courseshub.world · Lives in Mathematics and Physics

Charles Wheatstone

Sir Charles Wheatstone

A course in two strands: the life of the shy instrument maker from the Strand who became a professor, telegraph pioneer and knight, and the physics behind his rotating mirror, bridge, stereoscope, cipher and dynamo.

Born 6 February 1802, Barnwood, Gloucester Died 19 October 1875, Paris First railway telegraph Euston–Camden Town, 25 July 1837 Level advanced undergraduate
The bridge. No current flows through the galvanometer G when \(R_1/R_2=R_3/R_x\), so an unknown \(R_x\) is found from three known resistances by a null reading.
Lesson 00

How to use this course

Charles Wheatstone had no university education. He learned his physics in a music shop, and he remained above all a builder of instruments: devices that make sound visible, time an electric spark, show a solid shape from two flat drawings, or send letters down a wire. Several carry his name, and one of the most famous of them he did not invent. This course follows the instruments and the physics they embody.

Prerequisites for Part II
  • Ohm’s law and Kirchhoff’s circuit laws.
  • Simple harmonic motion and the superposition of vibrations.
  • Basic geometry and trigonometry; elementary combinatorics for Lessons 06 and 09.
  • For Lesson 10, Faraday’s law of induction.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The order roughly follows his career: acoustics (1820s), the speed of electricity (1834), telegraphy (from 1837), resistance measurement (1843), binocular vision (1838–1852), cryptography (1854) and electrical machines (1867).

Reference

Timeline

1802Born 6 February at Barnwood, Gloucester, son of a music seller.
c. 1816Apprenticed at 14 to his uncle Charles Wheatstone, maker and seller of musical instruments at 436 Strand, London.
1821September: exhibits the “Enchanted Lyre”, which plays music carried along a rod from hidden instruments.
1823His uncle dies; Charles and his brother William take over the business.
1827Invents the kaleidophone, which makes the vibrations of a rod visible.
182919 December: patents the symphonium, a free-reed instrument that prefigures the English concertina.
1832Mirror and prism stereoscopes built for him by the optician R. Murray.
1834Professor of experimental philosophy at King’s College London. Measures the speed of electricity with a rotating mirror.
1835Shows at the British Association in Dublin that sparks between different metals give different spectra.
1836Fellow of the Royal Society.
183727 February: William Fothergill Cooke visits him. Joint patent for a needle telegraph. 25 July: trial between Euston and Camden Town.
183821 June: paper on binocular vision and the stereoscope read to the Royal Society.
1839Telegraph working on the Great Western Railway from Paddington to West Drayton.
1840Royal Medal of the Royal Society. Patents the ABC (dial) telegraph.
1841Arbitration of the priority dispute with Cooke.
1843Bakerian Lecture describing the bridge first published by Samuel Hunter Christie in 1833. Second Royal Medal.
1844Underwater cable trial in Swansea Bay with J. D. Llewellyn.
18451 January: a telegraph message from Slough leads to the arrest of John Tawell.
184611 April: leaves the Royal Institution before his evening lecture; Michael Faraday speaks instead.
184712 February: marries Emma West at Christ Church, Marylebone.
1848The polar clock.
1852Second paper on binocular vision; the pseudoscope.
185426 March: describes the cipher later named after Lyon Playfair.
1866Emma dies, leaving five children.
1867Paper on self-excited dynamo-electric machines. Albert Medal.
1868Knighted. Copley Medal.
1875Dies 19 October in Paris; buried at Kensal Green Cemetery, London.
Part I · Formation
Lesson 01

Gloucester and the Strand, 1802–1823

Charles Wheatstone was born on 6 February 1802 at Barnwood, near Gloucester. Some sources give a house in Westgate Street, Gloucester, instead. His father was a shoemaker who also sold music, and the family later moved to London. Charles went to local schools and read widely, largely educating himself in science.

At fourteen he was apprenticed to his uncle and namesake, Charles Wheatstone, who made and sold musical instruments at 436 Strand. The apprentice showed little interest in selling and much interest in how instruments produce and carry sound. He experimented with the transmission of vibrations through solid rods and wires.

The Enchanted Lyre

In September 1821 Wheatstone exhibited a device he called the Enchanted Lyre. A lyre hung from the ceiling of a room and appeared to play by itself. The sound actually came from instruments played in another room, their vibrations carried through a thin rod into the lyre’s soundboard, which radiated them. The exhibition drew crowds and press notice. Wheatstone later proposed that rods could carry music between distant rooms, an early idea of sound transmission that he soon abandoned for electricity.

Lesson 02

The music business and acoustics, 1823–1834

When his uncle died in 1823, Charles and his brother William took over the business. Over the next decade Wheatstone combined trade with research. He published papers on acoustics and invented instruments both scientific and musical.

In 1827 he described the kaleidophone, a vertical steel rod with a silvered bead at its tip. Lit by a lamp and set vibrating, the bead traced bright curves that made the rod’s motion visible (Lesson 04). On 19 December 1829 he patented the symphonium, a small free-reed instrument blown by mouth and played with buttons. It led to the English concertina, which the Wheatstone firm made and sold through the century. He also worked on the transmission of sound through solids and on the vibrations of plates and air columns.

This work drew the attention of scientists at the Royal Institution and the Royal Society, among them Michael Faraday. In 1832 he had the optician R. Murray build mirror and prism stereoscopes, though he did not publish them for six years (Lesson 08).

Lesson 03

King’s College, 1834–1837

In 1834 King’s College London appointed Wheatstone professor of experimental philosophy, a remarkable step for a self-taught instrument maker. His first lecture course, on sound, went badly. He was tongue-tied before an audience and sometimes spoke to his diagrams rather than to the students. He soon limited himself to laboratory work and demonstrations, and King’s gave him space for experiments.

The same year he published the experiment that made his scientific name: a measurement of the speed of electricity along a wire half a mile long, using a mirror spinning hundreds of times a second (Lesson 05). In 1835, at the British Association meeting in Dublin, he showed that the light of an electric spark, seen through a prism, contains bright lines characteristic of the metal of the electrodes, and suggested that spectra could distinguish metals. He was elected a Fellow of the Royal Society in 1836.

By then he was thinking about sending signals by electricity. On 27 February 1837 William Fothergill Cooke, a former army officer who had built a telegraph after seeing a demonstration in Heidelberg, visited him. The two men were very different, and their partnership would make the electric telegraph a commercial reality in Britain (Lessons 06 and 11).

Part II · The physics
Lesson 04

Vibrating rods and the kaleidophone

For the general reader. Pluck a steel rod clamped at its base and its tip moves too fast to follow. Put a shiny bead on the tip, shine a light on it, and persistence of vision turns the moving point of light into a curve. Wheatstone’s kaleidophone of 1827 used this trick to show that a rod can vibrate in two directions at once, and that the two motions add to give ellipses, figure eights and more complex loops.

Goals
  • Compute the fundamental frequency of a clamped rod.
  • Explain why a rod of rectangular cross-section has two different frequencies.
  • Describe the curves traced by two perpendicular vibrations.

A clamped rod

A uniform rod of length \(L\), Young’s modulus \(E\), density \(\rho\), cross-section area \(A\) and second moment of area \(I\) bends according to the Euler–Bernoulli equation \(EI\,\partial_x^4 y+\rho A\,\partial_t^2y=0\). For a rod clamped at one end and free at the other, the frequencies are

\[ f_n=\frac{(\beta_nL)^2}{2\pi L^2}\sqrt{\frac{EI}{\rho A}},\qquad \beta_1L\approx1.875,\quad \beta_2L\approx4.694. \]

For a rectangular section of width \(w\) and thickness \(t\), bending across the thickness has \(I=wt^3/12\), so \(I/A=t^2/12\) and \(f\propto t/L^2\). Bending across the width has \(f\propto w/L^2\).

Superposition of perpendicular vibrations

If the tip moves as \(x=a\cos(\omega_1t)\), \(y=b\cos(\omega_2t+\varphi)\), the trace is a closed curve exactly when \(\omega_1/\omega_2=p/q\) is rational. For \(p/q=1\) it is an ellipse, which degenerates to a line when \(\varphi=0\) or \(\pi\) and is a circle when also \(a=b\) and \(\varphi=\pm\pi/2\). In general the curve fits in the rectangle \([-a,a]\times[-b,b]\), and for a generic phase the number of points where it touches a horizontal side divided by the number where it touches a vertical side equals \(\omega_2/\omega_1\).

A rod of square section has equal frequencies in the two directions, so a pluck in a slanting direction gives an ellipse or a line. A rod whose width is twice its thickness has frequencies in the ratio 2:1 and traces figure-eight curves. Wheatstone made such rods, and joined rods at right angles to obtain other ratios. The French physicist Jules Antoine Lissajous later produced the same curves with tuning forks and mirrors, and they are now named after him.

Exercise 4.1A steel rod (\(E=200\ \text{GPa}\), \(\rho=7800\ \text{kg/m}^3\)) of square section \(3\ \text{mm}\) and length \(25\ \text{cm}\) is clamped at its base. Find its fundamental frequency.

Show solution
\(\sqrt{EI/\rho A}=t\sqrt{E/12\rho}=0.003\cdot\sqrt{200\times10^9/93\,600}\approx0.003\cdot1462\approx4.39\ \text{m}^2/\text{s}\). Then \(f_1=3.516\cdot4.39/(2\pi\cdot0.0625)\approx39\ \text{Hz}\). The tip moves fast enough for the eye to see a continuous curve.

Exercise 4.2Eliminate \(t\) from \(x=\cos t\), \(y=\cos(2t)\) and describe the curve. What changes for \(y=\sin 2t\)?

Show solution
\(\cos2t=2\cos^2t-1\), so \(y=2x^2-1\) for \(|x|\le1\): an arc of a parabola traced back and forth. For \(y=\sin2t=2\sin t\cos t\), \(y^2=4x^2(1-x^2)\): a figure eight, touching the vertical sides \(x=\pm1\) once each and the horizontal sides \(y=\pm1\) twice each.
Lesson 05

The rotating mirror

For the general reader. How fast does electricity travel along a wire? In 1834 nobody knew. Wheatstone looked at sparks in a mirror spinning 800 times a second. A delay of a millionth of a second between two sparks would shift their images apart by a visible angle. The method was more important than his number: it became the standard way to measure very short times, and later the speed of light.

Goals
  • Derive the angular shift of an image in a rotating mirror.
  • Reconstruct Wheatstone’s 1834 estimates.
  • See how the method passed to optics.

The optical lever

Rotating-mirror principle

When a plane mirror turns through an angle \(\alpha\), a reflected ray turns through \(2\alpha\). If the mirror spins at \(n\) revolutions per second, two flashes separated by a time \(\Delta t\) are seen in directions differing by

\[ \Delta\theta=2\cdot360^\circ\cdot n\,\Delta t . \]

Wheatstone’s experiment

Wheatstone discharged a Leyden jar through a copper wire half a mile long, arranged so that three spark gaps sat side by side in a line: one near each end of the wire and one at its midpoint. He viewed them in a mirror that spun, by his estimate from the pitch of its whirring, about 800 times a second. If electricity took time to travel, the image of the middle spark would be displaced relative to the end ones. He saw the three images form a slight V, consistent with electricity entering from both ends at once, and judged that the displacement could not exceed half a degree.

Half a degree at 800 revolutions per second corresponds to \(\Delta t=0.5/(2\cdot360\cdot800)\approx1/1\,152\,000\ \text{s}\). The middle spark is a quarter-mile from either end, which gave him a speed of about 288,000 miles per second, roughly 460,000 km/s. The figure is too high, because the displacement was at the limit of what he could see, but it showed that the speed is of the order of the speed of light. He also found that the spark itself lasts less than a millionth of a second, and that sparks drawn out by long wires last longer.

In 1838 François Arago proposed using Wheatstone’s mirror to compare the speed of light in air and in water, a test between wave and particle theories. Léon Foucault and Hippolyte Fizeau carried out the comparison in 1850 and found light slower in water, as the wave theory predicts.

Exercise 5.1Wheatstone observed images of the spark elongated by up to about 24°. What spark duration does this imply, at 800 revolutions per second?

Show solution
\(\Delta t=24/(2\cdot360\cdot800)=24/576\,000\approx1/24\,000\ \text{s}\), about 42 microseconds.

Exercise 5.2In a Foucault-type measurement, light travels from a rotating mirror to a fixed mirror 20 m away and back. The rotating mirror spins at 500 revolutions per second. Find the angular deflection of the returning beam.

Show solution
\(\Delta t=40/(3\times10^8)\approx1.33\times10^{-7}\ \text{s}\). The beam turns by \(2\cdot2\pi\cdot500\cdot1.33\times10^{-7}\approx8.4\times10^{-4}\ \text{rad}\), about 0.05°. On a screen 5 m away the spot moves about 4 mm.
Lesson 06

Needle telegraphs and codes

For the general reader. A current in a coil deflects a magnetic needle, a discovery of Hans Christian Ørsted in 1820. Cooke and Wheatstone used five such needles on a diamond-shaped board of letters. Two needles moved at once and pointed along lines that crossed at the letter to be read, so an untrained clerk could send and read messages. Later instruments traded this simplicity for fewer wires and a code.

Goals
  • Count the symbols available to the five-needle telegraph.
  • Compare codes by number of wires and number of signal steps.
  • Understand the dial (ABC) telegraph.

The five-needle telegraph (1837)

Each needle could rest vertical or be deflected left or right. In the patented scheme exactly two needles moved for each letter, in opposite directions, so that their extensions met at one point of the board.

Counting

Choosing 2 needles out of 5 gives \(\binom52=10\) pairs, and each pair can turn in 2 opposite senses, giving \(20\) symbols. The board therefore omitted six letters: C, J, Q, U, X and Z.

Since the current for each letter went out along one needle’s wire and back along the other’s, the scheme needed five wires. When the buried cables of the Great Western line deteriorated, the companies moved to two-needle instruments and then to a single needle, which sent letters as short sequences of left and right deflections. Fewer wires meant cheaper lines but required trained operators. In 1840 Wheatstone patented the ABC or dial telegraph, in which a pointer stepped round a dial of letters; it needed no code and was used for decades on private lines.

Code capacity

A code using sequences of length at most \(k\) from an alphabet of \(s\) distinguishable signals provides \(s+s^2+\cdots+s^k=s(s^k-1)/(s-1)\) symbols. With \(s=2\) (left, right), \(k=4\) gives 30 symbols, enough for 26 letters.

Exercise 6.1If any subset of the five needles could be deflected, each independently deflected left, deflected right or left at rest, how many symbols would be available? Why did Cooke and Wheatstone not use them?

Show solution
\(3^5-1=242\) non-rest states. But readers would need a code book; the two-needle scheme could be read directly off the lettered board, which was the point for railway staff.

Exercise 6.2A single-needle code uses sequences of at most \(k\) deflections. Find the smallest \(k\) that covers 26 letters and 10 digits.

Show solution
We need \(2(2^k-1)\ge36\), so \(2^k\ge19\) and \(k=5\) (which gives 62 symbols). With \(k=4\) only 30 are available.
Lesson 07

The Wheatstone bridge

For the general reader. To measure a resistance accurately, do not measure a current; balance it. Arrange four resistances in a diamond, with a battery across one diagonal and a sensitive galvanometer across the other. Adjust a known resistance until the galvanometer reads zero. A zero reading is easy to detect precisely, so the method is very accurate. Samuel Hunter Christie published the arrangement in 1833; Wheatstone’s Bakerian Lecture of 1843 made it widely known, and it took his name.

Goals
  • Derive the balance condition.
  • Compute the galvanometer current off balance using Thévenin’s theorem.
  • Understand the linearized bridge used in strain gauges.

Balance

Label the bridge as in the hero diagram: \(R_1\) and \(R_2\) form the left arm from the top node \(A\) to the bottom node \(C\), with midpoint \(B\); \(R_3\) and \(R_x\) form the right arm, with midpoint \(D\). A source of emf \(V\) connects \(A\) and \(C\), and the galvanometer connects \(B\) and \(D\).

Balance condition

The galvanometer current is zero if and only if

\[ \frac{R_1}{R_2}=\frac{R_3}{R_x},\qquad\text{that is,}\qquad R_x=R_3\,\frac{R_2}{R_1}. \]

The condition does not involve \(V\), the internal resistance of the source, or the galvanometer resistance.

Proof. With no current through the galvanometer, each arm is a simple voltage divider. Measured from \(C\), \(V_B=V R_2/(R_1+R_2)\) and \(V_D=V R_x/(R_3+R_x)\). These are equal exactly when \(R_2(R_3+R_x)=R_x(R_1+R_2)\), which simplifies to \(R_2R_3=R_1R_x\). Since the source voltage cancels, fluctuations in the battery do not affect the result.

Off balance

Seen from the galvanometer terminals, the rest of the circuit (with an ideal source) is a Thévenin source with

\[ V_{\text{th}}=V\Big(\frac{R_2}{R_1+R_2}-\frac{R_x}{R_3+R_x}\Big),\qquad R_{\text{th}}=\frac{R_1R_2}{R_1+R_2}+\frac{R_3R_x}{R_3+R_x}, \]

so the galvanometer current is \(I_g=V_{\text{th}}/(R_{\text{th}}+R_g)\). For a bridge with all four arms near \(R\) and \(R_x=R(1+\varepsilon)\), \(V_{\text{th}}\approx-V\varepsilon/4\). This linear response is the basis of strain-gauge and resistance-thermometer bridges. Later variants include the Kelvin bridge for very small resistances and James Clerk Maxwell’s bridges for inductance.

Exercise 7.1With \(R_1=100\ \Omega\), \(R_2=1000\ \Omega\), the bridge balances when \(R_3=47.3\ \Omega\). Find \(R_x\).

Show solution
\(R_x=R_3R_2/R_1=47.3\cdot10=473\ \Omega\). The ratio arms act as a multiplier of 10.

Exercise 7.2All four arms are \(120\ \Omega\) and \(V=5\ \text{V}\). A strain gauge in the \(R_x\) position changes by \(+0.24\ \Omega\). Find the open-circuit voltage between \(B\) and \(D\), exactly and with the linear approximation.

Show solution
\(\varepsilon=0.002\). Exactly: \(V_{\text{th}}=5(0.5-120.24/240.24)=5(0.5-0.500500)\approx-2.50\ \text{mV}\). Linear: \(-5\cdot0.002/4=-2.5\ \text{mV}\). The approximation is excellent for small strains.

Exercise 7.3Show that interchanging the source and the galvanometer leaves the balance condition unchanged.

Show solution
After the swap the source drives \(B\) and \(D\), and the galvanometer joins \(A\) and \(C\). The arms from \(B\) are \(R_1\) (to \(A\)) and \(R_2\) (to \(C\)); from \(D\), \(R_3\) (to \(A\)) and \(R_x\) (to \(C\)). Balance requires \(V_A=V_C\), i.e. \(R_1/(R_1+R_3)=R_2/(R_2+R_x)\), which again gives \(R_1R_x=R_2R_3\). This is a case of the reciprocity of linear networks.
Lesson 08

Binocular vision and the stereoscope

For the general reader. Your two eyes see slightly different pictures of the world. Wheatstone showed in 1838 that the brain fuses these two pictures into a single solid view, and that two flat drawings, one for each eye, are enough to create the impression of depth. His mirror stereoscope proved the point. Before photography, he used line drawings of geometrical solids.

Goals
  • Describe the mirror stereoscope.
  • Derive the binocular disparity of a point at a given depth.
  • Explain the pseudoscope.

The instrument

In the paper read to the Royal Society on 21 June 1838, the first part of his Contributions to the Physiology of Vision, Wheatstone described two plane mirrors set at 45° to the line of sight, one before each eye, each reflecting a drawing placed to one side. The two drawings were perspective projections of the same object from the positions of the two eyes. When the reflections were superposed, observers saw a single object in relief. He earned a Royal Medal for the work in 1840. David Brewster’s lens stereoscope of 1849 made the effect portable and, combined with photography, popular. The two men disputed priority and the theory of fusion for years.

Disparity

Binocular disparity

Let the eyes be separated by \(b\) and fixate a point at distance \(D\) straight ahead. A second point straight ahead at distance \(D+\Delta D\) subtends at the two eyes angles differing by the relative disparity

\[ \delta=2\arctan\frac{b}{2D}-2\arctan\frac{b}{2(D+\Delta D)}\approx\frac{b\,\Delta D}{D^2}\qquad(\Delta D\ll D,\ b\ll D). \]

Disparity falls off as the square of distance, which is why stereoscopic depth is strong at arm’s length and weak across a valley. Stereo photographers increase \(b\) to exaggerate relief in distant scenes.

In 1852, in the second part of his paper, Wheatstone described the pseudoscope, which uses prisms to exchange the views of the two eyes. Disparities change sign, and some solid objects appear hollow, a test of his theory that depth comes from disparity.

Exercise 8.1With \(b=6.5\ \text{cm}\), find the disparity between points at 50 cm and 51 cm, in arcminutes. Human stereo acuity is about 0.3 arcminute; at what distance does a 1 cm depth difference become undetectable?

Show solution
\(\delta\approx0.065\cdot0.01/0.25=2.6\times10^{-3}\ \text{rad}\approx8.9'\). Setting \(b\,\Delta D/D^2=0.3'=8.7\times10^{-5}\ \text{rad}\) gives \(D^2=0.065\cdot0.01/8.7\times10^{-5}\approx7.5\ \text{m}^2\), so \(D\approx2.7\ \text{m}\).

Exercise 8.2A stereo camera photographs a mountain range 5 km away. What baseline gives the same disparity for a 500 m depth difference as the eyes get for a 1 cm difference at 50 cm?

Show solution
Require \(b\cdot500/5000^2=0.065\cdot0.01/0.25\), so \(b=2.6\times10^{-3}\cdot2.5\times10^7/500=130\ \text{m}\).
Lesson 09

The Playfair cipher

For the general reader. Telegraph messages passed through many hands, so privacy needed ciphers. In 1854 Wheatstone described a cipher that encrypts letters in pairs using a 5×5 square of letters built from a keyword. His friend Lyon Playfair promoted it to the government, and it bears Playfair’s name. It was simple enough for pencil and paper and stronger than single-letter substitution. British forces used it in the field in the Second Boer War and the First World War.

Goals
  • Encrypt and decrypt with the Playfair square.
  • Count the keys and see why digraph substitution resists simple frequency analysis.
  • Identify its weaknesses.
The rules
  1. Square. Write the keyword’s letters without repeats, then the rest of the alphabet, into a 5×5 grid, merging I and J.
  2. Pairs. Split the message into pairs, inserting X between doubled letters in a pair and at the end if needed.
  3. Same row: replace each letter by the one to its right, wrapping around. Same column: by the one below, wrapping around. Otherwise the letters span a rectangle: replace each by the letter in its own row at the other corner.

Decryption reverses the rules (left for right, up for down; the rectangle rule is its own inverse).

With the key PLAYFAIR EXAMPLE the square is

\[ \begin{array}{ccccc} P&L&A&Y&F\\ I&R&E&X&M\\ B&C&D&G&H\\ K&N&O&Q&S\\ T&U&V&W&Z \end{array} \]

The message HIDE THE GOLD IN THE TREE STUMP becomes HI DE TH EG OL DI NT HE TR EX ES TU MP and encrypts to BM OD ZB XD NA BE KU DM UI XM MO UV IF.

Strength and weakness

There are \(25!\approx1.6\times10^{25}\) squares, though each cipher is unchanged by cyclic shifts of rows and columns, so the number of distinct keys is smaller by a factor of 25. Because it substitutes pairs, the cipher spreads frequency over about 600 digraphs instead of 26 letters, so single-letter counts reveal little. But digraph frequencies survive, a pair and its reverse encrypt to reversed pairs (if AB→XY then BA→YX), and no letter ever encrypts to itself within a rectangle. Given a few hundred letters of ciphertext, a trained analyst can reconstruct the square.

Exercise 9.1Using the square above, encrypt WHEATSTONE.

Show solution
Pairs WH EA TS TO NE. WH: rectangle, W→Z, H→G gives ZG. EA: same column, each moves down: E→D, A→E gives DE. TS: rectangle gives ZK. TO: rectangle gives VK. NE: rectangle gives OR. Ciphertext ZG DE ZK VK OR.

Exercise 9.2Prove that in the Playfair cipher no plaintext pair can encrypt to itself, and that no letter in a rectangle pair encrypts to itself.

Show solution
Row and column cases shift each letter by one position in a cycle of length 5, so each letter changes. In the rectangle case each letter moves to a different column of its own row, so it changes too. Hence every letter of every pair changes, and in particular no pair is fixed. Analysts exploit this: a guessed plaintext word can be slid along the ciphertext and rejected wherever a letter would map to itself.
Lesson 10

The self-excited dynamo

For the general reader. Early generators used permanent magnets, which are weak. In early 1867 several inventors saw that a generator could feed part of its own output through the coils of its electromagnets. A trace of leftover magnetism produces a small current, which strengthens the magnet, which produces more current, until the iron saturates. Wheatstone described this principle to the Royal Society in February 1867, within days of a communication by C. W. Siemens. Werner von Siemens and Samuel Alfred Varley reached it independently in the same months. Self-excitation made large-scale electrical power possible.

Goals
  • Model self-excitation as an instability.
  • Find the critical speed for series and shunt machines.
  • Understand the role of saturation and residual magnetism.

A linear model

Let the armature turn at angular speed \(\omega\). In the unsaturated range the emf is proportional to the field current, \(\mathcal E=G\,\omega\,I_f\), with a machine constant \(G\) (units of henry). Wheatstone’s machine diverted part of the current through the field coil, an arrangement now called shunt winding; Siemens put the field coil in series with the load.

Self-excitation criterion

For a series-wound machine with total circuit resistance \(R\) and inductance \(L\), the current obeys

\[ L\,\frac{dI}{dt}=(G\omega-R)\,I . \]

A small residual current grows exponentially if and only if \(\omega>\omega_c=R/G\). For a shunt machine on open circuit, the field circuit (resistance \(R_f\), inductance \(L_f\)) obeys \(L_f\,\dot I_f=(G\omega-R_f)I_f\), with critical speed \(R_f/G\).

Above the critical speed the growth stops only when the iron saturates and \(\mathcal E\) ceases to grow linearly with \(I_f\). The operating point is the intersection of the magnetization curve \(\mathcal E(I_f)\) with the field-resistance line \(\mathcal E=R_fI_f\). Without residual magnetism, \(I=0\) is an equilibrium and nothing happens; a dynamo that has lost its remanence must be “flashed” with an external current.

Exercise 10.1A series machine has \(G=0.05\ \text{H}\), \(L=0.5\ \text{H}\) and circuit resistance \(2\ \Omega\). Find the critical speed in revolutions per minute, and the growth time constant at twice that speed.

Show solution
\(\omega_c=2/0.05=40\ \text{rad/s}\approx382\ \text{rpm}\). At \(\omega=80\), \(G\omega-R=2\ \Omega\), so \(\tau=L/2=0.25\ \text{s}\). A residual current grows by \(e\) every quarter-second until saturation.

Exercise 10.2A shunt machine’s magnetization curve is \(\mathcal E=E_s\tanh(I_f/I_s)\cdot\omega/\omega_0\), with \(E_s=150\ \text{V}\), \(I_s=2\ \text{A}\) at \(\omega=\omega_0\). The field resistance is \(R_f=50\ \Omega\). Does the machine self-excite at \(\omega_0\), and roughly at what voltage?

Show solution
The slope at the origin is \(E_s/I_s=75\ \Omega>R_f=50\ \Omega\), so it self-excites. The operating point solves \(150\tanh(I/2)=50I\), i.e. \(\tanh(I/2)=I/3\). With \(u=I/2\), \(\tanh u=2u/3\) gives \(u\approx1.29\), so \(I_f\approx2.6\ \text{A}\) and \(\mathcal E\approx129\ \text{V}\). At speeds below \(\omega_0\cdot50/75\) it would not excite.
Part III · Life and legacy
Lesson 11

Telegraph wars and a public life, 1837–1868

Cooke and Wheatstone agreed to work together in May 1837 and took out a joint patent for a needle telegraph that year; the formal deed of partnership was signed on 19 November 1837. On 25 July 1837 they tested a line along the London and Birmingham Railway from Euston to Camden Town, about a mile and a half, with Wheatstone sending from Euston and Cooke replying from Camden. The railway did not adopt it, but the Great Western Railway did: by 1839 a line ran thirteen miles from Paddington to West Drayton, and it was later extended to Slough. On 1 January 1845 a message from Slough to Paddington described a suspect in a murder, and John Tawell was arrested on arrival in London. The case made the telegraph famous.

The dispute

Cooke saw himself as the entrepreneur who had brought the telegraph to Britain; Wheatstone saw himself as the scientist whose knowledge made it work. Their quarrel over credit became public, and in 1841 it went to arbitration. Marc Isambard Brunel acted for Cooke and John Frederic Daniell, Wheatstone’s colleague at King’s, for Wheatstone. The arbitrators credited Cooke with introducing the telegraph as a useful undertaking and Wheatstone with the research that prepared the public to receive it. Neither was satisfied, and the pamphlet war continued into the 1850s.

Science and society

Wheatstone kept inventing. In 1844 he and J. D. Llewellyn signalled through an insulated wire laid in Swansea Bay, and the next year he proposed gutta-percha insulation for a cable under the Channel. He was consulted on the transatlantic cables of the 1850s and 1860s. On 11 April 1846 he was due to give an evening lecture at the Royal Institution but left the building at the last moment. Michael Faraday spoke in his place and, short of material, offered speculations that light might be a vibration of lines of force. James Clerk Maxwell later saw in them a foreshadowing of his electromagnetic theory of light.

On 12 February 1847 Wheatstone married Emma West; they had five children. In 1848 he presented the polar clock, which tells the time from the plane of polarization of light from the sky. His automatic telegraph, which sent messages punched on paper tape at about a hundred words a minute, was widely used on British lines. France made him a member of the Légion d’honneur. Emma died in 1866.

DiscussionThe arbitrators separated “introducing” an invention from the science that made it possible. Is this a fair way to divide credit? Compare the Cooke–Wheatstone case with Christie’s bridge, which carries Wheatstone’s name.

Lesson 12

Last years and legacy, 1868–1875

Wheatstone was knighted in 1868, after the completion of his automatic telegraph, and received the Copley Medal of the Royal Society the same year. The Society of Arts had given him its Albert Medal in 1867. In October 1875, on a visit to Paris, he fell ill with an inflammation of the lungs. He died there on 19 October 1875, aged 73. He was buried in Kensal Green Cemetery, London.

Legacy

  • Measurement: the bridge that bears his name remains the basic null method of resistance measurement, and its linear form underlies strain gauges and sensor circuits.
  • Timing: the rotating mirror turned very short time intervals into angles; through Arago, Fizeau and Foucault it became a tool for measuring the speed of light.
  • Communication: the Cooke and Wheatstone telegraph was the first commercial electric telegraph, and his automatic transmitter set the pattern for high-speed telegraphy.
  • Vision: the stereoscope founded the experimental study of binocular depth perception and, through photography, a Victorian mass medium.
  • Spectroscopy: his 1835 observation that spark spectra identify metals anticipated the spectrum analysis of Gustav Kirchhoff and Robert Bunsen.
  • Music: the English concertina, made by the family firm, is still played.
  • King’s College London named its physics laboratories after him, and keeps a collection of his instruments.

Final projectChoose one of Wheatstone’s instruments (kaleidophone, rotating mirror, bridge, stereoscope, dynamo). Write a two-page account for a strong undergraduate that explains what it measured or did, derives the physics in modern form, and compares it with a present-day instrument for the same purpose.

Reference

Connections

People linked to Charles Wheatstone across this collection. A name without a link has no page here yet. See Charles Wheatstone in the network of all 337 lives →

Influenced by

Influenced

Colleagues and collaborators

Rivals

Friends

Family

Also mentioned in this life

Fields

Reference

Sources and further reading

Biography

  1. Charles Wheatstone, Wikipedia (with references to the Royal Society obituary and the ODNB).
  2. J. J. O’Connor and E. F. Robertson, Charles Wheatstone, MacTutor History of Mathematics.
  3. Linda Hall Library, Scientist of the Day: Charles Wheatstone.
  4. B. Bowers, Sir Charles Wheatstone FRS, 1802–1875, HMSO / IEE, 1975; 2nd ed. 2001.
  5. The unrehearsed lecture that changed the physics of light, Mapping Ignorance (on 11 April 1846).

Physics

  1. C. Wheatstone, An account of some experiments to measure the velocity of electricity and the duration of electric light, Phil. Trans. R. Soc. 124 (1834). Text.
  2. C. Wheatstone, Contributions to the physiology of vision. Part the first: on some remarkable, and hitherto unobserved, phenomena of binocular vision, Phil. Trans. R. Soc. 128 (1838).
  3. C. Wheatstone, An account of several new instruments and processes for determining the constants of a voltaic circuit, Phil. Trans. R. Soc. 133 (1843) (Bakerian Lecture).
  4. C. Wheatstone, On the augmentation of the power of a magnet by the reaction thereon of currents induced by the magnet itself, Proc. R. Soc. 15 (1867).
  5. Cooke and Wheatstone telegraph; Wheatstone bridge; Stereoscope; Playfair cipher, Wikipedia.
  6. D. Kahn, The Codebreakers, Macmillan, 1967 (Playfair cipher).
  7. N. J. Wade (ed.), Brewster and Wheatstone on Vision, Academic Press, 1983.