Electromotive Force

Work on Each Coulomb

The glass jar on the left of the bench is a Leclanché cell: a zinc rod, and an unglazed porous pot packed with manganese dioxide and carbon round a carbon rod, all standing in a solution of sal ammoniac. Georges Leclanché devised it in 1866; a book of 1896 gives it about 1.5 V.

The chemistry in a cell pushes charge from one terminal to the other. The work it does on each coulomb is the electromotive force, E, measured in volts: one volt is one joule a coulomb. This cell does about 1.5 J of work on each coulomb it sends round. Which reactions give a cell its volts is on the page The Galvanic Cell.

The name says force, but it is not a force: it is work divided by charge. James Clerk Maxwell defined it that way in his Treatise of 1873: carry a charge along a path, and the whole work the electrical force does on it, divided by the charge, is the electromotive force along that path.

A Leclanché cell (1866) · about 1.5 V · 1 V = 1 J a coulomb

Reading It with No Current

A voltmeter needs a little current from the cell to move its needle, so what it reads is not quite the electromotive force. In 1841 the German physicist J. C. Poggendorff found another way: let the cell under test give no current at all, and oppose it with a stronger cell until the galvanometer’s needle does not stir at the moment of contact.

The metre of wire at the front is fed from the accumulator on the right, and its current gives every centimetre of the wire the same share of voltage: here, 2 V a metre. The cell’s positive terminal goes to the wire’s zero end, its negative through the galvanometer to the sliding contact, the jockey. At each stop the key is pressed and the galvanometer compares the cell’s electromotive force with the voltage on the wire from zero to the jockey; when they are equal the needle stays still and no current flows in the cell.

This cell balances at 75.0 cm: 1.50 V. In 1868 the English engineer Latimer Clark built the method into an instrument he called the potentiometer; by about 1900 laboratories often made it on a metre rule: a stretched wire, one or two accumulators and a sliding contact.

2 V a metre · balance at 75.0 cm: 1.50 V · no current in the galvanometer at balance

A Volt in a Glass H

Two volts a metre has to be set against a cell of known voltage. In 1872 Latimer Clark made a standard cell meant only for comparison, never for supplying current. The International Electrical Congress at Chicago in 1893 gave it 1.434 international volts at 15 °C, and the United States wrote that into law the next year.

The case at the far left of the bench holds Edward Weston’s cadmium standard cell, its front cut away. One limb of the glass H holds mercury under a paste of mercurous sulphate, the other an amalgam of cadmium; both lie under crystals of cadmium sulphate in its saturated solution. Weston applied for his patent in 1891. At 20 °C the cell gives about 1.0186 V and loses only about 0.04 mV for each degree warmer, about a thirtieth of the Clark cell’s change. The London conference of 1908 adopted it.

Put the jockey at 50.93 cm and turn the rheostat until the standard cell balances there: the wire then carries 2.000 V a metre. A standard cell has an internal resistance of hundreds of ohms and could not supply a current anyway; it is used only at balance. The United States has kept its volt with the Josephson effect since 1972, and every country has used the same agreed value since 1990.

Weston cell: about 1.0186 V at 20 °C · balance at 50.93 cm · 2.000 V a metre

Draw Current, Lose Volts

Put the plug in the key and a 5 Ω coil in the resistance box is connected across the cell under test, which starts to drive a current through it. The jockey finds a new balance at 62.5 cm: the cell now shows 1.25 V across its terminals, 0.25 V less.

Nothing is wrong with the cell. Pull the plug and the balance goes back to 75.0 cm. The missing 0.25 V is spent driving the current through the cell itself: there is resistance inside it.

So the 1.5 V printed on a cell holds only while no current is drawn. The brighter the lamp or the harder the motor works, the more current is taken and the lower the voltage across the cell.

A 5 Ω load · balance 75.0 → 62.5 cm · 1.50 → 1.25 V

The Resistance Inside

A cell can be pictured as two parts in series: an electromotive force E that does nothing but push charge, and an internal resistance r. With a current I through the cell, r takes I r, and the terminals are left with U = E − I r.

Two balances give r. With no load the voltage goes with 75.0 cm; with 5 Ω across the cell, with 62.5 cm. So r = R (l₀ − l₁)/l₁ = 5 × 12.5⁠/⁠62.5 = 1.0 Ω, and the current is 1.25 V over 5 Ω, 0.25 A.

In this cell the resistance lies in the solution, the porous pot and the ions’ slow way through the manganese dioxide. r is a model: it changes with temperature, age and the charge left, and even with the current. Cells of the 19th century differed widely: a book of 1868 gives a pint Grove cell usually under 1 Ω, and a Daniell cell 5 to 15 Ω.

r = R (l₀ − l₁)/l₁ = 1.0 Ω · I = 0.25 A · the page’s model values

A Straight Line

Try different resistances, note the current and the voltage across the cell each time, and the points fall on a straight line: U = E − I r. The line meets the voltage axis at E, and its slope is −r.

With no current the voltage is highest, equal to the electromotive force; the more current, the lower the voltage. The line reaches zero at the current the cell gives with no resistance outside, E/r: for this cell, 1.5 A.

The line is a model too. A real cell keeps close to it only over a range of currents: draw too much, or for too long, and the reactions at the electrodes fall behind and the voltage drops further. A book of 1896 already notes that a Leclanché cell’s electromotive force falls quickly on a strong current and comes back after a rest.

U = E − I r · meets the axis at 1.50 V · slope −1.0 Ω · the page’s model

The Lost Volts Become Heat

The chemistry does work at the rate E I. It splits in two: the resistance outside gets U I, and the cell’s own r gets I² r, as heat. E I = U I + I² r.

With 5 Ω outside, E I is 0.375 W: 0.3125 W goes to the coil and 0.0625 W stays in the cell. Five sixths is sent out, one sixth warms the cell.

The share inside grows with the square of the current. That is why a cell warms up on a heavy current, and more so once it is old and its r has grown.

E I = U I + I² r · at 5 Ω: 0.3125 + 0.0625 = 0.375 W

A Short Circuit

Put every plug of the resistance box in and its brass blocks join into one thick bar, so the cell’s terminals are all but joined. The current is the largest the cell can give, E/r = 1.5 A, and the voltage across it is almost nothing: the jockey backs off to near the zero end of the wire.

Nothing outside gets any power now. All the chemistry’s work, 2.25 J every second, turns to heat in the cell. The most current, and no power delivered.

A car battery has an internal resistance of a few milliohms, so a short circuit draws thousands of amperes: one car battery’s datasheet gives 3,⁠100 A, less than E/r predicts, because U = E − I r is only a model. The UK Health and Safety Executive warns that a spanner across the terminals gets very hot very quickly and can burst into a shower of molten metal, and that batteries give off hydrogen that a tiny spark can light. Twelve volts gives little risk of a shock; the danger is the current and the heat.

Every plug in · 1.5 A · about 0 V across the cell · all of it heat

A Load Equal to r

The power the load gets is P = E² R/(R + r)². A large resistance takes little current; a small one gets little voltage; both get little power. In between there is a peak, at R = r, of E²/(4r): for this cell, 0.5625 W.

Half the power then goes outside and half heats the cell: an efficiency of 50 %. At 5 Ω the power is smaller, but the efficiency is 83 %.

So batteries and grids are not run that way: a battery’s r is kept far below its load, and the US grid loses about 5 % in transmission and distribution. Load and source are matched only when the most power from a fixed source matters more than losing half of it, as between a radio transmitter and its antenna.

Most power at R = r · 0.5625 W · 50 % efficient · 83 % at 5 Ω

Cranking on a Cold Morning

A car battery is six lead-⁠acid cells in series, each just over 2 V: about 12.6 V when full. The glass box at the far right of the bench is one such cell, its brown lead-⁠dioxide plates between grey lead plates in dilute sulphuric acid. Gaston Planté showed this kind of cell to the Paris Academy of Sciences in 1860.

A starter turning an engine often draws more than 500 A. A healthy battery’s internal resistance is about 4 mΩ, so at 500 A it spends about 2 V inside, leaving about 10.6 V at the terminals and making about 1,⁠000 J of heat a second. In one measured start, the voltage fell from 12.2 V to about 9.5 V in 0.04 s, then held near 10 V.

In the cold the reactions slow and r grows. That is why the cold cranking current, CCA, is rated at −18 °C: the battery must give its rated current for 30 s and stay at 7.2 V or more. Reading a battery’s voltage with no load says little about its health; a tester puts a load on it and watches how far the voltage falls.

About 12.6 V · 500 A × 4 mΩ = 2 V · about 10.6 V left (computed here)

An AA Cell

The Leclanché cell became the dry cell: a book of 1896 says the first dry cell to give an appreciable current was made by Gassner in 1888, and that it was a form of Leclanché cell. Today’s alkaline AA still runs on zinc and manganese dioxide.

A new alkaline AA gives about 1.6 V with no current drawn, and its maker puts its internal resistance at 0.15 to 0.3 Ω. Across a 1 Ω load it drives about 1.2 to 1.4 A and shows about 1.23 to 1.39 V.

As it is used, its r keeps rising, and it rises in the cold too: on the maker’s curves, about 0.6 Ω near the end of its life and about 0.9 Ω at −40 °C. At 1 A, 0.6 Ω spends 0.6 V inside the cell. So the maker also says that reading the voltage with no load is a poor guide to how much is left.

About 1.6 V new · r 0.15⁠–⁠0.3 Ω · across 1 Ω: about 1.23⁠–⁠1.39 V (computed here)

The Bench in Ten Pieces

The bench comes apart in ten pieces: the leads, the jockey, the keys, the wire and rule, the galvanometer, the standard cell, the cell under test, the resistance box, the rheostat and the accumulator.

Point at one and it is lifted out on its own and named; at rest, all ten are named beside them.

Leads · jockey · keys · wire and rule · galvanometer · standard cell · cell under test · resistance box · rheostat · accumulator