How to use this course
Hans Christian Ørsted is remembered for one experiment. In 1820 he showed that a wire carrying an electric current deflects a magnetic needle, and so joined electricity and magnetism into one subject. Within months André-Marie Ampère, Jean-Baptiste Biot, Félix Savart and François Arago in Paris had turned the observation into a quantitative science, and within eleven years Michael Faraday had found the converse effect, induction. Ørsted did much more: he measured the compressibility of water, named thermoelectricity, made the first metallic aluminium, coined the word for a thought experiment, and built the institutions of Danish science.
Each technical lesson opens with a paragraph for the general reader, then gives precise statements in SI units, derivations and exercises with folded solutions. The lessons follow his own topics: the 1820 discovery and the field it revealed (Lessons 04 to 06), the thermoelectric experiments with Joseph Fourier (07), the piezometer (08), the acoustic figures that occupied him after 1805 (09), and his chemistry (10).
Timeline
| 1777 | Born 14 August in Rudkøbing on the island of Langeland, Denmark, son of the pharmacist Søren Christian Ørsted. His brother Anders Sandøe follows in 1778. |
| 1794 | Moves to Copenhagen with Anders to prepare for university. |
| 1796 | Gold medal of the university for a prize essay in aesthetics. |
| 1797 | Pharmacist’s examination with the highest grade; gold medal for a medical prize essay on the amniotic fluid. |
| 1799 | Doctorate with a dissertation on the metaphysics of nature in the work of Immanuel Kant. |
| 1800 | Unpaid adjunct in the medical faculty; runs the Lion Pharmacy (Løveapoteket). Begins experiments with Volta’s new pile. |
| 1801–04 | Travels in Germany, the Netherlands and France. Friendship with Johann Wilhelm Ritter in Jena. |
| 1806 | Extraordinary professor of physics at the University of Copenhagen. |
| 1808 | Member of the Royal Danish Academy of Sciences and Letters. |
| 1812 | Ansicht der chemischen Naturgesetze. Second European journey, 1812–14. |
| 1814 | Marries Inger Birgitte Ballum. |
| 1815 | Secretary of the Royal Danish Academy, a post he holds until his death. |
| 1817 | Ordinary (full) professor. |
| 1819 | First to isolate piperine, the pungent substance of black pepper. |
| 1820 | A lecture trial shows a small deflection of a compass needle by a current. July: systematic experiments; 21 July: the Latin memoir Experimenta circa effectum conflictus electrici in acum magneticam. Copley Medal of the Royal Society. |
| 1821 | Foreign Member of the Royal Society of London; Fellow of the Royal Society of Edinburgh. |
| 1822 | Builds a piezometer and measures the compressibility of water. |
| 1822–23 | Journey abroad; thermoelectric experiments with Joseph Fourier in Paris; visit to England. |
| 1824 | Founds the Society for the Dissemination of Natural Science. |
| 1825 | Presents the first metallic aluminium, impure, to the Danish Academy. |
| 1829 | Founds the Polytechnic Institute (Den Polytekniske Læreanstalt) and becomes its first director. |
| 1836 | Publishes the poem Luftskibet (The Airship). |
| 1842 | Prussian order Pour le Mérite for Sciences and Arts. |
| 1850 | November: jubilee for fifty years of service to the university. Aanden i Naturen (The Spirit in Nature) begins to appear. |
| 1851 | Dies 9 March in Copenhagen, aged 73. Buried in Assistens Cemetery. |
Rudkøbing and Copenhagen, 1777–1800
Hans Christian Ørsted was born on 14 August 1777 in Rudkøbing, a small port on the island of Langeland in the south of Denmark. His father, Søren Christian Ørsted, kept the town’s pharmacy. Hans Christian and his brother Anders Sandøe, a year younger, learned largely by themselves. A German wigmaker and his wife taught them German and the catechism, and other townspeople lent books and taught what they knew. From about the age of eleven Hans Christian helped in his father’s laboratory, where he learned practical chemistry by preparing medicines.
The brothers went to Copenhagen in 1794 and passed the university entrance examinations with distinction. They remained close all their lives. Anders became a great jurist and, in 1853–54, prime minister of Denmark.
Pharmacy, prizes and Kant
At the university Ørsted studied pharmacy, natural science and philosophy at once. He won the university’s gold medal in 1796 for an essay in aesthetics and in 1797 for a medical essay on the amniotic fluid, praised for its chemical analyses. The same year he passed the pharmacist’s examination with the highest grade. In 1799 he took his doctorate with a dissertation on the architectonics of the metaphysics of nature, a defence of Immanuel Kant’s dynamical theory of matter. For Kant, matter was not made of hard atoms but was the product of attractive and repulsive forces filling space. This conviction, that forces are primary and that the forces of nature are expressions of a single underlying activity, guided Ørsted’s experiments for the next twenty years.
In 1800 he became an unpaid adjunct in the medical faculty, teaching chemistry to pharmacy students, and managed the Lion Pharmacy (Løveapoteket) in Copenhagen for its owner, Ludvig Manthey. In the same year Alessandro Volta announced his pile, the first source of a steady electric current. Ørsted began experimenting with it at once.
The grand tour, 1801–1804
In 1801 a travel grant from the state allowed Ørsted to spend three years abroad. He visited the scientific centres of Germany, the Netherlands and France, including Berlin and Paris, met chemists, physicists and philosophers, and studied the new galvanic science of the voltaic pile.
The most important meeting was with Johann Wilhelm Ritter in Jena. Ritter was a brilliant and erratic experimenter. He had discovered ultraviolet light in 1801 by its chemical action, and he was convinced, like the philosophers of German Naturphilosophie, that electricity, magnetism, heat, light and chemical affinity were forms of one force. Ørsted became his friend and admirer, translated one of his works into French, and reported his results to the Paris savants. He absorbed Ritter’s belief that electricity and magnetism must be connected. He did not share Ritter’s readiness to see confirmations everywhere, and the French scientists’ coolness toward Ritter’s claims taught him that bold ideas needed careful demonstration.
In Halle he was given a book presenting the sound figures of Ernst Chladni, the patterns that sand forms on a vibrating plate. They seemed to him a visible image of how a force organizes matter, and he took them up after his return (Lesson 09). He came home to Copenhagen in 1804.
Professor in Copenhagen, 1804–1819
Back in Copenhagen, Ørsted gave private lectures on physics and chemistry that drew large audiences. In 1806 the university appointed him extraordinary professor of physics, with the duty of teaching both physics and chemistry. He was the first in Copenhagen to teach physical science as an experimental discipline, and he built up laboratories for it. In the autumn of 1806 he took into his home the young William Christopher Zeise, later the discoverer of one of the first organometallic compounds, as his lecture assistant. In 1808 he became a member of the Royal Danish Academy of Sciences and Letters.
Between 1804 and 1812 he worked out a dynamical theory of chemistry, in which chemical combination came from the meeting of two opposed forces, which he linked to positive and negative electricity. He published it in German in 1812 as Ansicht der chemischen Naturgesetze (View of the chemical laws of nature), with a French translation in 1813. In this period he also used, and named, a method of argument. Around 1812 he wrote of the Gedankenexperiment, the thought experiment, and in 1820 of the Gedankenversuch. He is the first modern author known to have named the method explicitly.
He travelled again in 1812–14. In 1814 he married Inger Birgitte Ballum, and they had several children. In 1815 he became secretary of the Academy, an office he held for the rest of his life, and he gave a university address presenting scientific research as a form of worship, which drew him into a public dispute with the theologian N. F. S. Grundtvig. In 1817 he was made ordinary professor. In 1819 he extracted the alkaloid piperine from black pepper.
Looking for the link
All this time he expected a connection between electricity and magnetism, and in his 1812 book he had speculated about one. The difficulty was to find the right arrangement. Early experimenters, including Ørsted, had looked for an effect of a current along its own direction, or for the effect of static charges on magnets, and found nothing convincing. In the winter of 1819–20 he lectured on electricity, galvanism and magnetism, and while preparing a lecture he reasoned that if a current acts on a magnet at all, it must act sideways, across the wire. Lesson 04 tells what happened next.
1820: the wire and the needle
For the general reader. Hold a wire over a compass and connect it to a battery. The needle swings away from north. Reverse the current and it swings the other way. Put the wire under the compass and the swing reverses again. This simple experiment, published by Ørsted in July 1820, showed that an electric current is surrounded by a magnetic effect, and that the effect goes around the wire rather than along it.
- Reconstruct Ørsted’s observations of 1820 and his reading of them.
- Compute the deflection of a compass needle by a straight current in the Earth’s field.
- Explain why the effect passes through glass and wood but does not act on a brass needle.
From lecture to memoir
By Ørsted’s own later account, he first tried the experiment in front of his students, with a thin platinum wire stretched over a compass, and saw the needle move. The motion was small and confused, and the audience was not impressed. The often-repeated story of an accidental discovery in a lecture misreads this account: he set up the test because he expected an effect across the wire. The trial is usually dated to April 1820. He did not follow it up for some three months.
In early July 1820 he repeated the experiments with a larger battery and with witnesses present. He announced the result in a short Danish note in the Copenhagen journal Dansk Litteratur-Tidende and then, on 21 July, in a four-page Latin memoir, Experimenta circa effectum conflictus electrici in acum magneticam (Experiments on the effect of the electric conflict on the magnetic needle). He had it printed at his own expense and sent copies at once to learned societies and scientists across Europe. Reprints and translations appeared in England, France, Germany and Italy within the year.
The memoir reports that the needle turns in one direction when the wire lies above it and in the other direction when the wire lies below; that reversing the current reverses the deflection; that the effect passes undiminished through glass, metal, wood, water, resin and stone placed between wire and needle; and that needles of brass or glass are not affected. Ørsted called the activity in the wire the electric conflict, a meeting of positive and negative electricity. His conclusion was that the conflict is not confined to the conductor but spreads into the space around it, and that it acts in circles about the wire.
The physics
In modern terms, a steady current \(I\) in a long straight wire produces a magnetic field whose lines are circles centred on the wire. Lesson 05 derives the field from the Biot–Savart law.
At distance \(d\) from a long straight wire carrying current \(I\) in vacuum (or air), the magnetic flux density has magnitude
\[ B=\frac{\mu_0 I}{2\pi d},\qquad \mu_0\approx4\pi\times10^{-7}\ \mathrm{T\,m\,A^{-1}}, \]and is tangent to the circle of radius \(d\) about the wire, in the sense given by the right-hand rule: with the thumb along the current, the fingers curl along \(\mathbf B\).
A compass needle is a magnetic dipole \(\mathbf m\). In a field \(\mathbf B\) it feels a torque \(\boldsymbol\tau=\mathbf m\times\mathbf B\) and comes to rest along \(\mathbf B\). A compass pivoted on a vertical axis responds only to the horizontal part of the field. Let the Earth’s horizontal field \(B_H\) point north, and stretch the wire north–south above the needle at height \(d\), with the current flowing north. Put the wire along \(\hat{\mathbf y}\) (north) and let \(\hat{\mathbf z}\) point up. The unit vector from the wire down to the needle is \(-\hat{\mathbf z}\), and the field there is along \(\hat{\mathbf y}\times(-\hat{\mathbf z})=-\hat{\mathbf x}\), that is, to the west, with magnitude \(B_w=\mu_0I/(2\pi d)\).
A horizontal compass needle a distance \(d\) below a north–south wire carrying current \(I\) northward comes to rest at an angle \(\theta\) west of north, where
\[ \tan\theta=\frac{B_w}{B_H}=\frac{\mu_0 I}{2\pi d\,B_H}. \]If the wire is above the needle with the current southward, or below the needle with the current northward, the needle turns east by the same angle.
The needle aligns with the resultant of two perpendicular horizontal fields, so the angle follows from the parallelogram of fields. Ørsted also placed the wire level with the needle, beside it. There the wire’s field at the needle is vertical, so a compass free to turn only horizontally hardly moves, while a dip needle tilts. This is what one expects of circles around the wire, and it is why earlier searches for an effect along the line of the current had failed.
Glass, wood, water and stone are almost perfectly non-magnetic: their permeability differs from \(\mu_0\) by a few parts per million, so they leave \(\mathbf B\) essentially unchanged. A brass or glass needle has no permanent magnetic moment and only a tiny induced one, so it feels almost no torque. Only ferromagnetic materials, such as the magnetized steel of a compass, respond strongly.
Exercise 4.1A wire carries 5 A northward, 2.0 cm above a compass. In Copenhagen the horizontal geomagnetic field is about \(1.7\times10^{-5}\) T. Find the deflection.
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Exercise 4.2The same compass, with the wire 5.0 cm above it, deflects by \(30^\circ\). What is the current?
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Exercise 4.3Show with the right-hand rule that moving the wire from above the needle to below it, with the same current direction, reverses the deflection. Then explain why a wire running east–west over the needle produces no deflection, only a change in the needle’s period of oscillation.
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The circular field of a current
For the general reader. Ørsted’s memoir gave a qualitative picture: the current’s action goes around the wire. Within three months, physicists in Paris had turned it into laws. Biot and Savart measured how the effect weakens with distance, Pierre-Simon Laplace broke it into contributions from each small piece of wire, and Ampère found a law relating the circulation of the field to the current. Together these give every magnetic field of a steady current.
- State the Biot–Savart law and derive the field of a straight wire and of a circular loop.
- State Ampère’s circuital law and use it for symmetric currents.
- See why the field lines of a straight current close on themselves.
The element law
On 30 October 1820 Biot and Savart reported that the action of a long straight current on a magnetic pole falls off as \(1/d\). Laplace showed that this follows from an element law in which each piece of wire contributes in proportion to \(\sin\theta/r^2\). In vector form:
A steady current \(I\) in a curve \(C\) produces at the point \(\mathbf x\) the field
\[ \mathbf B(\mathbf x)=\frac{\mu_0 I}{4\pi}\oint_C\frac{d\boldsymbol\ell'\times(\mathbf x-\mathbf x')}{|\mathbf x-\mathbf x'|^{3}}. \]Straight segment. Put the wire on the \(z\)-axis and the field point at distance \(s\) from it in the \(xy\)-plane. An element \(dz\) at height \(z\) contributes \(\frac{\mu_0I}{4\pi}\frac{s\,dz}{(s^2+z^2)^{3/2}}\), always in the azimuthal direction \(\hat{\boldsymbol\phi}\). With \(z=s\tan\psi\),
\[ B=\frac{\mu_0I}{4\pi s}\int_{\psi_1}^{\psi_2}\cos\psi\,d\psi=\frac{\mu_0I}{4\pi s}\big(\sin\psi_2-\sin\psi_1\big), \]where \(\psi_1,\psi_2\) are the angles to the ends of the segment, measured from the perpendicular. For an infinite wire \(\psi_{1,2}=\mp\pi/2\) and \(B=\mu_0I/(2\pi s)\), the law of Lesson 04.
The circuital law
For steady currents with density \(\mathbf J\), and any closed curve \(\gamma\) bounding a surface \(S\),
\[ \oint_\gamma\mathbf B\cdot d\boldsymbol\ell=\mu_0\int_S\mathbf J\cdot d\mathbf S=\mu_0I_{\text{enc}},\qquad\text{equivalently}\qquad\nabla\times\mathbf B=\mu_0\mathbf J,\quad \nabla\cdot\mathbf B=0. \]Sketch. Write the Biot–Savart field as \(\mathbf B=\nabla\times\mathbf A\) with \(\mathbf A(\mathbf x)=\frac{\mu_0}{4\pi}\int\frac{\mathbf J(\mathbf x')}{|\mathbf x-\mathbf x'|}d^3x'\), using \(\nabla\frac{1}{|\mathbf x-\mathbf x'|}=-\frac{\mathbf x-\mathbf x'}{|\mathbf x-\mathbf x'|^3}\). Then \(\nabla\cdot\mathbf B=0\) at once, and \(\nabla\times\mathbf B=\nabla(\nabla\cdot\mathbf A)-\nabla^2\mathbf A\). Since \(\nabla^2\frac{1}{4\pi|\mathbf x-\mathbf x'|}=-\delta^3(\mathbf x-\mathbf x')\), the second term gives \(\mu_0\mathbf J\); the first vanishes because \(\nabla\cdot\mathbf J=0\) for steady currents, after an integration by parts. Stokes’s theorem converts the differential form to the integral form. \(\square\)
For a straight wire, symmetry forces \(\mathbf B=B(s)\hat{\boldsymbol\phi}\), and the circuital law on a circle of radius \(s\) gives \(2\pi sB=\mu_0I\) in one line. The equation \(\nabla\cdot\mathbf B=0\) says that magnetic field lines never begin or end. For a straight current they close as circles, exactly Ørsted’s picture.
Exercise 5.1Find the field at the centre of a square loop of side \(a\) carrying current \(I\).
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Exercise 5.2Show that on the axis of a circular loop of radius \(R\), at distance \(z\) from its centre, \(B=\dfrac{\mu_0IR^2}{2(R^2+z^2)^{3/2}}\).
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Exercise 5.3A coaxial cable has an inner solid conductor of radius \(a\) carrying current \(I\) uniformly, and a thin outer sheath of radius \(b\) carrying \(I\) back. Find \(B(s)\) everywhere.
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Measuring currents with needles
For the general reader. Ørsted’s needle was the first meter for electric current. Within months of his memoir, instrument makers wound the wire many times around the needle to multiply the effect, and the galvanometer was born. For the rest of the century, needles and coils were how physicists measured currents, until Gauss and Wilhelm Weber learned to express those measurements in absolute units.
- Understand the multiplier and the tangent galvanometer.
- Find the deflection that gives the smallest relative error.
- Use oscillation periods to compare magnetic fields.
The multiplier
In the autumn of 1820 Johann Schweigger at Halle wound an insulated wire many times around a compass needle, so that the actions of all the turns add. He called the instrument a multiplier. Johann Christian Poggendorff in Berlin built a similar device independently. With a coil, a weak current gives a measurable swing; with a known geometry, the swing gives the current.
A flat circular coil of \(N\) turns and radius \(R\) stands in the magnetic meridian, with a short needle pivoted horizontally at its centre. A current \(I\) deflects the needle by \(\theta\), where
\[ \tan\theta=\frac{\mu_0NI}{2RB_H},\qquad\text{so}\qquad I=K\tan\theta,\quad K=\frac{2RB_H}{\mu_0N}. \]The coil’s field at its centre is \(\mu_0NI/(2R)\), perpendicular to the coil plane and so to the meridian (Exercise 5.2). The needle aligns with the resultant of this field and \(B_H\), exactly as in Lesson 04. The needle must be short compared with \(R\) so that it sits in a nearly uniform field.
Best working angle
From \(I=K\tan\theta\), a reading error \(\delta\theta\) gives \(\delta I=K\sec^2\theta\,\delta\theta\), so the relative error is
\[ \frac{\delta I}{I}=\frac{\sec^2\theta}{\tan\theta}\,\delta\theta=\frac{2\,\delta\theta}{\sin2\theta}, \]smallest at \(\theta=45^\circ\). Good practice was to choose the number of turns so that readings fall near \(45^\circ\).
Oscillations
A needle of magnetic moment \(m\) and moment of inertia \(J\), disturbed from equilibrium in a horizontal field \(B\), feels a restoring torque \(-mB\sin\alpha\approx-mB\alpha\). It oscillates with period
\[ T=2\pi\sqrt{\frac{J}{mB}}. \]Comparing periods compares fields, \(B_1/B_2=T_2^2/T_1^2\), without knowing \(m\) or \(J\). Biot and Savart used this method in 1820 to find the \(1/d\) law. In 1832 Gauss combined oscillations, which measure the product \(mB_H\), with deflections of a second needle, which measure the ratio \(m/B_H\), and obtained \(B_H\) itself in units of length, mass and time. That was the start of absolute electromagnetic measurement, developed by Wilhelm Weber.
Exercise 6.1A tangent galvanometer has 20 turns of radius 10 cm, and \(B_H=1.7\times10^{-5}\) T. Find the current for a \(45^\circ\) deflection.
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Exercise 6.2A needle makes 10 oscillations in 42.0 s in the Earth’s field alone. With a current-carrying wire placed so that its field adds to \(B_H\), 10 oscillations take 30.0 s. Find \(B_w/B_H\).
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Exercise 6.3If the angle can be read to \(\pm0.5^\circ\), what is the relative error in \(I\) at \(10^\circ\), \(45^\circ\) and \(80^\circ\)?
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Thermoelectricity
For the general reader. In 1821 Thomas Johann Seebeck in Berlin joined two different metals in a loop, heated one junction, and saw a compass needle beside the loop turn. He thought heat had produced magnetism directly. Ørsted argued that heat drives an electric current around the loop and that the current, as he had shown, deflects the needle. He named the effect thermoelectricity. In 1823, in Paris, he and Joseph Fourier built the first chains of thermoelectric couples.
- Define the Seebeck emf of a couple and of a chain of couples.
- Compute currents in thermoelectric circuits with internal resistance.
- See how these experiments prepared Ohm’s law.
Couples and chains
Seebeck called his effect thermomagnetism. Ørsted, who had shown that only currents deflect needles in this way, recognized the loop as a circuit with a source of electromotive force at the junctions and coined the term thermoelectric. During his journey of 1822–23 he worked with Fourier in Paris. Their joint paper, Sur quelques nouvelles expériences thermo-électriques, appeared in the Annales de chimie et de physique in 1823. They arranged several couples in a closed ring, heating alternate junctions and cooling the others, and found that the effects add. They concluded that the action depends only on the heating of the junctions. The work took about three weeks; the physicist Kirstine Meyer, who later edited his scientific works, described it as a precursor of Ohm’s law.
For a circuit of two conductors \(A\) and \(B\) whose junctions are held at temperatures \(T_1\) and \(T_2\), the open-circuit emf is
\[ \mathcal E=\int_{T_1}^{T_2}\big(S_A(T)-S_B(T)\big)\,dT\approx S_{AB}\,\Delta T, \]where \(S_A,S_B\) are the absolute Seebeck coefficients. A chain of \(n\) identical couples in series, with alternate junctions at \(T_2\) and \(T_1\), has emf \(n S_{AB}\Delta T\).
The sign convention for \(S\) varies between texts; what matters physically is that the emf of a couple changes sign when the hot and cold junctions are exchanged.
The circuit
Let each couple have internal resistance \(r\), and close the chain through an external resistance \(R\). By Ohm’s law, which Georg Simon Ohm published in 1827,
\[ I=\frac{nS_{AB}\Delta T}{nr+R}. \]If \(R\gg nr\), the current grows in proportion to \(n\); if \(R\ll nr\), it is independent of \(n\). Experimenters before Ohm found both behaviours and could not reconcile them. Thermoelectric couples, with their steady and reproducible emf, were the source Ohm himself used to establish his law.
Exercise 7.1A bismuth–antimony couple has \(S_{AB}\approx110\ \mu\mathrm V\,\mathrm K^{-1}\). Ten couples have their junctions at \(0^\circ\)C and \(100^\circ\)C. Each couple has internal resistance \(0.01\ \Omega\), and the external circuit \(0.1\ \Omega\). Find the emf and the current.
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Exercise 7.2For a fixed chain (\(\mathcal E,\ nr\)), show that the power delivered to the external resistance is greatest when \(R=nr\), and find that power.
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Exercise 7.3You have 12 identical couples (emf \(\varepsilon\), resistance \(r\) each) and a load \(R=3r\). Compare 12 in series; 2 parallel rows of 6; 3 rows of 4; 4 rows of 3. Which gives the largest current?
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The compressibility of water
For the general reader. Is water compressible? Experiments in the seventeenth century had squeezed water in metal spheres until it sweated through, and many concluded that it cannot be compressed at all. In 1822 Ørsted built an instrument, the piezometer, that measured the small change in volume directly. He found that water shrinks by about 45 parts in a million for each additional atmosphere of pressure, within a few per cent of the modern value.
- Define isothermal compressibility and bulk modulus.
- Understand the piezometer and the correction for the vessel.
- Connect compressibility to the speed of sound and to the ocean.
The piezometer
In a piezometer of the kind Ørsted introduced, a glass bulb filled with water ends in a fine graduated tube, closed by a small index of mercury. The bulb stands inside a strong glass cylinder full of water, so that it is pressed equally inside and out. A screw-driven piston raises the pressure in the cylinder. A tube of air closed at the top serves as a manometer: by Boyle’s law, the pressure is inversely proportional to the volume of trapped air. As the pressure rises, the mercury index moves down the graduated tube and measures the shrinkage of the water.
The isothermal compressibility of a substance is \(\kappa_T=-\dfrac{1}{V}\Big(\dfrac{\partial V}{\partial p}\Big)_T\), and its bulk modulus is \(K=1/\kappa_T\). For water at \(20^\circ\)C, \(\kappa_T\approx4.6\times10^{-10}\ \mathrm{Pa^{-1}}\), that is, about \(46\times10^{-6}\) per atmosphere (\(1\ \mathrm{atm}=101\,325\ \mathrm{Pa}\)).
The vessel correction
A solid vessel pressed equally inside and outside shrinks uniformly, as if it were solid glass: its inner volume decreases by the fraction \(\kappa_g\,\Delta p\), where \(\kappa_g\) is the compressibility of glass. The graduated tube therefore records only the apparent change
\[ \frac{\Delta V_{\text{app}}}{V}=-(\kappa_w-\kappa_g)\,\Delta p. \]Glass is about twenty times stiffer than water (\(\kappa_g\approx2\text{–}3\times10^{-11}\ \mathrm{Pa^{-1}}\)), so the correction is a few per cent. The equal pressure on both sides is the important design choice: a vessel pressed only from inside would swell by an amount that depends on its shape and wall thickness, which is much harder to calculate.
Sound
In a fluid, small pressure disturbances travel at \(c=1/\sqrt{\rho\kappa_S}\), where \(\kappa_S\) is the adiabatic compressibility. For water, \(\kappa_S\) and \(\kappa_T\) differ by less than one per cent at room temperature, because the ratio of heat capacities is close to 1. Thus a measurement of compressibility predicts the speed of sound in water, which Jean-Daniel Colladon and Charles Sturm measured in Lake Geneva in 1826.
Exercise 8.1Estimate the speed of sound in water from \(\kappa=4.6\times10^{-10}\ \mathrm{Pa^{-1}}\) and \(\rho=1000\ \mathrm{kg\,m^{-3}}\).
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Exercise 8.2The mean depth of the ocean is about 3700 m. Treating \(\kappa\) as constant, how much higher would sea level stand if water were incompressible?
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Exercise 8.3In a piezometer, the pressure is raised by 10 atm and the apparent shrinkage of the water is \(4.3\times10^{-4}\) of its volume. Taking \(\kappa_g=2.5\times10^{-11}\ \mathrm{Pa^{-1}}\), find the true compressibility of water.
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Sound figures on vibrating plates
For the general reader. Sprinkle sand on a metal plate and stroke its edge with a violin bow. The plate sings, and the sand dances away from the moving parts and gathers along lines that stay still. Ernst Chladni published these figures in 1787. Ørsted took them up after his return from Germany and found something Chladni had not: a very light powder does the opposite, collecting in small heaps on the parts that move most. Michael Faraday explained this in 1831.
- Find the normal modes of a vibrating square membrane and their nodal lines.
- See how degenerate modes combine into Chladni-like figures.
- Understand the difference between sand and light powder.
Ørsted and Chladni
From about 1805 Ørsted studied Chladni figures and the sounds of vibrating plates. He saw in them a visible form of the dynamical philosophy of Lesson 01: an invisible activity arranging matter into patterns. He found that lycopodium powder, the light spores of a club moss, collects not on the nodal lines but in little clouds over the regions of strongest vibration. In 1831 Faraday showed that the vibrating plate sets up air currents, which sweep light particles toward the antinodes; in a vacuum the light powder behaves like sand.
A model: the square membrane
Plates obey a fourth-order equation that makes exact calculation hard. A stretched square membrane, governed by the wave equation, shows the same phenomena with simpler mathematics.
Let \(u(x,y,t)\) solve \(u_{tt}=c^2(u_{xx}+u_{yy})\) on \([0,a]^2\) with \(u=0\) on the boundary. The normal modes are
\[ u_{mn}=\sin\frac{m\pi x}{a}\,\sin\frac{n\pi y}{a}\,\cos\omega_{mn}t,\qquad \omega_{mn}=\frac{c\pi}{a}\sqrt{m^2+n^2},\qquad m,n\ge1. \]Separation of variables, \(u=X(x)Y(y)\cos\omega t\), gives \(X''=-k_x^2X\), \(Y''=-k_y^2Y\) with \(k_x^2+k_y^2=\omega^2/c^2\); the boundary conditions force \(k_x=m\pi/a\), \(k_y=n\pi/a\). The nodal lines of \(u_{mn}\) form a grid. But \(\omega_{mn}=\omega_{nm}\), so any combination \(\alpha u_{mn}+\beta u_{nm}\) is also a mode at the same frequency, and its nodal set can be curved. These degenerate combinations produce the diagonals and curves of real Chladni figures.
Plates
For a thin elastic plate of thickness \(h\), density \(\rho\) and flexural rigidity \(D\), the transverse displacement \(w\) obeys \(D\,\nabla^4w+\rho h\,w_{tt}=0\), the equation Sophie Germain and Gustav Kirchhoff worked out in the nineteenth century. Because the operator is fourth order, frequencies scale as the square of the wavenumber, \(\omega\propto k^2\), and the overtones of a plate are not harmonic. Chladni found empirically that for circular plates the frequency rises roughly as \((m+2n)^2\), where \(m\) counts nodal diameters and \(n\) nodal circles.
Exercise 9.1Find the nodal lines of \(u_{12}+u_{21}\) and \(u_{12}-u_{21}\) on the square \([0,\pi]^2\).
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Exercise 9.2List the first four distinct frequencies of the square membrane as multiples of \(\omega_{11}\), with their degeneracies. Are they harmonic?
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Exercise 9.3A plate’s fundamental is 200 Hz. Using \(\omega\propto k^2\), what happens to the fundamental if the plate is made twice as large in each direction with the same thickness? If made twice as thick?
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Chemistry: aluminium and piperine
For the general reader. Ørsted trained as a pharmacist and thought of himself as a chemist as much as a physicist. He was the first to produce aluminium as a metal, by taking chlorine away from aluminium chloride with potassium. Aluminium is the most abundant metal in the Earth’s crust, yet it was so hard to free from its compounds that for decades it cost more than silver.
- Write the reduction reactions used by Ørsted and Wöhler.
- Understand why aluminium resisted isolation.
- Compare chemical reduction with electrolysis using Faraday’s laws.
The first aluminium
In 1824–25 Ørsted made anhydrous aluminium chloride by passing chlorine over a heated mixture of alumina and carbon. He then heated the chloride with an amalgam of potassium in mercury and distilled off the mercury, leaving small lumps of a metal with the colour and lustre of tin. He presented the result to the Danish Academy in early 1825. The metal was impure and he did not pursue it. In 1827 his friend Friedrich Wöhler in Germany took up the problem with Ørsted’s blessing, reduced the chloride with metallic potassium, and obtained aluminium powder; he obtained the metal in compact form in 1845.
Aluminium binds oxygen very strongly: forming one mole of \(\mathrm{Al_2O_3}\) releases about 1676 kJ. Carbon, which reduces iron ore in a blast furnace, cannot reduce alumina at practical temperatures. Only a more reactive metal, such as potassium or sodium, can take the chlorine or oxygen away. Industrial production became cheap only after 1886, when Charles Martin Hall and Paul Héroult independently found that alumina dissolved in molten cryolite can be electrolysed.
Faraday’s laws of electrolysis
The mass deposited by a charge \(Q\) is \(m=\dfrac{QM}{zF}\), where \(M\) is the molar mass, \(z\) the charge number of the ion and \(F\approx96\,485\ \mathrm{C\,mol^{-1}}\) the Faraday constant. For aluminium, \(z=3\) and \(M=26.98\ \mathrm{g\,mol^{-1}}\).
Piperine
In 1819 Ørsted extracted from black pepper (Piper nigrum) a crystalline substance responsible for its pungency and named it piperine. It is an alkaloid, \(\mathrm{C_{17}H_{19}NO_3}\), an amide of piperic acid and piperidine.
Exercise 10.1How much potassium is needed to make 1.00 g of aluminium by Wöhler’s reaction? (\(M_{\mathrm K}=39.10\ \mathrm{g\,mol^{-1}}\).)
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Exercise 10.2Find the charge needed to deposit 1.00 kg of aluminium by electrolysis, and the electrical energy if the cell runs at 4.5 V.
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Organizer of Danish science, 1820–1850
The memoir of July 1820 made Ørsted famous across Europe within weeks. Arago repeated the experiment in Paris in September, and the work of Ampère, Biot and Savart followed at once. The Royal Society of London awarded him the Copley Medal in 1820, and the French Academy of Sciences gave him a prize of 3000 francs. He became a Fellow of the Royal Society of Edinburgh and a Foreign Member of the Royal Society of London in 1821, and a foreign member of the Royal Swedish Academy of Sciences in 1822. The American Philosophical Society elected him in 1829.
He used his fame for Danish science. On his journey of 1822–23 he worked with Fourier in Paris and visited England, where the institutions for spreading science among a broad public impressed him. In 1824 he founded the Society for the Dissemination of Natural Science (Selskabet for Naturlærens Udbredelse), which organized public lectures across Denmark. In 1829 he founded the Polytechnic Institute, Den Polytekniske Læreanstalt, the ancestor of the Technical University of Denmark, and directed it until his death. He also helped found the bodies that became the Danish Meteorological Institute and the Danish patent office, and as secretary of the Academy for 36 years he shaped its work.
Ørsted cared about the Danish language. He coined some two thousand Danish words, among them ilt (oxygen), brint (hydrogen) and billedkunst (visual art), several of which are still standard. He reviewed literature for a monthly journal from 1829 to 1838, was a friend of the poet Adam Oehlenschläger, encouraged the young Hans Christian Andersen and predicted that his fairy tales would make him famous. In 1836 he published Luftskibet (The Airship), a long poem about the balloon flights of Étienne-Gaspard Robert. In 1842 he received the Prussian order Pour le Mérite for Sciences and Arts.
DiscussionGian Domenico Romagnosi reported in 1802, in an Italian newspaper that went largely unnoticed, an effect of a voltaic pile on a magnetic needle. Why did Ørsted’s 1820 memoir, and not Romagnosi’s note, start electromagnetism? Consider the arrangement of the experiment, the way each result was communicated, and what each author claimed.
The spirit in nature: death and legacy
In November 1850 the university and the city celebrated Ørsted’s fifty years of service with a jubilee. He was then preparing a collection of his philosophical essays, Aanden i Naturen (The Spirit in Nature), which presents natural science, art and morality as converging paths toward knowledge of God, whose reason is expressed in the laws of nature. It appeared in 1850–51 as the first volumes of his collected writings, with a German edition, and in an English translation by Leonora and Joanna B. Horner in 1852 as The Soul in Nature.
Early in March 1851 he caught a cold and declined quickly. He died in Copenhagen on 9 March 1851, aged 73, and was buried in Assistens Cemetery in Copenhagen.
Legacy
- Physics: the link between electricity and magnetism that led, through Ampère, Faraday and Maxwell, to electromagnetic field theory, electric motors, generators and the telegraph.
- Units: the oersted, the CGS unit of magnetic field strength \(H\), adopted in the early 1930s; \(1\ \mathrm{Oe}=1000/(4\pi)\ \mathrm{A\,m^{-1}}\approx79.6\ \mathrm{A\,m^{-1}}\).
- Method: the explicit naming of the thought experiment.
- Institutions: the Technical University of Denmark and the Society for the Dissemination of Natural Science, which has awarded the H. C. Ørsted Medal to Danish scientists since the early twentieth century.
- Honours: the American Association of Physics Teachers awards the Oersted Medal for physics teaching; the first Danish satellite, launched in 1999 to map the Earth’s magnetic field, is called Ørsted; Ørstedsparken in Copenhagen was named for him and his brother in 1879, and his statue was raised there in 1880.
Exercise 12.1A field of 1 Oe in vacuum corresponds to what flux density in tesla? Express the Earth’s horizontal field of Lesson 04 in oersted.
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Final projectWrite a two-page account of the six months from July 1820 to January 1821, as seen from Copenhagen, Geneva, Paris and London. Explain what each of Ørsted, Arago, Ampère, Biot and Savart, and Faraday added, and give one equation from Lessons 04 to 06 that each group could have written down.
Connections
People linked to Hans Christian Ørsted across this collection. A name without a link has no page here yet. See Hans Christian Ørsted in the network of all 337 lives →
Students
- William Christopher Zeise 1806
Influenced by
- Alessandro Volta
- Ernst Chladni
- Immanuel Kant
Influenced
- André-Marie Ampère
- Charles Wheatstone
- François Arago
- Georg Simon Ohm
- Jean-Baptiste Biot
- Joseph Henry
- Michael Faraday
- Johann Schweigger
Colleagues and collaborators
- Friedrich Wöhler 1827
- Joseph Fourier 1823
- Thomas Johann Seebeck
Friends
- Friedrich Wöhler 1827
- Johann Wilhelm Ritter 1801-1810
Family
- Anders Sandøe Ørsted
Also mentioned in this life
- Robert Boyle
- Humphry Davy
- Carl Friedrich Gauss
- Sophie Germain
- Gustav Kirchhoff
- Pierre-Simon Laplace
- James Clerk Maxwell
- George Gabriel Stokes
- Wilhelm Weber
Fields
- electromagnetism 12 lives
- physics 13 lives
- chemistry 21 lives
- acoustics 9 lives
- natural philosophy 12 lives
On CoursesHub
Sources and further reading
Biography
- Hans Christian Ørsted, Wikipedia.
- H.C. Ørsted, Den Store Danske (lex.dk).
- Hans Christian Ørsted, Encyclopaedia Britannica.
- National MagLab, Hans Christian Ørsted, Magnet Academy.
- D. C. Christensen, Hans Christian Ørsted: Reading Nature’s Mind, Oxford University Press, 2013.
- K. Meyer (ed.), H. C. Ørsted: Naturvidenskabelige Skrifter, 3 vols., Copenhagen, 1920.
- H. C. Ørsted, The Soul in Nature, trans. L. and J. B. Horner, London, 1852.
Physics
- H. C. Ørsted, Experimenta circa effectum conflictus electrici in acum magneticam, Copenhagen, 21 July 1820. English translation in Annals of Philosophy 16 (1820), 273–276. Bibliographical note on the first Danish announcement, Lynge & Søn.
- K. Jelved, A. D. Jackson and O. Knudsen (eds. and trans.), Selected Scientific Works of Hans Christian Ørsted, Princeton University Press, 1998.
- J. Fourier and H. C. Ørsted, Sur quelques nouvelles expériences thermo-électriques, Annales de chimie et de physique 22 (1823), 375–389. Bibliographical note.
- M. Faraday, On a peculiar class of acoustical figures, Philosophical Transactions of the Royal Society 121 (1831).
- Thermoelectric effect, Wikipedia (history section).
- J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999, chapter 5.
- D. J. Griffiths, Introduction to Electrodynamics, 4th ed., Cambridge University Press, 2017, chapter 5.