Heinrich Hertz

How to use this course

Heinrich Hertz is remembered for one unit and one discovery: the hertz, and the electromagnetic waves he generated and detected in Karlsruhe between 1886 and 1888. Those experiments turned James Clerk Maxwell’s field theory from a British speculation into accepted physics. Hertz did much more in a career of barely fourteen years. His 1882 theory of elastic contact is still the starting point of tribology and indentation testing. His work on evaporation gave the kinetic flux law now named after him and Martin Knudsen. He found the photoelectric effect by accident, and his last book recast mechanics so that force disappears from its foundations.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Formulas are given in modern SI notation. Hertz himself wrote in Gaussian units and, from 1890, in his own symmetric form of Maxwell’s equations.

Timeline

Hamburg, 1857–1875

Heinrich Rudolf Hertz was born in Hamburg on 22 February 1857. His father, Gustav Ferdinand Hertz, was a lawyer who later sat in the Hamburg Senate. The family had converted from Judaism to Lutheranism a generation earlier, and Heinrich was raised a Lutheran. His mother, Anna Elisabeth Pfefferkorn, was the daughter of a Frankfurt physician. Heinrich was the eldest of four sons and a daughter.

At six he entered a private school run by Richard Lange, where he stayed for about ten years under his mother’s close supervision. He was a strong student in every subject. At home he built apparatus with a lathe and other tools. A private tutor taught him technical drawing and, unusually, Arabic. He finished at the Johanneum, Hamburg’s classical grammar school, and passed the Abitur at Easter 1875.

The combination is characteristic. Hertz grew up between the classical education of a professional family and the workshop. Both mattered in Karlsruhe, where he built his own oscillators and described them in unusually clear prose.

From engineering to physics, 1875–1880

Hertz first chose engineering. He spent a year in practical civil-engineering training in Frankfurt am Main, began civil engineering at the Dresden Polytechnic in 1876, and served a year in a railway regiment in Berlin. In October 1877 he went to Munich to continue engineering, but soon wrote to his father that he wanted a scientific career. His father agreed to support him. In Munich he sought the advice of Philipp von Jolly, studied mathematics and mechanics, and took a laboratory course with the meteorologist Wilhelm von Bezold.

In October 1878 he moved to Berlin, then the strongest physics centre in Germany, to study with Hermann von Helmholtz and Gustav Kirchhoff. Helmholtz quickly noticed him. The university had set a prize question, on Helmholtz’s suggestion, asking whether moving electricity has inertia, like a moving mass. Hertz built an experiment with induction coils and showed that any such inertia was extremely small. He won the prize and a gold medal in 1879.

Helmholtz then proposed a much harder problem for the Prussian Academy’s prize: to show experimentally that changing polarization in an insulator has electromagnetic effects, as Maxwell’s theory required. Hertz judged that the effects would be too small to measure with the methods then available and declined to try. The question stayed with him. He returned to it in Karlsruhe seven years later with a fast enough source of oscillation.

Instead he wrote, in about three months, a theoretical thesis on electromagnetic induction in a sphere rotating in a magnetic field. He received his doctorate in March 1880, magna cum laude.

Helmholtz’s assistant and Kiel, 1880–1885

From 1880 to 1883 Hertz was Helmholtz’s assistant at the Berlin Physical Institute. He published about fifteen papers in these three years, on a remarkable spread of topics: electric discharges, the evaporation of liquids, a hygrometer, the hardness of materials and the elastic contact of solids. The contact paper (Lesson 04) grew from observations of Newton’s rings between glass lenses pressed together, and it is the most cited work of his early career.

He also studied the glow discharge in gases. In 1883 he tried to deflect cathode rays with an electric field, saw no deflection, and concluded that the rays were not streams of charged particles. The conclusion was wrong. His deflecting plates sat inside the conducting gas of the discharge, which screened the field, as J. J. Thomson showed in 1897 with a better vacuum.

Promotion in German universities required habilitation. In 1883 Hertz moved to Kiel, habilitated in May with work on the glow discharge, and became a Privatdozent in theoretical physics. Kiel offered little laboratory support, and he turned largely to theory. He studied Maxwell’s treatise closely and in 1884 published a paper showing that the older action-at-a-distance electrodynamics of Wilhelm Weber and Franz Neumann, made consistent, leads to Maxwell’s equations (Lesson 06). He also wrote on the graphical treatment of moist air in the atmosphere and on floating elastic plates.

In 1885 he accepted the chair of physics at the Technische Hochschule in Karlsruhe. His inaugural lecture, on 20 April 1885, dealt with the energy balance of the Earth. Karlsruhe gave him a well-equipped laboratory and the large lecture hall in which he would find electric waves. On 31 July 1886 he married Elisabeth Doll, daughter of the Karlsruhe geodesist Max Doll.

The contact of elastic solids

For the general reader. Press two marbles together and they flatten slightly where they touch. How large is the flat patch, how hard is the pressure at its centre, and how far do the marbles approach? Hertz answered these questions in 1881–82, aged 24. His formulas are used every day to design ball bearings, gears, railway wheels and the probes of atomic force microscopes.

Setting

Two smooth, linearly elastic bodies with radii of curvature \(R_1,R_2\), Young’s moduli \(E_1,E_2\) and Poisson ratios \(\nu_1,\nu_2\) are pressed together by a force \(F\). Assume the contact patch is small compared with the radii and that there is no friction or adhesion. Define the effective radius and modulus

Derivation sketch

Hertz’s insight was to guess the pressure. A semi-ellipsoidal pressure distribution on a half-space produces a surface displacement that is quadratic in \(r\) inside the loaded region. For the distribution above, potential theory gives the normal displacement of the combined surfaces

Before deformation the gap between the surfaces is \(r^2/2R\) near the axis. Contact requires \(u_z(r)=\delta-r^2/2R\) for every \(r\le a\). Matching the \(r^2\) terms gives \(p_0=2aE^*/(\pi R)\), and matching the constants gives \(\delta=\pi a p_0/(2E^*)=a^2/R\). Integrating the pressure,

Hertz compared the theory with experiments on glass lenses, measuring the contact patch through the interference rings it produced. He neglected adhesion, which matters for soft materials; the Johnson–Kendall–Roberts (1971) and Derjaguin–Muller–Toporov (1975) theories add it and reduce to Hertz’s formulas when adhesion vanishes. Hertz also treated general curved bodies, for which the contact patch is an ellipse, and proposed defining hardness by the pressure that first causes permanent deformation.

Evaporation and moist air

For the general reader. How fast can a liquid evaporate? Kinetic theory sets a ceiling: molecules cannot leave the surface faster than vapour molecules would strike it at equilibrium. Hertz measured the evaporation of mercury in 1882 and found the real rate well below that ceiling. Two years later, interested in weather, he drew a chart for the cooling of rising moist air that meteorologists used for decades.

The wall flux

In an ideal gas of number density \(n\) at temperature \(T\), the number of molecules striking unit area per unit time is \(\Phi=\tfrac14n\bar v\), where \(\bar v=\sqrt{8k_BT/\pi m}\) is the mean speed of the Maxwell distribution. With \(p=nk_BT\),

At equilibrium between a liquid and its saturated vapour, the evaporation flux equals the condensation flux. If every molecule that strikes the liquid sticks, the evaporation flux is \(\Phi\) evaluated at the saturation pressure \(p_s(T)\). The net flux into a vapour at pressure \(p\) is then bounded by the difference.

Rising air

A parcel of dry air rising without exchanging heat expands and cools. With \(c_p\,dT=dp/\rho\) and hydrostatic balance \(dp=-\rho g\,dz\), the temperature falls at the dry adiabatic lapse rate \(\Gamma_d=g/c_p\approx9.8\) K/km, and

Once the parcel saturates, condensation releases latent heat and the cooling slows. Hertz’s 1884 paper handled this second stage, and the later stages where water freezes, with a graphical chart from which a meteorologist could read off the state of the parcel. Adiabatic charts of this kind became standard tools of synoptic meteorology.

Maxwell against Weber

For the general reader. In 1880 most German physicists explained electricity by forces acting directly at a distance between moving charges, following Wilhelm Weber. Maxwell’s British theory placed the action in a field filling space and predicted waves travelling at the speed of light. Hertz showed in 1884 that a consistent action-at-a-distance theory turns into Maxwell’s, and in 1890 he wrote Maxwell’s equations in the clean symmetric form we use today.

The field equations

Maxwell’s own presentation of 1873 mixed potentials and fields and ran to twenty equations. Hertz, in 1890, and Oliver Heaviside, independently in the same years, eliminated the potentials and wrote the theory directly in terms of the field vectors. In the introduction to his collected papers Hertz wrote that Maxwell’s theory is Maxwell’s system of equations. The test of the theory was its predictions, not the mechanical pictures behind it.

Why the displacement current

Take the divergence of the last equation without the term \(\partial\mathbf D/\partial t\): it gives \(\nabla\cdot\mathbf J=0\), which contradicts charge conservation \(\nabla\cdot\mathbf J+\partial\rho/\partial t=0\) whenever charge accumulates, as on a capacitor plate. With the term, \(\nabla\cdot(\mathbf J+\partial_t\mathbf D)=\nabla\cdot\mathbf J+\partial_t\rho=0\). In an insulator the displacement current \(\partial\mathbf D/\partial t\) produces a magnetic field exactly as a conduction current does. This was the effect the Berlin prize asked to be detected.

Waves

In vacuum with \(\rho=0\), \(\mathbf J=0\), take the curl of Faraday’s law and use \(\nabla\times\nabla\times\mathbf E=\nabla(\nabla\cdot\mathbf E)-\nabla^2\mathbf E\):

Hertz’s 1884 argument

Hertz started from a principle of the unity of electric force: an electric force produced by induction should act on charges and currents in the same way as one produced by static charges. Applied to Neumann’s and Weber’s electrodynamics, this demands corrections to the induced forces. Each correction produces further induced effects, which need further corrections. Hertz summed the series and found that the result satisfies Maxwell’s field equations, with propagation at the speed \(c\). The paper did not convince him that Maxwell was right; it showed that the question was experimental.

Making and detecting electric waves

For the general reader. In the autumn of 1886, in a Karlsruhe lecture room, Hertz noticed that discharging a coil made a small spark jump in a second, separate coil nearby. He realized he had a source of very fast electrical oscillations and a way to detect them. Over two years he showed that these oscillations sent waves through the room, measured their wavelength, and showed they travel at a finite speed. It was the first deliberate transmission and reception of radio waves.

The oscillator

Hertz’s transmitter was a straight conductor broken by a short spark gap, with large metal spheres or plates at its ends acting as capacitance. A Ruhmkorff induction coil charged the two halves until the gap broke down. While the spark conducted, charge sloshed back and forth between the ends. In circuit terms this is an \(LC\) circuit with a resistance \(R\) supplied by the spark and by radiation.

The formula \(T=2\pi\sqrt{LC}\) for the period goes back to William Thomson in 1853 and had been confirmed for Leyden-jar discharges by Feddersen’s rotating-mirror photographs. Hertz estimated \(L\) and \(C\) for his oscillators and found periods of order \(10^{-8}\) s, about a hundred times faster than anything measured before. The secondary circuit, a wire loop broken by an adjustable micrometer spark gap, responded strongly only when its own natural period matched that of the primary. On 13 November 1886 he observed sparks across a distance of about 1.5 m, and in early December he demonstrated resonance between the two circuits.

Standing waves

In 1888 Hertz placed his oscillator in front of a large zinc sheet on the far wall of the lecture room and moved the detector along the line between them. Reflected waves combine with incident waves to form a standing wave

with nodes of the electric field at the conducting wall and every half-wavelength from it. From the node spacing he found wavelengths of a few metres for his large oscillator. Combined with his calculated frequency, this gave a finite propagation speed of the order of the speed of light.

A horrible conflict

Hertz also sent the oscillations along wires and compared the interference of waves in air with waves on the wire. His wire measurements gave a speed different from the speed in air, which contradicted Maxwell’s theory, and Hertz said so openly. George Francis FitzGerald’s group in Dublin reproduced his results, while Édouard Sarasin and Lucien de la Rive in Geneva, working in a larger space, found equal speeds. The discrepancy came from reflections in Hertz’s room and from errors in his calculated frequency. Later work confirmed that waves on thin wires and in air travel at the same speed \(c\).

The oscillating dipole

For the general reader. After the experiments, Hertz calculated exactly what field his oscillator should produce according to Maxwell. His 1889 paper drew the electric field lines at successive instants and showed loops pinching off the source and moving outward. The pictures, reproduced in nearly every textbook since, showed for the first time how a wave detaches from the circuit that makes it.

The Hertz vector

Hertz reduced the problem to a single vector function. In SI form, given a polarization density \(\mathbf P\), define \(\boldsymbol\Pi\) by

These potentials satisfy the Lorenz gauge automatically, and the fields are

For a point dipole \(\mathbf p(t)=p(t)\hat{\mathbf z}\) at the origin, the retarded solution is \(\boldsymbol\Pi=\dfrac{p(t-r/c)}{4\pi\varepsilon_0 r}\hat{\mathbf z}\).

Near the source, \(kr\ll1\), the \(1/r^3\) terms dominate and the electric field is the static dipole field following \(p(t)\). Far away, \(kr\gg1\), only the \(1/r\) terms survive: \(\mathbf E\) and \(c\mathbf B\) are equal in size, perpendicular to each other and to \(\hat{\mathbf r}\), in phase, and largest at the equator. Averaging the Poynting flux over a sphere gives

The strong \(\omega^4\) dependence explains why Hertz needed oscillations about a hundred million times per second before radiation became detectable across a room.

Rays of electric force and the photoelectric effect

For the general reader. In late 1888 Hertz shortened his waves to about 66 cm and shaped them into beams with metal mirrors. He reflected them, bent them through a prism, and blocked or passed them with a grid of wires, exactly as one does with light. Light was an electromagnetic wave. In the course of the same experiments he noticed that ultraviolet light from one spark made another spark jump more easily, the first observation of the photoelectric effect.

Mirrors, grids and a prism

The transmitter for these experiments was a pair of brass cylinders, each about 13 cm long, with a spark gap between them, placed along the focal line of a cylindrical parabolic mirror of sheet zinc about 2 m high. An identical mirror focused the beam onto a receiving dipole. The wavelength was about 66 cm, a frequency near 450 MHz.

• Reflection. A flat metal sheet reflected the beam with equal angles of incidence and reflection.
• Polarization. The receiver responded only when its dipole was parallel to the transmitter’s. A frame strung with parallel wires stopped the beam when the wires were parallel to the electric field and passed it when they were perpendicular.
• Refraction. A large prism cast from pitch deflected the beam. Hertz deduced a refractive index of about 1.7 for pitch at this wavelength.

The wire grid passes the component perpendicular to the wires because currents driven along the wires reradiate a field that cancels the parallel component. A dipole receiver is itself a polarizer: its response is proportional to the component of \(\mathbf E\) along it.

The effect of ultraviolet light

In 1887 Hertz noticed that the receiver’s spark was shorter when it was shielded from the light of the transmitter spark. By interposing plates of glass, quartz and other materials, he showed that the cause was ultraviolet light falling on the negative electrode of the gap. He reported the effect in a paper of 1887 without attempting an explanation. Wilhelm Hallwachs, Philipp Lenard and others followed it up. Lenard showed in 1902 that the energy of the emitted electrons depends on the frequency of the light and not on its intensity, and Albert Einstein explained this in 1905 with light quanta:

Mechanics without force

For the general reader. Newton’s mechanics rests on force, a concept Hertz found logically murky. In his last book, finished weeks before his death, he built mechanics from space, time and mass alone. Every system moves along the straightest possible path its constraints allow. What we call forces arise from hidden masses in motion that we do not see. The programme did not replace Newton, but its rigour shaped the philosophy of science, and its geometry anticipated ideas later central to relativity.

The fundamental law

Hertz describes a system of \(N\) mass points with masses \(m_i\) and Cartesian coordinates \(x_\alpha\) (\(\alpha=1,\dots,3N\)). He measures displacements with the mass-weighted metric

and allows constraints linear in the velocities, \(\sum_\alpha a_{\rho\alpha}(x)\,\dot x_\alpha=0\). A path is straightest if at each point its curvature in this metric is as small as the constraints permit.

Hertz’s law is equivalent to Gauss’s principle of least constraint for a free system: among all accelerations compatible with the constraints, the actual one minimizes \(\sum m_\alpha\ddot x_\alpha^2\). For holonomic constraints it follows from Hamilton’s principle with \(L=T\). Hertz noted that for nonholonomic systems, such as a ball rolling without slipping, the straightest path is generally not the shortest, so Hamilton’s principle in its usual form fails. This observation about rolling constraints is one of the lasting technical contributions of the book.

Hidden masses

Hertz replaced potential energy by the kinetic energy of concealed motions. Suppose a visible coordinate \(q\) is coupled to a hidden cyclic coordinate \(\varphi\):

Since \(\varphi\) does not appear in \(L\), its momentum \(p_\varphi=b\dot\varphi\) is constant. Routh’s reduction replaces \(L\) by \(R=L-p_\varphi\dot\varphi=\tfrac12a\dot q^2-\dfrac{p_\varphi^2}{2b(q)}\), and the visible motion obeys the Euler–Lagrange equation for \(R\). To an observer who sees only \(q\), the system moves under a potential \(V(q)=p_\varphi^2/2b(q)\), which is the hidden kinetic energy.

Bonn, 1889–1893

Rudolf Clausius died in August 1888, and Hertz, now famous, was called to his chair at Bonn. He arrived in April 1889 and moved into Clausius’s house. In September 1889 he spoke to the German Association of Natural Scientists and Physicians at Heidelberg on the relations between light and electricity, presenting optics as a branch of electromagnetism to a broad audience.

Honours followed: the Matteucci Medal of the Italian Society of Sciences in 1888, and the Rumford Medal of the Royal Society of London in 1890. In 1890 he published his two papers on the fundamental equations of electrodynamics, for bodies at rest and in motion. The first gave the equations their modern symmetric form and did much to win German physicists to Maxwell. The second, on moving bodies, assumed the aether is carried along completely by matter. That assumption conflicts with optical experiments, and the problem was resolved only by Hendrik Lorentz and Einstein.

In the laboratory he returned to cathode rays. In 1892 he showed that they pass through thin films of aluminium and gold. His assistant Philipp Lenard used this to let the rays out of the tube through a thin window, a step toward the discovery of X-rays and of the electron. The Norwegian Vilhelm Bjerknes worked with him in Bonn on resonance in Hertzian oscillators, and later founded dynamic meteorology. In 1892 Hertz gathered his wave papers into a book; D. E. Jones’s English translation, Electric Waves, appeared in 1893. The young Guglielmo Marconi and others soon began turning the waves into wireless telegraphy. Hertz himself saw no practical use for them.

Death and legacy

In 1892 Hertz began to suffer from severe pain and swelling in the nose, ears and throat. He underwent several operations, including surgery on the bone behind the ear, and took cures on the Riviera and at Bad Reichenhall without lasting relief. He continued to work on the mechanics book. He sent the manuscript to the publisher on 3 December 1893, gave his last lecture on 7 December, and died in Bonn on 1 January 1894, aged 36. He was buried in the Ohlsdorf cemetery in Hamburg. His biographer Albrecht Fölsing has suggested that the illness was granulomatosis with polyangiitis, a vasculitis; the diagnosis cannot be confirmed.

Helmholtz, who outlived him by eight months, wrote the preface to Die Prinzipien der Mechanik, published in 1894, and Lenard edited his collected works. Hertz left his wife, Elisabeth, and two daughters, Johanna and Mathilde; Mathilde became a biologist. Because the family had Jewish ancestry, the Nazi regime removed Hertz’s name from public places, and Mathilde lost her post in Berlin. She, her sister and their mother emigrated to England. His nephew Gustav Hertz shared the 1925 Nobel Prize in Physics with James Franck.

Legacy

• Electromagnetism: the experiments of 1886–1888 confirmed Maxwell’s theory and established that light is electromagnetic. Radio, radar and wireless communication descend directly from them.
• Mechanics of materials: Hertzian contact theory underlies the design of bearings and gears and the interpretation of indentation and atomic force microscopy.
• Quantum physics: the photoelectric effect, found by chance in 1887, became one of the founding experiments of the quantum theory of light.
• Philosophy of science: the introduction to the Prinzipien, on scientific theories as images whose consequences must match the consequences of things, influenced Ludwig Boltzmann, Ludwig Wittgenstein and later philosophers of science.
• Honours: the SI unit of frequency, the hertz, adopted by the IEC in 1930 and by the General Conference on Weights and Measures in 1960; the Heinrich Hertz Institute in Berlin, founded in 1928; a lunar crater.

Connections

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

Teachers and advisors

• Hermann von Helmholtz 1878-1883

Students

• Philipp Lenard
• Vilhelm Bjerknes

Influenced by

• Gustav Kirchhoff 1878-1880
• James Clerk Maxwell
• Joseph Henry
• Wilhelm Weber

Influenced

• Albert Einstein
• Ferdinand Braun
• Guglielmo Marconi
• Hendrik Lorentz
• Ludwig Boltzmann
• Nikola Tesla 1889

Colleagues and collaborators

• Oliver Heaviside
• Rudolf Clausius
• George Francis FitzGerald

Rivals

• William Crookes

Family

• Gustav Hertz

Also mentioned in this life

• André-Marie Ampère
• Leonhard Euler
• Michael Faraday
• Joseph von Fraunhofer
• Carl Friedrich Gauss
• William Rowan Hamilton
• J. J. Thomson
• William Thomson, Lord Kelvin
• Joseph-Louis Lagrange
• Étienne-Louis Malus
• Isaac Newton
• Siméon Denis Poisson
• Count Rumford

Fields

• electromagnetism 12 lives
• mechanics 27 lives
• experimental physics 16 lives

On CoursesHub

• Classical Electrodynamics
• Classical Mechanics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Heinrich Rudolf Hertz, MacTutor History of Mathematics.
2. Heinrich Hertz, Wikipedia.
3. Hertz, Heinrich, Neue Deutsche Biographie.
4. Fraunhofer Heinrich Hertz Institute, Biography of Heinrich Hertz.
5. Heinrich Hertz, Encyclopaedia Britannica.
6. A. Fölsing, Heinrich Hertz: Eine Biographie, Hoffmann und Campe, 1997.
7. J. Hertz (ed.), Heinrich Hertz: Erinnerungen, Briefe, Tagebücher, 1927; English edition 1977.
8. J. Z. Buchwald, The Creation of Scientific Effects: Heinrich Hertz and Electric Waves, University of Chicago Press, 1994.

Physics

1. H. Hertz, Über die Berührung fester elastischer Körper, J. reine angew. Math. 92 (1882) 156–171.
2. H. Hertz, Über die Verdunstung der Flüssigkeiten, insbesondere des Quecksilbers, im luftleeren Raume, Ann. Phys. 253 (1882).
3. H. Hertz, Über einen Einfluss des ultravioletten Lichtes auf die electrische Entladung, Ann. Phys. 267 (1887).
4. H. Hertz, Die Kräfte electrischer Schwingungen, behandelt nach der Maxwell’schen Theorie, Ann. Phys. 272 (1889).
5. H. Hertz, Electric Waves, trans. D. E. Jones, Macmillan, 1893.
6. H. Hertz, The Principles of Mechanics Presented in a New Form, trans. D. E. Jones and J. T. Walley, Macmillan, 1899; Dover reprint 1956.
7. J. G. O’Hara, A “horrible conflict with theory” in Heinrich Hertz’s experiments on electromagnetic waves, European Review 15 (2007) 545–559.
8. K. L. Johnson, Contact Mechanics, Cambridge University Press, 1985.
9. J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999, chapter 9.