Oliver Heaviside

How to use this course

Oliver Heaviside left school at sixteen, worked six years as a telegraph operator, and spent the rest of his life as an independent researcher with almost no income. From that position he gave Maxwell’s electromagnetic theory the four-equation vector form in which it is now taught, discovered how energy flows through the field, wrote down the force on a moving charge, solved the problem of distortion on long telephone lines, and invented an operational calculus that engineers used for decades before mathematicians could explain why it worked. This course presents that work alongside a life marked by deafness, poverty, bitter quarrels and late recognition.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Formulas use SI units unless stated; Heaviside himself preferred the rationalized Gaussian units now named after him and Hendrik Lorentz.

Timeline

Camden Town, 1850–1867

Oliver Heaviside was born on 18 May 1850 at 55 King Street, Camden Town, in north London. His father, Thomas Heaviside, was a wood engraver and watercolourist. His mother, Rachel Elizabeth West, had been a governess and, when money was short, ran a school and let rooms. The family was poor.

Scarlet fever in early childhood damaged his hearing. Partial deafness cut him off from other boys and shaped the solitary habits of his whole life. When he was thirteen the family moved to a better part of Camden, and he attended Camden House School. In 1865 he placed fifth among about 500 candidates in a public examination, but did poorly at geometry, which he disliked as taught in the Euclidean style. His parents could not keep him at school beyond sixteen. He spent a year studying alone, then left formal education for good.

A famous uncle

His mother’s sister had married Charles Wheatstone, professor at King’s College London, co-inventor of the electric telegraph and the man after whom the Wheatstone bridge is named. Wheatstone took an interest in the boy, encouraged him to learn Morse code, German and Danish, and helped him into the telegraph business. His elder brother Arthur also entered telegraphy and later worked for the Post Office.

The telegraph, 1867–1874

In 1867 Heaviside went to Newcastle upon Tyne, where his brother Arthur was working in telegraphy. In 1868 he joined a Danish telegraph company at Fredericia in Denmark, which was soon absorbed into the Great Northern Telegraph Company, operators of the submarine cable between Denmark and Newcastle. Around 1870 he moved to the company’s station at Newcastle, where he worked as an operator and then as an electrician.

The work made him think about what actually happened to a signal in a cable. He began publishing in 1872. His 1873 paper in the Philosophical Magazine on the best arrangement of the Wheatstone bridge caught the attention of Maxwell, who cited its result in the second edition of his Treatise. The same year his application to join the Society of Telegraph Engineers was turned down, reportedly on the ground that he was a mere telegraph clerk. He was admitted after William Thomson, later Lord Kelvin, supported him.

In 1874, aged 24, Heaviside resigned. His growing deafness made the job harder, and he had found a better use for his time. He moved back into his parents’ home in London. He never held another salaried position.

Alone with Maxwell’s Treatise, 1874–1887

Heaviside read Maxwell’s Treatise on Electricity and Magnetism soon after it appeared in 1873 and decided to master it. The book was long, hard, and written in a heavy notation of components and potentials. He spent several years working through it, before he felt he understood it.

For the next decade he lived in his parents’ house in Camden, worked in a small room by gaslight, and published in technical journals, chiefly The Electrician, which from 1882 paid him a small fee for a regular series. His articles mixed deep theory with sarcastic commentary on the electrical establishment. In 1880 he took out a patent on the coaxial cable. Between 1880 and 1887 he developed the operational calculus of Lesson 07. In 1884–85 he rewrote Maxwell’s theory in the vector form of Lesson 04, discovering along the way the law of energy flow that John Henry Poynting had published months earlier.

The quarrel with Preece

In 1887 Heaviside and his brother Arthur wrote a paper on telephony which Arthur’s superior at the Post Office, William Henry Preece, blocked from publication. Preece was the most powerful electrical engineer in Britain, and his practical rule held that self-inductance was harmful on telegraph and telephone lines. Heaviside’s theory showed the opposite: adding inductance could remove distortion (Lesson 06). The dispute became public and personal. The Electrician stopped his series that year. Within two years, however, Heaviside had powerful allies. Heinrich Hertz’s detection of electromagnetic waves in 1887–88 made Maxwell’s theory urgent, and the physicists who championed it, George Francis FitzGerald and Oliver Lodge above all, recognized Heaviside as one of their own.

Maxwell’s equations in four lines

For the general reader. Maxwell’s theory of electricity and magnetism, as he left it, was a sprawl of about twenty equations in twenty unknowns. Heaviside reorganized it around the electric and magnetic fields themselves, wrote it in the language of vectors, and arrived at four short equations. Those are the “Maxwell’s equations” on physics students’ T-shirts today.

From potentials to fields

Maxwell’s Treatise worked mainly with the scalar potential \(\phi\) and the vector potential \(\mathbf A\), which he regarded as physically basic. Heaviside regarded them as unobservable scaffolding. He eliminated them and took the electric force \(\mathbf E\) and magnetic force \(\mathbf H\), with \(\mathbf D=\varepsilon\mathbf E\), \(\mathbf B=\mu\mathbf H\), as the primary quantities. He used the vector notation of dot and cross products which he developed independently of Josiah Willard Gibbs.

The equations of Gauss, Faraday and Ampère, with Maxwell’s displacement current \(\partial\mathbf D/\partial t\), are all there. Heinrich Hertz reached a similar form independently in 1890, and the system was for a time called the Hertz–Heaviside equations. Albert Einstein’s 1905 relativity paper used it as its starting point.

Rational units

In the older Gaussian units, factors of \(4\pi\) appear in the field equations, such as \(\nabla\cdot\mathbf E=4\pi\rho\). Heaviside argued that this was an accident of definition: \(4\pi\) belongs in the field of a point charge, where spherical symmetry puts it, and should not appear in the general laws. His “rational” units removed it. Heaviside–Lorentz units are standard in particle physics, and SI units are rationalized in his sense.

Energy flow and the force on a charge

For the general reader. When a battery lights a bulb, where does the energy travel? Not inside the wire, said Poynting and Heaviside, but through the space around it, carried by the fields. Heaviside also wrote down how the fields push on a moving charge, the law that underlies every electric motor and particle accelerator.

Proof. By the vector identity \(\nabla\cdot(\mathbf E\times\mathbf H)=\mathbf H\cdot(\nabla\times\mathbf E)-\mathbf E\cdot(\nabla\times\mathbf H)\) and the circuital laws,

Poynting published first. Heaviside found the result independently from his own reformulation and gave a derivation that extended to moving media. He was the first to stress that energy in a circuit flows through the dielectric around the conductors and enters the wire sideways.

The force on a moving charge

J. J. Thomson had attempted the magnetic part in 1881 and obtained half the correct value. Heaviside gave the correct expression in 1889. Lorentz made it the basis of his electron theory in 1892–1895, and it is usually called the Lorentz force.

The telegrapher’s equations and the distortionless line

For the general reader. Early undersea cables blurred signals: each dot and dash spread out until they merged, and messages had to be sent slowly. Kelvin had explained this with a model in which the signal diffuses like heat. Heaviside saw that Kelvin had left out inductance, and that with the right amount of it a line would carry a signal undistorted. The cure, adding inductance, was the opposite of what engineers of the day believed.

The equations

Model a line by series resistance \(R\) and inductance \(L\) per unit length, and shunt conductance \(G\) and capacitance \(C\) per unit length. Applying Kirchhoff’s laws to a short segment gives

Eliminating \(I\),

With \(L=G=0\) this is Kelvin’s diffusion model of 1855; with \(R=G=0\) it is the lossless wave equation with speed \(1/\sqrt{LC}\).

Proof. For a sinusoid \(e^{i\omega t-\gamma x}\), the equations give \(\gamma^2=(R+i\omega L)(G+i\omega C)\). If \(R/L=G/C=k\), then \(\gamma^2=LC(k+i\omega)^2\), so \(\gamma=\sqrt{LC}\,(k+i\omega)=\sqrt{RG}+i\omega/v\). The attenuation \(\sqrt{RG}\) and the speed \(v\) do not depend on \(\omega\), and Fourier superposition gives the stated form.

Loading coils

Real cables have \(R/L\gg G/C\). Heaviside proposed to raise \(L\), at first continuously and then by inserting inductance coils at intervals. Preece’s opposition delayed the idea in Britain. Around the turn of the century George Campbell at AT&T and Michael Pupin at Columbia, both building on Heaviside’s papers, worked out the spacing rules for discrete loading coils, and AT&T bought Pupin’s patents. Loaded lines made long-distance telephony practical in the early twentieth century. Heaviside, who had published the principle a decade earlier, received nothing, and he refused AT&T’s later offer of money that came without full acknowledgment.

Operational calculus

For the general reader. Heaviside replaced differentiation by a symbol \(p\) and then treated \(p\) as an ordinary number: he divided by it, expanded in powers of it, even took its square root. Differential equations became algebra. The results were right, but he could not prove it, and the Royal Society refused to print the third part of his paper on the method. Thirty years later mathematicians explained it with the Laplace transform.

The method

Write \(p=d/dt\). Heaviside considered systems at rest for \(t\lt 0\) and switched on at \(t=0\). He used the unit step function, which he wrote as \(1\) and is now written \(H(t)\) in his honour: \(H(t)=0\) for \(t\lt 0\) and \(1\) for \(t\gt 0\). A circuit equation becomes \(Z(p)\,I=V\), and he wrote the solution as \(I=Z(p)^{-1}V\), interpreting \(p^{-1}\) as integration from 0:

Justification

In 1916 Thomas Bromwich showed that Heaviside’s rules follow from the contour integral now called the Bromwich inversion formula. In modern terms, if \(\hat f(s)=\int_0^\infty e^{-st}f(t)\,dt\), the step response of \(Y(p)\) is the inverse Laplace transform of \(Y(s)/s\). The expansion theorem is the residue theorem applied to \(Y(s)e^{st}/s\): the residue at \(s=0\) gives \(N(0)/D(0)\), and the residue at \(s=p_k\) gives the \(k\)-th term. John Carson and others carried this programme further in the 1920s, and Laurent Schwartz’s theory of distributions later gave rigorous meaning to Heaviside’s derivative of the step, the Dirac delta.

Fractional operators

Heaviside’s boldest rule was

It follows from the Laplace pair \(\int_0^\infty e^{-st}(\pi t)^{-1/2}dt=s^{-1/2}=s^{1/2}\cdot\frac1s\). He used it to solve Kelvin’s cable and heat-conduction problems in a few lines.

Fields of moving charges

For the general reader. A charge at rest is surrounded by a spherical electric field. In 1888 Heaviside calculated the field of a charge moving at constant speed and found that it flattens in the direction of motion. The result helped lead FitzGerald and Lorentz to the idea that moving bodies contract. Heaviside also predicted that a charge outrunning light in a medium would radiate a cone of waves, the effect observed by Pavel Cherenkov in 1934.

Ahead of and behind the charge (\(\theta=0\)) the field is reduced by \(1-\beta^2\); at right angles (\(\theta=\pi/2\)) it is enhanced by \((1-\beta^2)^{-1/2}\). The surfaces of equal potential are ellipsoids flattened along the motion by the factor \(\sqrt{1-\beta^2}\), sometimes called Heaviside ellipsoids. FitzGerald’s 1889 contraction hypothesis was inspired in part by this result. In special relativity the formula follows from a Lorentz boost of the Coulomb field.

Faster than light in a medium

Heaviside also considered a charge moving faster than the speed of waves \(c/n\) in a medium. Wavelets emitted along the path then pile up on a cone, as sound does behind a supersonic aircraft. The effect was forgotten until Cherenkov observed the blue glow in 1934; Ilya Frank and Igor Tamm explained it in 1937.

Skin effect, cables and the upper atmosphere

For the general reader. Rapidly alternating current crowds toward the surface of a wire, so a thick conductor is partly wasted. Heaviside explained why: the electromagnetic wave enters the metal from outside and dies away within a short distance. The same view of waves guided by conductors led him to patent the coaxial cable and to propose that a conducting layer high in the atmosphere guides radio waves around the Earth.

Derivation. Neglecting displacement current, \(\nabla\times\mathbf H=\sigma\mathbf E\) and \(\nabla\times\mathbf E=-\mu\partial_t\mathbf H\) give \(\nabla^2\mathbf E=\mu\sigma\,\partial_t\mathbf E\), a diffusion equation. For \(E\propto e^{i\omega t-kz}\), \(k^2=i\mu\sigma\omega\), so \(k=(1+i)/\delta\).

The coaxial cable

Heaviside’s 1880 patent placed one conductor inside a tubular other. The fields are confined to the space between them, so the line neither radiates nor picks up interference. For inner radius \(a\), outer radius \(b\) and a dielectric of permittivity \(\varepsilon\):

The conducting layer

In 1901 Guglielmo Marconi sent radio signals across the Atlantic. Straight-line waves should have been blocked by the curvature of the Earth. In 1902 Heaviside, suggested that a conducting layer in the upper air could guide the waves between itself and the sea. Arthur Kennelly in the United States made the same proposal that year. The ionosphere was confirmed experimentally in the 1920s, and its lower region, the E layer, is still called the Kennelly–Heaviside layer.

Vectors, impedance and gravity

For the general reader. Heaviside needed new tools and new words for his work, and many of both survive. He wrote electromagnetism in a vector algebra he built for the purpose, and he named the quantities of alternating-current engineering: impedance, inductance, conductance, admittance. He also asked whether gravity might have a magnetic-like partner, a question general relativity answered decades later.

Vector algebra

The third chapter of Electromagnetic Theory, volume 1 (1893), is a self-contained course in vector algebra and analysis. Like Gibbs, Heaviside took the scalar and vector products from William Rowan Hamilton’s quaternion product and discarded the rest. He wrote vectors in heavy type, and his \(\mathbf{VAB}\) for the vector product and \(\mathbf{AB}\) for the scalar product are notational cousins of the modern \(\mathbf A\times\mathbf B\) and \(\mathbf A\cdot\mathbf B\). He called the quaternionists’ insistence on the full product an obstacle for physicists, and he argued the point at length with Peter Guthrie Tait.

Impedance

For a sinusoidal steady state, every quantity varies as \(e^{i\omega t}\), and Heaviside’s operator \(p\) acts as multiplication by \(i\omega\). The operational impedance \(Z(p)\) of Lesson 07 becomes a complex number. Heaviside introduced the word impedance in 1886, along with inductance; admittance followed in 1887.

Gravitational analogue

In 1893 Heaviside asked what would follow if gravity, like electricity, propagated at a finite speed and had a magnetic-like companion produced by moving masses. He wrote field equations of the same form as Lesson 04, with mass density in place of charge and the sign of the source term reversed, since like masses attract. Heaviside did not pursue the idea far. Its modern descendant, gravitoelectromagnetism, emerges as the weak-field, slow-motion limit of general relativity. The frame-dragging it predicts was measured by the Gravity Probe B satellite.

Recognition and retreat to Devon, 1888–1908

Hertz’s experiments of 1887–88 transformed Heaviside’s position. The British Maxwellians, chiefly FitzGerald, Lodge and Heaviside, with Hertz as a correspondent, became the leading interpreters of field theory. Kelvin, despite their scientific disputes, became a friend. In 1891 Heaviside was elected a Fellow of the Royal Society. The Electrician resumed his series, which became the three volumes of Electromagnetic Theory (1893, 1899, 1912), and his collected Electrical Papers appeared in 1892.

The mathematicians were less welcoming. In 1892–93 the Royal Society published two parts of his paper “On operators in physical mathematics” but a further part was refused. The mathematician William Burnside was among the referees who objected to its lack of rigour. Peter Guthrie Tait attacked his vectors from the quaternion side. Heaviside answered in print with wit and contempt. He asked whether he should refuse his dinner because he did not fully understand digestion.

In June 1889 he moved with his parents to Paignton in Devon, near his brother Charles, who had a music business in the Torbay area. His parents died in 1894 and 1896. In 1896 FitzGerald and John Perry arranged a Civil List pension, which he accepted. In 1897 he moved alone to Newton Abbot. He also worked on a gravitational analogue of electromagnetism (1893) and, against Kelvin, argued that the Earth could be much older than Kelvin’s cooling estimate allowed. He received an honorary doctorate from Göttingen in 1905.

Torquay, death and legacy, 1908–1925

In 1908 Heaviside moved to Torquay, to Homefield in Lower Warberry Road, where Mary Way, the sister of his brother Charles’s wife, kept house for him. From 1916 he lived alone, and for more than a year in 1921–22 he could not pay for gas for heat or light. He grew more eccentric and suspicious, quarrelled with neighbours, signed letters with the initials W.O.R.M., and let the house fall into disrepair. He also opposed Einstein’s theory of relativity, although his own work had helped prepare it.

Honours still arrived. He became an honorary member of the Institution of Electrical Engineers in 1908, and of the American Institute of Electrical Engineers. In 1922 he was the first recipient of the IEE’s Faraday Medal; the Institution’s president travelled to Torquay to present it. Some weeks before his death he was found unconscious at Homefield after a fall and taken to a nursing home. He died there on 3 February 1925, aged 74, and was buried in his parents’ grave in Paignton cemetery. The grave was restored and rededicated in 2014.

Legacy

• Electromagnetism: the four-equation vector form of Maxwell’s theory, the energy-flux theorem, the force law and rationalized units.
• Engineering: transmission-line theory, characteristic impedance, the distortionless condition, loading, the coaxial cable and the vocabulary of impedance, inductance, conductance and admittance, much of which he coined.
• Mathematics: operational calculus, the step function and an early use of the delta function, which led through Bromwich and Carson to the Laplace-transform methods of control theory and, through Schwartz, to distributions.
• Physics: the field of moving charges, the anticipation of Cherenkov radiation, and the ionospheric layer that bears his name.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Oliver Heaviside, MacTutor History of Mathematics.
2. Oliver Heaviside, Wikipedia (with extensive references).
3. Institution of Engineering and Technology, Oliver Heaviside 1850–1925, IET Archives biography.
4. Torbay Civic Society, Oliver Heaviside.
5. P. J. Nahin, Oliver Heaviside: Sage in Solitude, IEEE Press, 1988; 2nd ed. Johns Hopkins University Press, 2002.
6. B. Mahon, Oliver Heaviside: Maverick Mastermind of Electricity, IET, 2009.
7. B. J. Hunt, The Maxwellians, Cornell University Press, 1991.

Physics and mathematics

1. O. Heaviside, Electrical Papers, 2 vols., Macmillan, 1892.
2. O. Heaviside, Electromagnetic Theory, 3 vols., The Electrician, 1893, 1899, 1912. Volume 1 at the Internet Archive.
3. O. Heaviside, On operators in physical mathematics, parts I and II, Proc. Roy. Soc. London 52 (1892–93) and 54 (1893).
4. T. J. I’A. Bromwich, Normal coordinates in dynamical systems, Proc. London Math. Soc. 15 (1916).
5. J. R. Carson, Electric Circuit Theory and the Operational Calculus, McGraw-Hill, 1926.
6. I. Yavetz, From Obscurity to Enigma: The Work of Oliver Heaviside, 1872–1889, Birkhäuser, 1995.
7. J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999: the Poynting theorem, fields of moving charges and Cherenkov radiation.