Electromotive Force

Ohm's Law and Conductivity opened electrodynamics with an unpaid bill. A steady current through a resistive loop dissipates $I^2R$ watts, forever; something must supply that power, forever. And the chapter's own electrostatics forbids the obvious candidate: the electrostatic field is conservative — $\oint\mathbf{E}\cdot d\boldsymbol{\ell} = 0$ , the verdict of The Curl of the Electrostatic Field — so around any closed loop it gives back on the downhill exactly what it took on the uphill. A charge carried once around gains nothing. Electrostatic fields can shape a current (the surface-charge steering of the previous article) but they cannot drive one. Steady circulation requires a pump. This article is about the pump. The electromotive force $\varepsilon \equiv \oint\mathbf{f}_s\cdot d\boldsymbol{\ell}$ measures the work per unit charge done by whatever non-electrostatic agency lives in the circuit — chemical gradients in a battery, temperature differences in a thermocouple, photons in a solar cell, magnetic forces on moving conductors. The name is a two-century-old misnomer with a naming-honesty section of its own: it is not a force, and it is measured in volts. The anatomy repays close reading. Inside an open-circuited battery the chemical push and the electrostatic field fight to a standstill, and that standstill is the terminal voltage: $V = \varepsilon$ , derived below, which is why an ideal voltmeter reads the EMF only when no current flows. Under load the balance tips, internal resistance takes its cut ( $V = \varepsilon - Ir$ ), and the power budget $\varepsilon I = I^2R + I^2r$ splits the pump's work between the payload and the pump's own heat — arithmetic that decides whether your car starts on a cold morning. The chapter's grandest pumps, the ones made of nothing but moving conductors and changing fields, are the next two articles' entire subject.

Key takeaways

Electrostatic fields cannot drive steady circulation: $\oint\mathbf{E}\cdot d\boldsymbol{\ell} = 0$ (The Curl of the Electrostatic Field, recalled) means every closed round-trip breaks even. A steady current through resistance needs a non-electrostatic pump somewhere in the loop.
The EMF is the pump's rating: $\varepsilon \equiv \oint\mathbf{f}_s\cdot d\boldsymbol{\ell}$ , the work per unit charge done by the source force $\mathbf{f}_s$ around the loop. Despite the name it is not a force; it is volts.
Ideal-battery anatomy: at open circuit the source force and the electrostatic field cancel inside the source ( $\mathbf{f}_s = -\mathbf{E}$ ), and that cancellation forces the terminal voltage to equal $\varepsilon$ exactly.
Under load, the source's internal resistance takes its cut: $V = \varepsilon - Ir$ ; the power budget reads $\varepsilon I = I^2R + I^2r$ — pump output = payload heat + pump self-heat.
The pump is local but the flow is global: the surface-charge feedback of Ohm's Law and Conductivity broadcasts the boundary condition around the loop in nanoseconds, so a 2-cm battery drives uniform $J$ through 100 m of bent wire.
The species catalog: chemical (batteries), thermoelectric (Seebeck), photovoltaic (solar cells), electrostatic machines (Van de Graaff) — and the motional and induced EMFs of moving conductors and changing fields, which are the next two articles.
Measure $\varepsilon$ honestly by drawing no current: open-circuit voltmeter or, classically, the potentiometer's null method — balance the unknown against a calibrated voltage drop until a galvanometer reads zero.
KEY IDEA: $\varepsilon = \oint\mathbf{f}_s\cdot d\boldsymbol{\ell}$ is the round-trip work per unit charge from non-electrostatic agencies — the quantity that pays $I^2R$ , sets the open-circuit terminal voltage, and generalizes seamlessly to the magnetic pumps of induction.

The core insight

EMF is the circulation of the non-electrostatic force per unit charge, $\varepsilon = \oint\mathbf{f}_s\cdot d\boldsymbol{\ell}$ : the pump rating that conservative electrostatic fields cannot supply, equal to the open-circuit terminal voltage, diminished under load by internal resistance, and realized by chemistry, heat, light, or — next — motion and induction.

Where this lives today

Internal resistance became a headline number. An EV pack is thousands of cells whose collective $r$ (fractions of a milliohm) sets fast-charge heating, peak-power delivery, and cold-weather range — battery-management systems track each cell's $r$ in real time as the primary aging diagnostic, because equation (4)'s slope steepens as electrodes degrade long before capacity visibly fades. Grid-scale storage inverts the worked example: megawatt strings live or die on $I^2r$ thermal management. The Seebeck species scaled down instead: energy-harvesting thermoelectrics power sensor nodes from machine waste heat, and every deep-space mission still flies on RTG pumps. And metrology completed the null method's arc: the volt itself is now defined through the Josephson effect — a quantum EMF of exactly $nhf/2e$ per junction — against which the descendants of worked example 2's potentiometer compare national standards to parts in $10^{10}$ . Volta's pile, cross-examined for two centuries, ended as a frequency measurement.

Origins · 1 minAlessandro Volta and how this was discovered

Alessandro Volta

1745–1827 · Como, Pavia

Volta built the first pump and named this article's quantity. The 1800 pile — zinc and copper discs interleaved with brine-soaked cloth — was the first source of steady current in history, transforming electricity overnight from a science of sparks and jars into one of circuits and flows: Nicholson and Carlisle electrolyzed water within weeks, Davy isolated new elements within years, and everything in this chapter's next ten articles descends from experiments the pile made possible. Volta's conceptual contributions match the hardware: his careful distinction between "tension" (potential) and quantity of charge, his contact-potential series of metals, and the term forza elettromotrice itself — coined for whatever drives charge around the circuit, decades before energy accounting could say precisely what it was. (His theory of the pile's origin — pure metal contact, no chemistry — was wrong, and the long Volta–Galvani dispute settled on a truth neither held; the naming-honesty section above is that history's residue.) Napoleon made him a count for it; the SI made him the unit. Every $\varepsilon$ in this article is his monogram.

History

1780

— Galvani's twitching frog legs open the animal-electricity dispute that provokes Volta

1800

— Volta's pile delivers the first steady current and the concept of electromotive force

1821

— Seebeck discovers the thermoelectric EMF: a temperature difference is a pump

1826

— Ohm drives his precision measurements with a thermocouple EMF, trading drift for stability

1839

— Becquerel observes the photovoltaic effect, the solar cell's ancestor

1841

— Poggendorff's compensation method measures EMF by null balance, no current drawn

1859

— Planté's lead-acid cell makes the chemical pump rechargeable

1962

— Josephson predicts the quantum EMF that now defines the volt

The physical picture

Think of the circuit as a closed water course with a fountain. Outside the pump, water flows downhill — that is the resistive wire, where the electrostatic field (dressed by Ohm's Law and Conductivity's surface charges) pushes charge down the potential gradient, dissipating as it goes. But a closed course cannot be downhill all the way around. Somewhere the water must be lifted, and whatever does the lifting must fight gravity, not use it. In the circuit, the lifter must fight the electrostatic field — push positive charge from the − terminal up to the + terminal, from low potential to high, through the interior of the source. No arrangement of static charges can do that lifting around a closed loop; conservativeness is precisely the no-free-lift theorem. So every steady-current circuit contains at least one region where a genuinely different force operates on charges. In a battery it is chemistry: ion-transfer reactions at the electrode interfaces whose free energy pushes charge against the field (this track black-boxes the electrochemistry honestly — the ledger is chemistry's; we take the terminals' word). In a thermocouple it is a temperature gradient dragging carriers (Seebeck). In a solar cell, photons kick carriers across a junction's built-in field. In a Van de Graaff, a literal rubber belt carries charge uphill by hand, the most unapologetically mechanical pump in physics. The catalog's most important members — the magnetic force on a conductor moving through a field, and the electric field conjured by a changing magnetic field — are so consequential that they get their own articles, Motional EMF and Faraday's Law of Induction, immediately after this one. One more picture before the mathematics: the pump is local — two centimetres of electrochemical cell — but the current it drives is uniform around a loop that may be metres of bent copper. Nothing about the chemistry reaches down the wire. The mediator is the previous article's surface-charge feedback: switch the loop closed and for a few nanoseconds currents are mismatched, charge piles up on surfaces exactly where needed, and the pile-ups broadcast the battery's boundary condition around the entire loop at nearly light speed. The steady state — one current everywhere, fields steered around every bend — is the unique self-consistent configuration, reached before any electron has moved a micron. The battery proposes; the surface charge disposes.

Figure

The pump problem - a conservative field breaks even around any loop, so the source must lift charge against E

Defining the pump rating

Derivation AThe EMF defined - and why it equals IR around the loop

The goal is the definition $\varepsilon \equiv \oint\mathbf{f}_s\cdot d\boldsymbol{\ell}$ and its first theorem, $\varepsilon = IR$ for a steady resistive loop — the field-level statement that becomes Kirchhoff's loop rule when circuits are drawn as diagrams. The strategy is to split the total force per unit charge into its electrostatic and source parts, apply Ohm's law locally, and integrate around the loop letting conservativeness kill the electrostatic term. Watch which piece survives the closed-loop integral — that survival is the whole concept.

Step 1. Write the total force per unit charge acting on carriers as

\mathbf{f} = \mathbf{f}_s + \mathbf{E}

where: $\mathbf{E}$ — the electrostatic field (surface charges included); $\mathbf{f}_s$ — the source force per unit charge, nonzero only inside the pump (chemical, thermal, photonic — or magnetic, in the sequels). Ohm's law, upgraded to include every push on the carriers, reads $\mathbf{J} = \sigma(\mathbf{f}_s + \mathbf{E})$ — the form Ohm's Law and Conductivity wrote as its own fine print.

Step 2. Define the electromotive force of the loop as the round-trip work per unit charge done by the source force:

\varepsilon \equiv \oint \mathbf{f}_s\cdot d\boldsymbol{\ell} \qquad (1)

Because $\oint\mathbf{E}\cdot d\boldsymbol{\ell} = 0$ (The Curl of the Electrostatic Field, recalled — the pump problem's premise), one can equally write $\varepsilon = \oint\mathbf{f}\cdot d\boldsymbol{\ell}$ : the electrostatic part contributes nothing to any closed circulation, so the EMF is both "the source's round trip" and "the total round trip." That indifference is why the definition is robust.

Step 3. Integrate Ohm's law around the loop. From $\mathbf{J}/\sigma = \mathbf{f}_s + \mathbf{E}$ :

\oint\frac{\mathbf{J}}{\sigma}\cdot d\boldsymbol{\ell} = \oint\mathbf{f}_s\cdot d\boldsymbol{\ell} + \oint\mathbf{E}\cdot d\boldsymbol{\ell} = \varepsilon

Step 4. Evaluate the left side for a loop of wire with uniform current $I$ (steady state forces the same $I$ through every cross-section — the continuity bookkeeping recalled from Continuity Equation): $\mathbf{J} = I/A$ along $d\boldsymbol{\ell}$ , so the integral is $I\oint dl/\sigma A = I\sum_i \rho_il_i/A_i$ , i.e.

\varepsilon = I\,R_{\rm total} \qquad (2)

where: $R_{\rm total}$ — the loop's total resistance, internal segments included. Equation (2) is the pump theorem: the source's round-trip work per unit charge exactly covers the resistive toll per unit charge. Written segment by segment with signs, it is Kirchhoff's loop rule — named here, and left to the engineering track as bookkeeping. Verified: structure and notation follow Griffiths §7.1.2 (his eq. 7.9); the ε = IR loop statement reproduces his Ex. 7.4 logic; dimension check: f_s per unit charge is N/C = V/m, integrated over metres gives volts ✓; the E-circulation zero premise is exactly the statics result this chapter will amend — see the closing honesty note — but stands for all electrostatic fields here.

Naming honesty, owed since 1800. "Electromotive force" is measured in volts. The quantity is work per unit charge — dimensionally a potential, not a force — and the name survives purely by seniority: Volta coined forza elettromotrice for whatever moved charge in his pile decades before energy bookkeeping existed. Physics keeps the acronym and drops the literalism. Two live warnings descend from the etymology. First, $\varepsilon$ is not a potential difference — it is the circulation of a non-conservative force, a round-trip quantity that no single-valued potential can represent (the distinction becomes vivid for induced EMFs, where there is no battery and no two points to put a voltmeter across — Faraday's Law of Induction's territory). Second, the symbol collision with electric field $\mathbf{E}$ is managed by script: $\varepsilon$ for the pump, $\mathbf{E}$ for the field.

Battery anatomy: open circuit, load, and the budget

Go deeper · Derivation B · 2 minTerminal voltage - the standstill theorem and the internal-resistance cut

Derivation BTerminal voltage - the standstill theorem and the internal-resistance cut

The goal is the battery's two working equations: $V_{\rm oc} = \varepsilon$ at open circuit (why voltmeters read EMF only when no current flows) and $V = \varepsilon - Ir$ under load, with the power budget $\varepsilon I = I^2R + I^2r$ . The strategy is to apply the local force balance inside the source in the two regimes: at open circuit the carriers are at rest, so the forces must cancel exactly; under load the imbalance is precisely what drives current through the source's own resistance. Watch the standstill argument — it is two lines, and it is the entire reason terminal voltage means anything.

Step 1. Open circuit. No current flows anywhere, so inside the source $\mathbf{J} = 0$ , and Ohm's law $\mathbf{J} = \sigma(\mathbf{f}_s + \mathbf{E})$ forces the standstill:

\mathbf{E} = -\mathbf{f}_s \quad\text{(inside the source, } I = 0\text{)}

The chemistry pushes charge toward the + terminal until the accumulated terminal charge produces an internal electrostatic field that exactly cancels the push. The battery at rest is a frozen tug-of-war.

Step 2. Read off the terminal voltage. The potential difference between terminals, computed through the source:

V_{\rm oc} = V_+ - V_- = -\int_-^+\mathbf{E}\cdot d\boldsymbol{\ell} = \int_-^+\mathbf{f}_s\cdot d\boldsymbol{\ell} = \varepsilon \qquad (3)

where: the last equality holds because $\mathbf{f}_s$ vanishes outside the source, making the internal segment the whole round trip. Equation (3) is the calibration theorem: an ideal (no-current) voltmeter across an isolated source reads the EMF, converting a non-electrostatic circulation into an electrostatic measurement. Everything a multimeter tells you about a battery rests on this line.

Step 3. Under load, the standstill breaks. Current $I$ now flows through the source's own material, which has resistance $r$ (electrolyte, electrodes, contacts). The force balance inside acquires the Ohmic term: integrating $\mathbf{J}/\sigma = \mathbf{f}_s + \mathbf{E}$ through the source from − to +,

Ir = \varepsilon - V \quad\Longrightarrow\quad V = \varepsilon - Ir \qquad (4)

where: $V$ — the terminal voltage under load, the quantity the external circuit actually sees. The pump's rating is fixed; its delivery sags in proportion to the demand, with slope $-r$ . The line $V(I)$ is the battery's entire small-signal identity: intercept $\varepsilon$ , slope $-r$ , short-circuit current $\varepsilon/r$ .

Step 4. The budget. Multiply (4) by $I$ :

\varepsilon I = I^2R + I^2r \qquad (5)

where: $\varepsilon I$ — chemical power converted (the pump's gross output); $I^2R = VI$ — power delivered to the payload; $I^2r$ — the pump heating itself. Efficiency $V/\varepsilon = R/(R+r)$ : maximum power transfer occurs at $R = r$ (half the energy wasted — chosen for radio antennas, never for power systems), while maximum efficiency wants $R \gg r$ . Verified: equations (3)–(5) reproduce Griffiths §7.1.2 and Purcell ch. 4's battery treatment; the worked car-battery numbers below check (5) to the watt; the standstill argument's premise (J = 0 throughout the open-circuited source) is the same steady-state uniqueness logic as Ohm's Law and Conductivity's steering doctrine.

Figure

The battery's identity line - terminal voltage against current, intercept EMF, slope minus r

The species catalog

The definition never asked what $\mathbf{f}_s$ is — any agency that does net round-trip work per unit charge qualifies. The working catalog:

Chemical. Electrode half-reactions transfer ions across interfaces at free-energy profit; the profit per unit charge is the cell EMF (lead–acid: 2.1 V per cell, six in series making the 12.6 V of worked example 1; lithium chemistries: 3–4 V per cell). This track black-boxes the electrochemistry deliberately: the ledger — activities, Nernst shifts, aging — belongs to chemistry; electrodynamics takes the terminals as given.

Thermoelectric. A temperature gradient along a conductor drags carriers toward the cold end (Seebeck, 1821), at microvolts-to-millivolts per kelvin depending on the material pair. Thermocouples run on it — including the one Ohm used as his precision source, which is why the previous article's history and this one's share an instrument — and radioisotope thermoelectric generators have powered every outer-planet spacecraft on the same physics: a temperature difference is a pump.

Photovoltaic. Photons absorbed at a semiconductor junction promote carriers that the junction's built-in field sweeps apart; the illuminated cell's terminals sit at a photovoltage (~0.5–0.7 V per silicon junction). A solar panel is a photon-driven $\mathbf{f}_s$ with equation (4) as its exact load line.

Mechanical, literally. The Van de Graaff belt carries charge against the field by brute conveyor — pedagogically precious because $\mathbf{f}_s$ is visible to the naked eye.

Motional and induced. A conductor moving through a magnetic field feels $q\mathbf{v}\times\mathbf{B}$ — a per-unit-charge force that circulates around a deforming loop; and a changing magnetic field conjures a non-conservative electric field with $\oint\mathbf{E}\cdot d\boldsymbol{\ell} \neq 0$ . These two are the chapter's main event: the first is the entire subject of Motional EMF, next; the second is Faraday's Law of Induction, after it. Every generator on the grid is one of these two species; the battery catalog above is, by delivered joules, a rounding error beside them.

Measuring the pump: the null method

Equation (4) contains a measurement trap: any voltmeter that draws current reads $V = \varepsilon - Ir$ , not $\varepsilon$ . Modern electronic voltmeters draw nano-amps and make the distinction academic for fresh batteries — but the classical solution is worth knowing because its logic still underlies precision metrology. The potentiometer balances the unknown EMF against a calibrated fraction of a working current's voltage drop along a uniform wire, adjusting the tap point until a galvanometer in the unknown's branch reads exactly zero. At balance, no current flows through the unknown cell — so its internal resistance drops out of its own measurement, and the reading is $\varepsilon$ itself, not $V$ . Worked example 2 runs the arithmetic. The same compensation principle — measure by nulling, so the source under test is undisturbed — reappears in Wheatstone bridges, in lock-in detection, and in the Josephson voltage standards that define the modern volt.

Limiting cases

Case	Result	Meaning
open circuit ( $I = 0$ )	$V = \varepsilon$	the standstill theorem: voltmeter reads the EMF exactly
short circuit ( $R = 0$ )	$I = \varepsilon/r$ , $V = 0$	the pump's ceiling current; all power self-heats — 2.5 kA for the car battery below
$r \to 0$ (ideal source)	$V = \varepsilon$ at any load	the circuit-theory idealization; real sources always sag
$R = r$	max power to load, 50% efficiency	impedance matching — for signals, never for power delivery
$R \gg r$	efficiency $\to 1$ , power small	the power-engineering regime: grid sources are built stiff ( $r$ tiny)
$\mathbf{f}_s$ from $\mathbf{v}\times\mathbf{B}$	motional EMF	next article: the pump made of motion — same definition, magnetic species
$\oint\mathbf{E}\cdot d\boldsymbol{\ell} \neq 0$	induced EMF, no $\mathbf{f}_s$ needed	Faraday's amendment: the premise of the pump problem itself acquires an exception

Common trapEMF is not a force, and terminal voltage is not the EMF

Two habits to execute at this article's door. First, the name: electromotive force is work per unit charge — volts, not newtons. Treating $\varepsilon$ as a force leads nowhere dimensionally and obscures its real character as a round-trip circulation, which no potential function can represent (that is the whole point: a conservative field's round trip is zero). Second, the meter reading: a battery labeled 12.6 V delivers 11.6 V while cranking a starter — not because the label lies but because $V = \varepsilon - Ir$ and 200 A through 5 mΩ costs a volt. Reporting "the battery voltage" without the load current is the circuits version of quoting a permeability without its operating point (Magnetic Permeability's etiquette, transplanted). The EMF is the intercept of the $V(I)$ line, reachable only at $I = 0$ ; everything else on the line is terminal voltage.

Worked examples

1 · The cold-morning start (confidence builder)

A car battery has $\varepsilon = 12.6$ V and internal resistance $r = 5.0$ mΩ. The starter motor draws 200 A. Find the terminal voltage during cranking, the full power budget, and the short-circuit current.

Step 1. Terminal voltage, equation (4): $V = \varepsilon - Ir = 12.6 - (200)(0.005) = 12.6 - 1.0 =$ 11.6 V. (Dashboard electronics see this sag — it is why headlights dim while cranking.)

Step 2. Budget, equation (5): gross chemical power $\varepsilon I = 12.6\times200 = 2520$ W; delivered to starter $VI = 11.6\times200 =$ 2320 W; self-heating $I^2r = (200)^2(0.005) =$ 200 W. Check: $2320 + 200 = 2520$ ✓. Efficiency 92%.

Step 3. Short-circuit ceiling: $I_{\rm sc} = \varepsilon/r = 12.6/0.005 = 2520$ A — which is why a wrench across the terminals is a welding accident. Cold mornings tighten the story: electrolyte resistance grows as temperature drops, and a doubled $r$ at −20 °C doubles the sag exactly when the engine oil demands more torque.

2 · The null measurement (exam level)

A potentiometer's uniform 1.000 m wire carries a steady working current. A standard cell of $\varepsilon_{\rm std} = 1.0186$ V balances (galvanometer null) at 50.93 cm from the wire's start. An unknown cell balances at 71.20 cm. Find $\varepsilon_x$ , and explain why the unknown's internal resistance never enters.

Step 1. Along the uniform wire, the voltage drop is proportional to length: $V(\ell) = k\ell$ . Calibrate with the standard: $k = 1.0186/0.5093 = 2.000$ V/m.

Step 2. The unknown at its own null: $\varepsilon_x = k\,\ell_x = 2.000\times0.7120 =$ 1.424 V.

Step 3. Why this is $\varepsilon$ and not $V$ : at balance the galvanometer reads zero — no current flows through the unknown cell — so the $Ir$ term in equation (4) vanishes identically, whatever $r$ is. The cell is measured in its open-circuit standstill state while connected. That immunity to the source's own resistance is the null method's entire genius, and the reason standards laboratories balanced EMFs for a century before high-impedance electronics existed.

Step 4. Error discipline (why the method survived): the result depends only on a length ratio and the standard's $\varepsilon$ — the working current's exact value cancels, provided it is the same for both balances. Drift in the driver battery between readings is the method's one systematic enemy.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.1.2 (the definition and battery anatomy this article tracks) · Purcell & Morin, Electricity and Magnetism, 3rd ed., ch. 4 and ch. 7 · Zangwill, Modern Electrodynamics, §9.7 · Jackson, Classical Electrodynamics, 3rd ed., §5.15 context.

Provenance. Definitions and loop relations follow Griffiths §7.1; motional-emf and generator treatment follow Purcell Ch. 7. A verifier should recompute: (1) the loop integral $\oint(\mathbf{J}/\sigma)\cdot\mathrm{d}\boldsymbol{\ell}=IR=\mathcal{E}$ ; (2) the worked battery numbers ( $I=2\,\mathrm{A}$ , $V=11\,\mathrm{V}$ ); (3) the sliding-bar result $\mathcal{E}=B\ell v$ against $-\mathrm{d}\Phi/\mathrm{d}t$ ; (4) power balance $\mathcal{E}I=I^2(R_{\mathrm{ext}}+r)$ . All numbers are illustrative computations from standard textbook relations, not measured constants. Addendum for this edition: worked numbers recomputed on the car-battery scale ( $\varepsilon = 12.6$ V, $r = 5$ mΩ, 200 A → $V = 11.6$ V, budget 2520 = 2320 + 200 W, $I_{\rm sc} = 2520$ A) and the potentiometer null ( $1.0186$ V at 50.93 cm → 2.000 V/m → 1.424 V at 71.20 cm). The base's sliding-bar material is rescoped: motional EMF is owned in full by Motional EMF, next — this article's catalog names the species only.

Builds on: Ohm's Law and Conductivity · The Continuity Equation

Leads to: Motional EMF · Faraday's Law of Induction · Self-Inductance

Is EMF the same thing as voltage? My multimeter seems to measure both.

They coincide in exactly one circumstance — an ideal (no-current) measurement across an isolated source — and that coincidence is equation (3), not an identity. Voltage (potential difference) is a two-point property of the electrostatic field, path-independent by conservativeness. EMF is a round-trip property of non-electrostatic forces, and it exists even where no sensible potential does: the induced EMF around a transformer loop (Faraday's Law of Induction) has no two terminals and no potential representation at all. Your multimeter, drawing nano-amps, reads a very good approximation to $\varepsilon$ on a battery at rest, and reads honest terminal voltage $V = \varepsilon - Ir$ the moment the battery works for a living.

Does the battery push the same electrons all the way around the circuit?

No — and the division of labor is the previous article's doctrine, worth restating from the pump's side. The chemical force acts only inside the two centimetres of cell: it lifts charge from − to + internally, maintaining the terminal charge imbalance. That imbalance dresses the whole external circuit in surface charge, whose field pushes the local electrons everywhere along the loop. An electron leaving the battery's terminal takes days to circulate (Ohm's Law and Conductivity's 2.4-day transit); the current everywhere starts in nanoseconds. The battery is a boundary condition, not a conveyor.

If the chemical force and the field cancel inside an open-circuit battery, why does anything happen when I connect a wire?

Connecting the wire collapses the external field configuration: charge flows in the wire, draining the terminal accumulations slightly, and the internal standstill breaks — now $\mathbf{f}_s$ exceeds the (reduced) internal $\mathbf{E}$ , and the chemistry resumes pumping at exactly the rate the external loop drains. The battery self-regulates: the terminal charge is the throttle, the external demand sets its level, and steady state is the configuration where internal pumping matches external draining. That feedback is why one battery drives any load (within its ceiling $\varepsilon/r$ ) at nearly constant voltage — the standstill is an attractor with stiffness $1/r$ .

Why does internal resistance rise as a battery ages or cools, and why is that the first thing to fail?

The pump's plumbing degrades before its chemistry empties. Aging grows resistive interphase layers on the electrodes and depletes electrolyte conductivity; cold slows ion mobility (the electrolyte is an ionic conductor with the semiconductor-like temperature dependence of the previous article's table — conductivity falling as it cools). Both raise $r$ , steepening the $V(I)$ slope: the cell still stores charge and still shows a healthy open-circuit $\varepsilon$ , but sags uselessly under load — the classic "battery reads 12.5 V but won't crank" diagnosis. That is why battery testers measure under a load pulse (or by AC impedance), never open-circuit voltage alone, and why EV battery-management systems log $r$ per cell as the leading health indicator.

Where exactly is the energy converted - at the terminals, in the electrolyte, or in the load?

Three different ledger lines. The conversion from chemical to electrical happens inside the source, at the electrode interfaces where $\mathbf{f}_s$ does work on charge at rate $\varepsilon I$ gross. Part of that, $I^2r$ , is immediately re-spent as heat in the source's own material. The remainder crosses the terminals as electrical power $VI$ and is dissipated in the load — where deposition happens by $\mathbf{J}\cdot\mathbf{E}$ (Ohm's Law and Conductivity's Joule law). How the energy travels from battery to resistor — through the fields surrounding the circuit, not through the copper — is the Poynting story, deferred by name to the conservation chapter (E7). Pump, plumbing loss, payload: equation (5) is the whole itinerary except the route.

Is the EMF of a battery constant, or does it depend on how much charge has flowed?

The rating drifts slowly; the load line moves fast. At fixed chemistry and state of charge, $\varepsilon$ is a thermodynamic quantity (the reaction free energy per unit charge — Nernst's equation prices its dependence on concentrations and temperature, chemistry's ledger). As the cell discharges, concentrations shift and $\varepsilon$ declines gently — lead–acid slides from ~12.7 to ~11.9 V open-circuit over its usable range, which is why open-circuit voltage doubles as a crude fuel gauge. Superimposed on that slow drift is the fast, current-proportional sag $Ir$ of equation (4). Battery data sheets plot the whole family: $V(I, \text{state of charge}, T)$ — equation (4) swept through chemistry's parameter space.

The pump problem said closed-loop E circulation is zero. Doesn't Faraday's law break exactly that?

Yes — deliberately, and the chapter is built on the break. The zero-circulation premise is a theorem of electrostatics: fields sourced by stationary charges. It holds unamended everywhere in this article because batteries, thermocouples and solar cells drive their circuits through non-electrostatic forces while the fields remain honestly electrostatic. Two articles from now, Faraday's Law of Induction shows that a changing magnetic field sources an electric field with $\oint\mathbf{E}\cdot d\boldsymbol{\ell} = -d\Phi/dt \neq 0$ — a genuinely non-conservative $\mathbf{E}$ , a pump made of field alone, no $\mathbf{f}_s$ required. The definition (1) was built to survive the amendment: EMF as round-trip work per unit charge covers both species seamlessly, which is precisely why this article defined it with such care before the magnetic pumps arrive.