $EL=30$ MV, so the field pays $P = I\,EL = 10^{-3}\times3.0\times10^{7} = \mathbf{30\ kW}$ into the metre. (As a $\mathbf{J}\cdot\mathbf{E}$ integral: $J=I/A$ over beam area $A$ cancels in $\int \mathbf{J}\cdot\mathbf{E}\,\mathrm{d}\tau = (I/A)(E)(AL) = IEL$ — geometry drops out, as it must.)

Step 2. Fate audit: vacuum, no collisions — Derivation B's collisionless branch. Every joule lands as kinetic energy: each electron gains $\Delta K = eEL = \mathbf{30\ MeV}$ crossing the metre. This is the energy per particle. Cross-check the ledgers: particles per second $I/e=6.2\times10^{15}$ , times $30\ \mathrm{MeV} = 4.8\times10^{-12}$ J each: $6.2\times10^{15}\times4.8\times10^{-12}=\mathbf{30\ kW}$ ✓ — hence, if we calculate the total energy carried by all the electrons, it matches the energy pumped into the beam as expected.

Step 3. Where the RF plant's power went: klystrons pump energy into cavity fields (their drive currents run at $\mathbf{J}\cdot\mathbf{E}<0$ — sources, Derivation C), the fields bank it as $u$ (next article), and the beam withdraws it as $+\mathbf{J}\cdot\mathbf{E}$ . One transfer term, three appearances, ledger closed.

Step 4. The magnetic contrast that keeps everyone honest: the same beam threading a quadrupole magnet ten times stronger in force terms gains zero energy — focusing is pure steering. Accelerator physics states this article's theorem operationally: magnets shape the beam; only $\mathbf{E}$ feeds it.

Origins · 1 minJames Prescott Joule and how this was discovered

James Prescott Joule

1818–1889 · Salford, Manchester — the brewer who metered the field's payments

Joule ran his family's brewery by day and, in a home laboratory, re-founded the concept of energy by night. At 22 he established that a current's heating of a conductor goes as $I^2R$ — announced in 1841, dismissed initially by the Royal Society, and now the operating identity of every resistor on Earth (in the SI's old units, the joule per second through an ohm). His obsession with precision — thermometers read to 1/200 of a degree, the famous honeymoon spent measuring waterfall temperatures — let him show by 1843 that electrical heat, mechanical work (the paddle-wheel experiments), and chemical action were one currency, convertible at fixed rates: 4.18 joules, as we now say, per calorie. Kelvin's collaboration made the ideas respectable; Helmholtz's 1847 memoir made them a law of nature. This article's $\mathbf{J}\cdot\mathbf{E}$ is Joule's $I^2R$ grown up: localized to the cubic metre, signed to handle sources, and waiting to be conscripted as the transfer term of field theory — a promotion he anticipated in spirit, having insisted all along that heat was not a substance but bookkeeping of motion.

History

1841

— Joule's law: resistive heating goes as I²R — the first quantitative field-to-matter energy rate

1843

— Joule's mechanical equivalent of heat: electrical, mechanical, thermal energy interconvert at fixed rates

1847

— Helmholtz's conservation memoir makes energy a universal ledger

1873

— Maxwell's Treatise establishes electromagnetic field energy and lays the groundwork for local energy conservation.

1884

— Poynting and Heaviside identify where the transferred energy travels: J·E gets its partner term

1900

— Drude's electron gas derives Ohmic dissipation microscopically: J·E = J²/σ from collisions

1951

— first ohmically heated fusion plasmas discover the T^(-3/2) resistivity ceiling

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §8.1.1 (Poynting's theorem) and §5.1.2 (magnetic forces do no work) · Jackson, Classical Electrodynamics, 3rd ed., §6.7 (Poynting's theorem and energy conservation) · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 4 (energy dissipation, Joule heating) and Ch. 6 · Zangwill, Modern Electrodynamics, §15.2 (work-energy theorem for fields). Key identities used: the triple-product identity $\mathbf{v}\cdot(\mathbf{v}\times\mathbf{B})=0$ ; the volume-integral result $P=\int\mathbf{J}\cdot\mathbf{E}\,\mathrm{d}\tau\to I^2R$ for a uniform Ohmic wire; and the source audit $\int\mathbf{J}\cdot\mathbf{E}\,\mathrm{d}\tau=-\mathcal{E}I$ . No numerical constants beyond $\varepsilon_0,\mu_0$ (standard SI vacuum constants) are used. Worked numbers: copper 1 mm²/10 A → $E=0.17$ V/m, $\mathbf{J}\cdot\mathbf{E}=1.7\times10^6$ W/m³ = 1.7 W/m = $I^2R$ ✓; linac 30 MV/m × 1 mA → 30 kW/m = 30 MeV per electron × $6.2\times10^{15}$ /s ✓.

Builds on: The Lorentz Force Law · Electric Current and Current Density · Charge Conservation and the Continuity Equation.

Leads to: Poynting's Theorem · The Poynting Vector · Energy Density of the Electromagnetic Field · Hidden Momentum in Magnetic Systems.

If magnetic forces do no work, how does an electromagnet lift a car?

By brokering. Lifting engages induced electric fields and internal stresses: as the car rises, the magnet's flux through the car's magnetized matter changes, back-EMFs appear in the magnet's supply circuit, and the power supply does extra J·E work in the windings at exactly the lifting rate (plus losses). The magnetic field redirects that electrically delivered energy into gravitational potential energy while doing zero net work itself — the crank-arm audit of the induction chapter, which balanced such a ledger to the milliwatt. The theorem constrains the accounting, not the usefulness.

Why doesn't the thermal motion of carriers contribute to J·E?

Because power is linear in velocity. Each carrier contributes qv·E; splitting v into drift plus thermal fluctuation, the fluctuation averages to zero and only the drift survives: J·E with J built from mean velocities. Contrast kinetic energy, quadratic in v, where thermal motion contributes enormously — copper's electrons carry huge thermal (Fermi) kinetic energy and almost no drift energy. The field pays the drift; collisions convert the payment into the thermal pool; the pool never pays back through J·E directly.

Ohm's law says J = σE, so isn't J·E = σE² always?

Only where Ohm's law holds — a material relation, not a law of nature. The exact statement is J·E, full stop. In vacuum beams σ is meaningless; in sources the current runs against E (J·E < 0, manifestly not σE² ≥ 0); in plasmas the relation is tensorial and frequency-dependent; in superconductors J flows with E = 0. The chain "J·E, then insert your medium's constitutive law" keeps the bookkeeping honest in all of them.

What does negative J·E in a generator's armature actually look like microscopically?

Carriers being pushed uphill by the lattice. The armature wire moves through B; the motional pump v×B drives electrons along the wire while the circuit's electrostatic field opposes them inside the source region. The lattice — spun by the turbine — drags the carriers through that opposing field via collisions and constraint forces; carrier by carrier, mechanical work is converted to electromagnetic energy. J·E < 0 is the volume-density receipt of that conversion, and integrating it over the armature gives exactly the εI the turbine must supply (plus I²r heat) — the motional-EMF power balance, localized.

Does a charge sitting still in a strong field exchange any energy with it?

None. J·E requires both current and field at the same point: a static charge in a megavolt field has J = 0 and trades nothing (it stores potential energy — a configuration statement, not a flow). Equally, a field region with no charge trades nothing there, however energetic the field. This locality of the transaction is what makes J·E fit the chapter's template: transfers happen at points, through the local values, or not at all.

Is J·E frame-independent?

No — and it shouldn't be. Change frames and E, J, even the split between "field energy" and "kinetic energy" all transform; a purely magnetostatic configuration in one frame (J·E = 0) shows electric fields and nonzero transfer in another. What survives is the law: in every inertial frame, that frame's J·E is exactly the field-to-matter transfer rate in that frame's ledger, and each frame's energy conservation closes without help. The covariant packaging (the transfer as part of ∂μT^μν) belongs to relativistic electrodynamics, by name — where "conserved but not invariant," flagged two articles ago, becomes machinery.