EMF and Internal Resistance

Turn a car's ignition key on a cold morning and watch the headlights sag. Nothing outside the battery changed — the lamps still have their resistance, the wiring is the same — yet the moment the starter motor demands its hundred-and-fifty amperes, the whole electrical system's voltage droops. The battery is telling you, in the most direct way possible, that it is not an ideal source. Every real pump of charge carries an internal toll booth, and the busier the traffic, the larger the toll. The track's Electromotive Force article built the concept of the pump itself: EMF as energy handed to each coulomb, $\varepsilon = \oint \mathbf{f}_s\cdot d\mathbf{l}$ , with the standstill theorem and the terminal-voltage cut $V = \varepsilon - Ir$ derived from the force balance inside the source. This article takes that result and does what a lab or a design office does with it: treats $V(I) = \varepsilon - Ir$ as the battery's load line, a straight-line fingerprint you can measure with two readings, intersect with a load to predict the operating point, and optimise — including the famous maximum-power-transfer theorem and its widely misunderstood 50% fine print. What mastering this buys you is the ability to read any real source — AA cell, car battery, lab supply, solar panel — as two numbers, $\varepsilon$ and $r$ , and to reason instantly about sag, short-circuit current, battery stacking, and why power grids and audio amplifiers make opposite design choices about matching. It is also the gateway to Kirchhoff's rules: once sources have internal structure, circuits acquire real bookkeeping, and that is exactly where this chapter goes next.

Key takeaways

A real source is an ideal EMF $\varepsilon$ in series with an internal resistance $r$ : terminal voltage $V = \varepsilon - Ir$ falls linearly as the drawn current rises.
The load line's two intercepts are measurable: open circuit gives $V = \varepsilon$ (no current, no toll), and the slope of $V$ versus $I$ gives $-r$ .
Connected to a load $R$ , the circuit runs at $I = \varepsilon/(R+r)$ — the single-loop equation every battery problem starts from.
A car battery: $\varepsilon = 12.6$ V, $r \approx 0.02\ \Omega$ . Cranking at 150 A drops the terminals to 9.6 V — that is the headlight sag, computed.
Maximum power reaches the load when $R = r$ , delivering $P_{\max} = \varepsilon^2/4r$ — but at 50% efficiency, which is why the theorem rules signal electronics and is deliberately avoided by power engineering.
Series cells add EMFs and add internal resistances; parallel identical cells keep $\varepsilon$ and divide $r$ by the count — and mismatched parallel cells silently circulate current between themselves.
KEY IDEA: the pair ( $\varepsilon$ , r) is a complete terminal description of any real DC source, and the load line $V = \varepsilon - Ir$ is how you measure it, load it, and optimise it.

The core insight

Every real source presents a straight load line $V = \varepsilon - Ir$ : EMF is the zero-current intercept, internal resistance the slope, and every application — sag, short circuit, matching — is a point on that line.

Where this lives today

Internal resistance has quietly become one of the most-monitored quantities in modern technology. Every electric vehicle's battery-management system tracks the effective $r$ of each cell group in real time — from voltage sag under known current pulses, exactly Derivation A — because rising internal resistance is the earliest honest signal of cell ageing, and because $I^2r$ heating limits fast-charge rates. The dreaded "battery health" percentage on a phone is largely an $r$ statement dressed for consumers. The load-line habit generalises far beyond chemistry. Solar photovoltaic panels are sources with a bent (non-linear) $V$ – $I$ characteristic, and their inverters run maximum power point tracking — continuously hunting the knee of the curve, the working generalisation of Derivation B's $R = r$ . Radio-frequency engineers match amplifier stages to antenna impedances for exactly the theorem's reason; grid engineers hold source impedance far below load precisely to escape it. One theorem, two industries, opposite verdicts — both correct.

The physical picture

Think of the battery as a water tower with a narrow outlet pipe inside the fence. The tower's height — the EMF — is fixed by chemistry: how much energy the reactions hand each coulomb they lift. But every drop that leaves must squeeze through the internal pipe, and the pressure you measure at the gate is the tower height minus the pressure lost inside. Draw nothing, and the gate pressure equals the full height. Open the valve wide, and the internal pipe eats a growing share. The internal resistance is not a design flaw bolted on; it is the physical interior of the source. In a chemical cell it is the electrolyte's ionic resistance, the electrode surfaces, the separators — matter the current must physically traverse. That is why $r$ is a state of health indicator: as a cell ages or runs down, its chemistry clogs, $r$ creeps upward, and the terminal sag under load grows even while the open-circuit reading still looks healthy. A tired battery lies to a voltmeter and confesses to a load.

Figure

The battery load line — terminal voltage falls linearly with drawn current, from the EMF at open circuit to zero at the short-circuit current, with the operating point set by the load

The picture to carry away is the load line diagram above. The battery's entire terminal behaviour is one straight line: intercept $\varepsilon$ on the voltage axis, intercept $\varepsilon/r$ on the current axis, slope $-r$ . Any load you connect is another line through the origin, $V = IR$ . The circuit runs where the two lines cross. Every question in this article — how much sag? what short-circuit current? which load extracts maximum power? — is geometry on this diagram.

The loaded source

The Electromotive Force article derives, from the force balance inside the source, the terminal relation this article takes as its starting point:

V = \varepsilon - I r \qquad (1)

where: $V$ — terminal voltage, what a voltmeter across the posts reads; $\varepsilon$ — the EMF, the chemistry's fixed energy-per-coulomb; $r$ — internal resistance, ohms; $I$ — the current being drawn. Read it as a budget: of the $\varepsilon$ joules given to each coulomb, $Ir$ are spent crossing the source's own interior, and only $V$ survive to the outside world. At $I = 0$ the toll vanishes and the terminals show the full EMF — which is why EMF is measured open-circuit, and why the potentiometer (a null device meeting us in Kirchhoff's Rules) was invented to make that measurement exactly. Close the loop with an external resistance $R$ and the same coulomb must also pay $IR$ outside. The budget balances only at one current:

I = \frac{\varepsilon}{R + r} \qquad (2)

where: $R + r$ — the total series resistance of the loop, internal plus external. This is the master equation of every single-loop battery problem. Two limits orient it immediately: with $R \gg r$ (a light load) the current is nearly $\varepsilon/R$ and the source behaves ideally; with $R \to 0$ (a short circuit) the current saturates at $I_{sc} = \varepsilon/r$ — for a car battery, $12.6/0.02 = 630$ A, which is why dropping a spanner across battery terminals is a welding event, not a mishap. Multiplying equation (1) by $I$ gives the power ledger $\varepsilon I = I^2R + I^2r$ — the pump's full output split between payload and internal heat. That ledger is derived and interpreted in Electromotive Force; here we will optimise it (Derivation B) and use it for engineering judgement.

Derivation AThe load line as a measuring instrument — extracting EMF and internal resistance from two readings

We want a practical procedure that turns equation (1) from a description into a measurement: given two (current, voltage) readings from a working battery, determine both $\varepsilon$ and $r$ , and understand why the naive method — read a voltmeter, call it the EMF — fails for a tired battery. The strategy is to treat $V(I) = \varepsilon - Ir$ as a straight line in the $(I, V)$ plane and extract its two parameters from two points. The watch-out: a voltmeter draws almost no current, so it always reads a point near the $V$ -axis intercept — informative about $\varepsilon$ , silent about $r$ .

Step 1. Write equation (1) at two measured operating points: $V_1 = \varepsilon - I_1 r$ and $V_2 = \varepsilon - I_2 r$ . Two equations, two unknowns.

Step 2. Subtract to isolate the slope: $r = (V_1 - V_2)/(I_2 - I_1)$ . The internal resistance is minus the slope of the terminal voltage against drawn current. Why subtraction first: the EMF cancels, so the $r$ measurement needs no knowledge of the chemistry — only two honest load points.

Step 3. Back-substitute either point: $\varepsilon = V_1 + I_1 r$ .

Step 4. Run the car-battery numbers. With headlights only: $I_1 = 10$ A, $V_1 = 12.4$ V. Cranking: $I_2 = 150$ A, $V_2 = 9.6$ V. Then $r = (12.4 - 9.6)/(150 - 10) = 2.8/140 = 0.020\ \Omega$ and $\varepsilon = 12.4 + 10(0.020) = 12.6$ V. The 3-volt sag during cranking is not a fault; it is $I_2 r = 3.0$ V of internal toll at 150 A, and it is exactly why the headlights dim.

Step 5. Note what the procedure exposes that a voltmeter alone cannot. An aged battery may still show $\varepsilon \approx 12.5$ V open-circuit while its $r$ has tripled; it passes the voltmeter test and fails the cranking test. Battery testers therefore apply a load and watch the sag — they are measuring the slope, precisely this derivation. The growth of $r$ with depletion and age (electrolyte concentration falling, electrode surfaces degrading) also means $r$ is honestly a function of state, and the pair $(\varepsilon, r)$ is a snapshot, not a constant of nature. Verified: r = 2.8/140 = 0.0200 Ω and ε = 12.6 V recomputed; short-circuit projection ε/r = 630 A consistent with the load line's current intercept.

Figure

The matching trade-off — load power peaks at R equals r, but efficiency climbs right past it: signal engineers sit at the peak, power engineers far up the efficiency curve

Go deeper · Derivation B · 2 minMaximum power transfer — R equals r, and the 50 percent fine print

Derivation BMaximum power transfer — R equals r, and the 50 percent fine print

We want the load resistance that extracts the most power from a source of given $\varepsilon$ and $r$ , and — just as important — the efficiency at that optimum, because the theorem is misapplied more often than applied. Strategy: write the load power as a function of $R$ , maximise by calculus, then audit the energy budget at the optimum. The power ledger $\varepsilon I = I^2R + I^2r$ from Electromotive Force is our starting line; the optimisation is the new content.

Step 1. The load power at the operating current of equation (2) is

P_R(R) = I^2 R = \frac{\varepsilon^2 R}{(R + r)^2}

Both limits vanish: $R \to 0$ wastes everything inside the source ( $P_R \to 0$ though the current is huge), and $R \to \infty$ draws no current. Somewhere between sits a maximum.

Step 2. Differentiate and set to zero:

\frac{dP_R}{dR} = \varepsilon^2\,\frac{(R+r)^2 - R\cdot 2(R+r)}{(R+r)^4} = \varepsilon^2\,\frac{r - R}{(R+r)^3} = 0

The numerator flips sign at $R = r$ : positive below (power still rising), negative above (falling). The stationary point is a genuine maximum. Why the answer is "match the source": below $r$ , the internal toll dominates; above $r$ , the current starves. The optimum balances the two losses — the load and the source dissipate equally.

Step 3. The maximum itself: $P_{\max} = \varepsilon^2 r/(2r)^2 = \varepsilon^2/4r$ . For an AA cell ( $\varepsilon = 1.5$ V, $r = 0.15\ \Omega$ ): $P_{\max} = 2.25/0.6 = 3.75$ W into a matched 0.15 Ω load.

Step 4. Audit the efficiency at the optimum: the load's share of the total is $R/(R+r) = 1/2$ . Exactly half the chemistry's energy heats the battery. Check the neighbours numerically for the AA cell: $R = 0.05\ \Omega$ gives 2.81 W at 25% efficiency; $R = 0.15$ gives 3.75 W at 50%; $R = 0.45$ gives 2.81 W again but at 75% efficiency.

Step 5. Hence the design split. When the source is a signal — an antenna, a sensor, a microphone — the watts are tiny and unrepeatable, so you match ( $R = r$ ) and gladly pay the 50%. When the source is a power plant, efficiency is the entire business: grids run with load resistance vastly exceeding source impedance, transferring far less than the theoretical maximum power but at efficiencies above 90%. The theorem tells power engineers what to avoid. (The AC generalisation — match the complex conjugate of the source impedance — belongs to AC Circuits and Phasors and beyond.) Verified: dP/dR sign change at R = r rechecked symbolically; AA-cell table (2.8125 W / 3.75 W / 2.8125 W at 25% / 50% / 75%) recomputed numerically.

Go deeper · Derivation C · 2 minBatteries in combination — series stacks, parallel banks, and the mismatch trap

Derivation CBatteries in combination — series stacks, parallel banks, and the mismatch trap

We want the effective $(\varepsilon, r)$ of cells combined in series and in parallel, and the failure mode of parallel banks with unequal EMFs — a real engineering hazard invisible to ideal-battery thinking. Strategy: apply energy-per-charge bookkeeping along a series chain, then current bookkeeping at a parallel junction, keeping every internal resistance explicit. This is single-loop and single-junction reasoning; circuits needing more machinery are exactly why Kirchhoff's Rules comes next.

Step 1. Series: a coulomb traversing $n$ cells in a row is lifted $\varepsilon$ by each pump and pays $I r$ inside each. The chain behaves as one source with

\varepsilon_{\text{eq}} = n\varepsilon, \qquad r_{\text{eq}} = n r

A 6-cell lead-acid battery is exactly this: $6 \times 2.1\ \text{V} = 12.6$ V, with the cells' resistances summing to the $0.02\ \Omega$ we measured in Derivation A.

Step 2. Parallel, identical cells: $n$ equal cells share the load current evenly by symmetry, $I_{\text{each}} = I/n$ . Each cell's terminal voltage is $\varepsilon - (I/n)r$ , and since the terminals are common this is the bank's voltage:

\varepsilon_{\text{eq}} = \varepsilon, \qquad r_{\text{eq}} = \frac{r}{n}

Parallel banks buy stiffness (less sag) and capacity, not voltage — why power-tool packs parallel groups of cells inside a series stack.

Step 3. Parallel, mismatched cells: connect a fresh cell $\varepsilon_1 = 12.8$ V to a tired one $\varepsilon_2 = 12.0$ V, each with $r = 0.05\ \Omega$ , no external load at all. Going around the internal loop, the net driving EMF is $\varepsilon_1 - \varepsilon_2 = 0.8$ V across total resistance $r_1 + r_2 = 0.1\ \Omega$ :

I_{\text{circ}} = \frac{\varepsilon_1 - \varepsilon_2}{r_1 + r_2} = \frac{0.8}{0.1} = 8\ \text{A}

Why this is a trap: eight amperes flow with nothing connected — the fresh cell force-charges the tired one, both heat at $I^2r = 3.2$ W each, and the bank drains itself. This is why chargers and battery-management systems balance cells before paralleling them, and why you never jump-start by connecting mismatched banks casually.

Step 4. The mixed rule of thumb follows: series to raise voltage (accepting summed $r$ ), parallel to lower $r$ and raise capacity (requiring matched $\varepsilon$ ). Every commercial pack — from a 3.7 V phone cell to an 800 V vehicle pack of thousands of cylindrical cells — is a series-of-parallel lattice designed with exactly these two rules. Verified: circulating-current arithmetic 0.8/0.1 = 8.0 A and per-cell heating (8²)(0.05) = 3.2 W recomputed; series/parallel reductions consistent with the resistor combination laws derived independently in Series and Parallel Networks.

Limiting cases

case	result	meaning
$I \to 0$ (open circuit)	$V \to \varepsilon$	voltmeter reads the EMF; the toll booth is idle
$R \to 0$ (short circuit)	$I \to \varepsilon/r$	current capped by the source itself — 630 A for a car battery
$r \to 0$	$V = \varepsilon$ at any load	the ideal source of circuit diagrams
$R \gg r$	$I \approx \varepsilon/R$ , efficiency $\to 1$	the power-engineering regime
$R = r$	$P_R = \varepsilon^2/4r$ , efficiency $= 1/2$	maximum power transfer — the signal-engineering regime
$n$ cells series	$n\varepsilon$ , $nr$	voltage stacks, so does the toll
$n$ identical cells parallel	$\varepsilon$ , $r/n$	stiffness and capacity, not voltage

Worked examples

Example 1 — flashlight honesty. Two AA cells ( $\varepsilon = 1.5$ V, $r = 0.15\ \Omega$ each) in series drive a bulb of resistance $R = 3.0\ \Omega$ . Find the current, the bulb's actual voltage and power, and the fraction of the chemical energy wasted internally.

Step 1. Series combination: $\varepsilon_{\text{eq}} = 3.0$ V, $r_{\text{eq}} = 0.30\ \Omega$ .

Step 2. Operating current from equation (2): $I = 3.0/(3.0 + 0.30) = 0.91$ A.

Step 3. Bulb voltage: $V = IR = 0.91 \times 3.0 = 2.73$ V — the bulb never sees the nominal 3 V; it sees the load line's operating point.

Step 4. Powers: bulb $P = I^2R = 2.48$ W; internal loss $I^2r_{\text{eq}} = 0.25$ W. Efficiency $R/(R + r_{\text{eq}}) = 3.0/3.3 =$ 91% — a well-designed load, far from the matched point, exactly as Derivation B recommends for power delivery.

Example 2 — sizing the sag before it happens. A 12.6 V, 0.020 Ω car battery must run a 60 W headlight pair (drawing about 5 A) while the starter takes 150 A. What voltage do the headlights see during cranking, and how much power does the battery itself dissipate?

Step 1. Total demand during cranking: $I \approx 155$ A.

Step 2. Terminal voltage from equation (1): $V = 12.6 - 155 \times 0.020 = 12.6 - 3.1 = 9.5$ V. The lamps, built for 12.6 V, run at three-quarters voltage — visibly dim, since incandescent output falls steeply with voltage.

Step 3. Internal heating: $P_{\text{int}} = I^2 r = (155)^2(0.020) = 480$ W inside the battery case for the few seconds of cranking — why repeated failed starts warm (and stress) a battery, and why $r$ 's further rise in extreme cold (slower electrolyte chemistry) can push the terminal voltage below what the ignition system tolerates. Both examples recomputed numerically: I = 0.909 A, V = 2.727 V, P_bulb = 2.479 W, efficiency 0.909; cranking V = 9.50 V, internal 480.5 W.

Origins · 1 minThomas Edison and how this was discovered

Thomas Edison

1847–1931 · Menlo Park and West Orange, New Jersey

Edison meets this article not as the inventor of the bulb but as the first engineer to fight internal resistance at industrial scale. His Pearl Street station (1882) delivered DC to Manhattan, and its economics lived and died by source resistance and line drop: generators wound for low $r$ , feeder cables costed against $I^2R$ loss, load lines drawn in copper and dollars. He then spent a full decade (1899–1909) on the storage battery itself, testing thousands of chemistries to produce the nickel–iron alkaline cell — heavy and modest in EMF, but with a hardy chemistry whose internal resistance stayed serviceable through years of abuse; it powered delivery trucks, mine lamps, and railway signals for half a century. Edison's DC empire lost the grid war to high-voltage AC (a story told at Transformers), but the discipline he pioneered — treat every source as chemistry plus internal ohms, and engineer both — is precisely this article's subject.

History

1780

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

1800

— Volta's pile stacks zinc, copper, and brine into the first steady source: EMF enters physics

1836

— Daniell's cell tames polarisation, giving telegraphy a source whose EMF holds steady in use

1841

— Poggendorff's compensation (null) method measures EMF with no current drawn — the open-circuit intercept made exact

1859

— Planté's lead–acid cell makes the pump rechargeable; its descendants still crank engines today

1882

— Edison's Pearl Street station engineers generators and feeders around internal resistance and line drop

1909

— Edison's nickel–iron alkaline cell ships after a decade of chemistry — durability engineered into r

Going further

Within this chapter: Kirchhoff's Rules is the immediate sequel — once circuits hold several sources and branches, the load line gives way to systematic bookkeeping, and the potentiometer there finally measures EMF with literally zero drawn current, perfecting this article's open-circuit idea. RC Circuits then adds time: the same $(\varepsilon, r)$ source charging a capacitor.

Deeper in the track: Electromotive Force (EMAG.MAXW.EMF.0550) owns the concept's foundations — the source as a charge pump, $\varepsilon = \oint \mathbf{f}_s\cdot d\mathbf{l}$ , the standstill theorem behind "open circuit reads EMF," the derivation of $V = \varepsilon - Ir$ from the internal force balance, and the running car-battery energy budget. Current, Resistance, and Resistivity (this chapter) supplies the $P = IV$ family used throughout. Motional EMF and Faraday's Law of Induction extend the pump concept to generators — sources whose $\varepsilon$ comes from motion and changing flux rather than chemistry.

Provenance. Authored fresh for the SuperSTEM E&M track's circuits chapter (recon-§d gap heal); sources: v3 circuit pilots (rewritten to UG register) + legacy wiki circuit articles (narrative) — no AllQFT base exists for this topic.

Is EMF actually a force? The name says so.

No — it is a historical misnomer that stuck. EMF is energy delivered per unit charge, measured in volts, exactly like potential difference. The distinction is origin: potential differences come from the electrostatic field, while an EMF comes from something non-electrostatic (chemistry, changing magnetic flux, temperature gradients) that can drive charge around a complete loop, which a conservative electrostatic field can never do. The Electromotive Force article makes this precise with $\varepsilon = \oint \mathbf{f}_s\cdot d\mathbf{l}$ .

Where physically is the internal resistance? Is it a component someone put inside?

It is the source's own interior seen as a conductor. In a chemical cell: the electrolyte's ionic conduction, the porous electrode surfaces, separators, and terminals. Nobody installs it; the current has no way around it, because every coulomb the pump lifts must physically traverse the cell. That is also why $r$ changes with temperature (ion mobility), with state of charge (electrolyte concentration), and with age (electrode degradation) — it is chemistry's plumbing, not a resistor.

Why does terminal voltage drop only when current flows?

Because the internal loss is $Ir$ — proportional to traffic. With no current there is no drop across $r$ (Ohm's law with $I = 0$ ), so the terminals show the full EMF. The moment a load draws $I$ , each coulomb pays the internal toll $Ir$ before reaching the outside. This is also the honest content of "open circuit reads EMF": it is the standstill limit of the load line, not a separate fact.

Can terminal voltage ever exceed the EMF?

Yes — during charging. Force current backwards through a battery (as an alternator or charger does) and the sign of the internal drop flips: the charger must supply $V = \varepsilon + Ir$ to push against the pump. That is why a car's electrical bus sits near 14 V while the engine runs, comfortably above the battery's 12.6 V EMF: the excess covers the internal drop of the charging current.

Does the maximum power transfer theorem mean I should always match my load to the source?

Only when the source's power is scarce and the goal is extracting the most of it — antennas, sensors, audio signal stages. For energy delivery the theorem marks the point to avoid: matched operation wastes exactly half the energy inside the source. Grids, chargers, and power supplies are engineered so the source impedance is tiny compared with the load, giving efficiencies above 90% at the cost of never approaching $P_{\max}$ — a trade they happily make, since their limit is heat and cost, not available wattage.

Why do batteries perform worse in the cold?

Chiefly because internal resistance rises: the electrolyte's ion mobility falls with temperature, so $r$ can multiply severalfold near freezing. The EMF barely moves (it is set by reaction thermodynamics), but the load line steepens dramatically — and a starter motor demanding 150 A through a doubled $r$ drops the terminals below the threshold the ignition electronics need. Same chemistry, same open-circuit reading, failed start: a pure slope effect.

What is the difference between a battery's capacity and its EMF?

EMF (volts) is the energy per coulomb; capacity (ampere-hours) is how many coulombs the chemistry can deliver before depletion. A tiny watch cell and a car battery can share an EMF near 1.5 V per cell chemistry-for-chemistry, while their capacities differ ten-thousand-fold. Energy stored is roughly the product: $\varepsilon \times$ charge. Internal resistance is a third, independent axis — how fast the stored energy can be withdrawn without unacceptable sag.

When two mismatched batteries are paralleled, the fresh one charges the tired one — is that not a good thing?

It is uncontrolled, and that is the problem. The circulating current $I = \Delta\varepsilon/(r_1 + r_2)$ is set by the mismatch and the internal resistances alone — 8 A in this article's example — with no regulator limiting it. Lead-acid tolerates modest abuse; lithium chemistries can overheat dangerously. Battery-management systems do deliberately balance cells, but through controlled, monitored paths of known resistance. The trap is the unintended loop, hot and invisible because nothing external is connected.