Experiments: Electromagnetism
- The predictions under test
- Coulomb: the torsion balance (1785)
- Cavendish: the concentric spheres
- Galvanic circuits: Ohm and Kirchhoff
- Ørsted: the current and the needle (1820)
- Ampère: the force between currents (1820)
- Faraday: electromagnetic induction (1831–1832)
- Weber and Kohlrausch: the ratio of the units (1856)
- Hertz: electromagnetic waves (1888)
- Millikan: the quantization of charge (1911–1913)
- Modern repetitions and precision
Electromagnetism is the first branch of physics that was built, experiment by experiment, into a field theory, and this chapter collects the measurements that did the building: Coulomb's torsion balance and Cavendish's null test for the inverse-square law [Coulomb:1785] [Maxwell:1879], Ørsted's deflected needle [Oersted:1820], Ampère's force between currents [Ampere:1820], Faraday's induction [Faraday:1832], the Weber–Kohlrausch measurement that first tied the two systems of units to the speed of light [Weber:1856], Hertz's electromagnetic waves [Hertz:1888], and Millikan's quantization of charge [Millikan:1913]. Each is the evidence for a specific structural claim of Electrostatics and Magnetostatics and of The Maxwell Equations and Electromagnetic Waves and Optics, and each is cross-referenced from the theory it tests.
When written out, every experiment below will use the structured
experiment environment, with apparatus, procedure, observations
(numbers and uncertainties in SI units), interpretation and primary
references given in full. The chapter closes with the modern precision
descendants: laboratory bounds on a deviation from the Coulomb exponent
[Williams:1971] and the resulting photon-mass limits compiled by
the Particle Data Group [Navas:2024].
Experiments: Electromagnetism: all derivations of this chapter are pending.
The predictions under test
[Reserved: the chain of claims this chapter must test, stated before any apparatus: that the electrostatic force is central and inverse-square (Electrostatics); that currents produce and feel magnetic fields with a definite force law (Magnetostatics); that a changing magnetic flux drives an electromotive force; that the completed equations propagate transverse waves at a speed fixed by electric and magnetic measurements alone, the prediction of Maxwell's dynamical theory [Maxwell:1865]; and that charge comes in integer multiples of a smallest unit.]
Coulomb: the torsion balance (1785)
Apparatus and procedure
[Reserved: Coulomb's torsion balance [Coulomb:1785]: the silver suspension wire as a calibrated torque standard, the charged pith balls, the twist-angle protocol for repulsion, and the separate torsion-pendulum timing method of the second memoir for attraction. The instrument is the same one that weighs the gravitational constant in Experiment: The Cavendish Torsion Balance, and the shared metrology of torsion fibres will be treated once, there, and cited from here.]
Observations
[Reserved: the tabulated twist angles versus separation from [Coulomb:1785], converted to SI — forces of order \(10^{-4}\,\mathrm{N}\) at separations of a few \(\mathrm{cm}\) — with Coulomb's own three-point verification that halving the distance quadruples the force, and a modern uncertainty analysis of the instrument's systematics (charge leakage, induction on the enclosure).]
Interpretation
[Reserved: the inference from the data to the force law of Electrostatics, written as an exponent bound \(F\propto r^{-2\pm\delta}\) with Coulomb's own \(\delta\) of order a few percent; why a torsion measurement tests the law only at laboratory separations; and the statement that the modern form of this inference is the photon-mass bound of Section 63.11.2.]
Two small bodies at rest carrying static electric charge act on one another along the line joining them, with a force proportional to the product of the charges and falling as the inverse square of the separation. The torsion balance establishes the exponent directly, by showing that halving the distance quadruples the force, to within a few percent [Coulomb:1785]. The null method establishes it far more sharply: if the exponent is written \(2+\delta\), then and only then does the field inside a closed charged conductor vanish for \(\delta=0\), so a sensitive detector inside a charged shell that registers nothing bounds \(\delta\). That line of experiment runs from Cavendish, who reached \(\abs{\delta}<0.02\) a century before the result was printed [Maxwell:1879], through Plimpton and Lawton at \(2\times 10^{-9}\) [Plimpton:1936], to lock-in detection between concentric icosahedral shells at \(\abs{\delta}\lesssim3\times 10^{-16}\) [Williams:1971]. Read as a photon mass, the same null record gives \(m_{\gamma}<10^{-18}\,\mathrm{eV}/c^{2}\) [Navas:2024].
The Coulomb exponent: derive the interior field of a uniformly charged spherical shell for a force law with exponent \(2+\delta\), show that it vanishes identically only when \(\delta\) is zero, and give the conversion of a measured null voltage between concentric conductors into a bound on \(\delta\). Then relate \(\delta\) to a photon rest mass through the Yukawa form that a massive vector field gives in place of the Coulomb potential.
Cavendish: the concentric spheres
[Reserved: Cavendish's unpublished null experiment, performed around 1772 and printed only in Maxwell's 1879 edition of the electrical researches [Maxwell:1879]: a charged outer sphere, an inner sphere connected and then isolated, and the null reading on the pith-ball electrometer that bounds the exponent deviation to \(\abs{\delta}<0.02\) — a sharper test than [Coulomb:1785], a century earlier than its publication. The null method — zero signal inside a closed conductor if and only if the exponent is exactly \(2\) — is the template for every later test in Section 63.11.1, and its theory is Gauss's law from Electrostatics.]
Galvanic circuits: Ohm and Kirchhoff
[Reserved: the quantitative laws of steady currents: Ohm's measurements with thermocouple sources and calibrated wires and the linear relation of current to electromotive force [Ohm:1827], and Kirchhoff's circuit laws with their field-theoretic justification [Kirchhoff:1845]. The material evidence — galvanometer deflections versus wire length in SI units — and the reduction of circuit theory to the field description of Electrodynamics in Matter are to be written here.]
Ørsted: the current and the needle (1820)
[Reserved: Ørsted's observation that a galvanic current deflects a nearby compass needle, circulating around the wire rather than pointing along it [Oersted:1820] — the first coupling of electricity to magnetism, and the first force in physics that is neither central nor along the line joining its sources. The immediate quantitative sequel of Biot and Savart, measuring the field of a long straight current through the oscillation period of a suspended magnet [Biot:1820], gives the \(1/r\) law of Magnetostatics. Apparatus, procedure and the deflection data in SI units are to be tabulated when written.]
Ampère: the force between currents (1820)
[Reserved: Ampère's current balance and his four null experiments, establishing within weeks of Ørsted's announcement that parallel currents attract and antiparallel currents repel [Ampere:1820], and the mature force law of the 1826 treatise deduced, as its title claims, uniquely from experiment [Ampere:1826]. The force per unit length between long parallel wires — \(2\times 10^{-7}\,\mathrm{N}/\mathrm{m}\) at \(1\,\mathrm{A}\) and \(1\,\mathrm{m}\) — defined the SI ampere until the 2019 redefinition fixed the elementary charge instead [BIPM:2019]; both definitions and the continuity between them belong here, with Magnetostatics carrying the theory.]
Faraday: electromagnetic induction (1831–1832)
Apparatus and procedure
[Reserved: the induction ring — two coils on a soft iron torus — the galvanometer as ballistic charge integrator, and the three protocols of the First Series of the Experimental Researches [Faraday:1832]: switching the primary current, moving a magnet through a coil, and rotating a copper disc in a magnetic field (the homopolar generator).]
Observations
[Reserved: which quantities are tabulated when written — ballistic galvanometer throws proportional to flux change, in SI \(\mathrm{V}\,\mathrm{s}\); the transient nature of the effect (current flows only while the flux changes); the sign reversal on reversal of motion; and the proportionality of integrated electromotive force to the change of flux rather than to its rate history [Faraday:1832].]
A ballistic galvanometer in a circuit linked to a changing magnetic flux gives a throw proportional to the total charge that has passed, and that charge depends only on the total change of flux and on the resistance of the circuit — not on how quickly the change was made. Switching the primary current of an induction ring slowly or abruptly gives the same throw; so does moving a magnet through a coil quickly or slowly [Faraday:1832]. Only the endpoints matter:
Derivation. The circuit has resistance \(R\) and carries the current \(I=\mathcal{E}/R\) driven by the induced electromotive force. Integrating over the whole of the transient, and using the flux rule \(\mathcal{E}=-\dd\Phi_{B}/\dd t\) (The Maxwell Equations),
by the fundamental theorem of calculus. The integrand's time dependence has been integrated away: whatever path \(\Phi_{B}\) took between the two endpoints, and however long it took, only the difference survives. This is Equation (63.1).
The result is what makes the ballistic galvanometer the right instrument, because such an instrument responds to the impulse delivered to its coil — the time integral of the current — rather than to the current itself, provided the transient is short compared with its swing period. It is also why the measurement is a flux meter: Equation (63.1) inverted gives \(\Delta\Phi_{B}=QR\) in \(\mathrm{V}\,\mathrm{s}\), from a charge and a resistance.
∎Interpretation
[Reserved: the induction law as the experimental content of the curl-E Maxwell equation of The Maxwell Equations; Faraday's own field-line language as the origin of the field concept; the motional/transformer distinction and its resolution in the flux rule, whose two mechanisms unify only under the relativity of Lorentz Transformations.]
Weber and Kohlrausch: the ratio of the units (1856)
[Reserved: the measurement that connected electricity to light before Maxwell wrote a field equation: the ratio of the electrodynamic to the electrostatic unit of charge, measured by discharging a Leyden jar of known electrostatic charge through a ballistic galvanometer [Weber:1856], yielding a velocity within a few percent of the speed of light. Maxwell's identification of that velocity with the propagation speed of electromagnetic waves [Maxwell:1865] turned this number into the central prediction that Section 63.9 tests.]
The quantity of electricity on a charged conductor can be measured in two entirely independent ways: electrostatically, from the mechanical force it exerts on another charged body, and electrodynamically, from the deflection produced when it is discharged through a galvanometer whose constant has been fixed magnetically. The two measures are not the same quantity, and their ratio has the dimensions of a velocity. Weber and Kohlrausch measured that ratio by discharging a Leyden jar of known electrostatic charge, and the velocity that came out agreed, to within the accuracy of the experiment, with the speed of light determined optically [Weber:1856] — a number obtained with no light anywhere in the apparatus, and the first quantitative link between electromagnetism and optics [Maxwell:1865].
The ratio of the units: show that the electrostatic and electromagnetic definitions of the unit of charge differ by a factor with the dimensions of a speed, equal in SI terms to the reciprocal square root of the product of the electric and magnetic constants, and hence that this measurement determines exactly the combination that reappears as the propagation speed in the vacuum wave equation. The error budget of the original determination, and its agreement with the contemporary optical values, are to be reconstructed here.
Convection currents: Rowland (1876)
[Reserved: Rowland's demonstration in Helmholtz's laboratory that a moving electrostatic charge produces the same magnetic field as a conduction current [Rowland:1878] — charge in motion is current, with no distinction of mechanism — closing the conceptual gap between Section 63.2 and Section 63.6 and anticipating the relativity of the field decomposition treated in Lorentz Transformations.]
An insulating disc carrying a static electrostatic charge, spun rapidly about its axis, deflects a magnetometer needle exactly as a circular conduction current of the same magnitude would. The deflection reverses with the sense of rotation and with the sign of the charge, and it is proportional to the product of the charge and the angular velocity [Rowland:1878]. There is therefore no distinction of mechanism in the magnetic effect: it is the motion of charge that matters, not whether that motion is conduction along a wire or the carriage of a charge on a moving body.
Derivation. The current density that appears as the source in the field equations is \(\vect{J}=\rho\vect{v}\), and this is a definition of what a current is — charge crossing unit area per unit time — with nothing in it about the mechanism of the motion. A ring of the disc at radius \(r\), carrying surface charge density \(\sigma\) and turning at angular velocity \(\Omega\), therefore constitutes a surface current
whose total, integrated over the radius of a disc of radius \(a\) carrying total charge \(Q=\sigma\pi a^{2}\), is \(I=\int_{0}^{a}\sigma\Omega r\,\dd r=\sigma\Omega a^{2}/2 =Q\Omega/2\pi\) — the whole charge passing any radial line once per revolution, as it must. Since the field equations see only \(\vect{J}\), the field produced is the field of that current, and it is proportional to \(Q\Omega\) and reverses with either factor, as observed.
The result looks trivial and is not: before it, “current” was a property of galvanic circuits, and it was an open question whether the magnetic action belonged to the conduction process itself. What the experiment shows is that the source term of magnetostatics is the convective flux of charge, which is exactly the statement that relativity later makes structural, since a static charge in one frame is a current in another (Lorentz Transformations).
∎Hertz: electromagnetic waves (1888)
Apparatus and procedure
[Reserved: the spark-gap dipole oscillator driven by an induction coil, the resonant loop receiver with its micrometer spark gap, and the standing-wave protocol: reflecting the radiation from a zinc sheet and mapping nodes and antinodes along the laboratory [Hertz:1888].]
Observations
[Reserved: the tabulated node positions giving wavelengths of several metres at an oscillator frequency near \(50\,\mathrm{MHz}\), hence a propagation speed consistent with the optical \(3\times 10^{8}\,\mathrm{m}/\mathrm{s}\); the demonstrations of rectilinear propagation, reflection, refraction in a pitch prism, and polarization by a wire grid [Hertz:1888].]
A spark-gap oscillator radiates a disturbance that crosses the laboratory, is reflected from a metal sheet, and forms with its own reflection a stationary pattern whose nodes and antinodes can be located by carrying a small resonant loop through it and watching its spark appear and vanish. The node spacing gives wavelengths of several metres, which with the oscillator's frequency yields a propagation speed agreeing with the optical speed of light to within the accuracy of the determination. The same radiation is reflected from conductors, refracted by a prism of pitch, and extinguished by a grid of parallel wires held one way and transmitted by it held the other [Hertz:1888]. It is therefore a transverse wave, and it behaves in every tested respect as light does.
Derivation. Only the standing-wave measurement needs a derivation; the rest is direct observation. A conducting sheet cannot support a tangential electric field at its surface, so the reflected wave must cancel the incident one there. Superposing an incident wave travelling towards the sheet at \(x=0\) with its reflection,
which vanishes for all \(t\) wherever \(\sin(kx)=0\), that is at \(x=0,\;\pi/k,\;2\pi/k,\dots\). The nodes are therefore spaced by
so the measured spacing gives \(\lambda=2\Delta x\) with a metre rule and nothing else. The propagation speed then follows from the universal relation \(v=f\lambda\), with \(f\) the frequency of the oscillator, fixed by its capacitance and inductance. The magnetic field has its antinodes where the electric field has its nodes, which is why the receiving loop, sensitive to the changing magnetic flux through it, reads the pattern shifted by a quarter wavelength from a detector sensitive to \(\vect{E}\) — a check on the interpretation, not a complication of it.
∎Interpretation
[Reserved: the verdict — electromagnetic disturbances propagate at finite speed as transverse waves, as [Maxwell:1865] predicted and as Electromagnetic Waves and Optics derives; light is such a wave, and the optical evidence for that identification is the subject of Experiment: Wave Optics. Action-at-a-distance electrodynamics is dead from this table onward.]
Millikan: the quantization of charge (1911–1913)
Apparatus and procedure
[Reserved: the oil-drop chamber: atomizer, parallel plates at kilovolt potentials, the microscope and stopwatch protocol of timed rises and falls of a single drop, and the correction to Stokes drag at low pressure worked out in the 1911 paper [Millikan:1911]. The prehistory — the electron's \(e/m\) from cathode-ray deflection [Thomson:1897] — fixes what the measured \(e\) implies for the electron mass.]
Observations
[Reserved: charge changes of single drops always integer multiples of one unit; the 1913 value \(e=1.592\times 10^{-19}\,\mathrm{C}\) (in modern units, with its uncertainty dominated by the then-current viscosity of air) [Millikan:1913]; the table of drop radii, fall times and inferred charges to be reproduced in SI when written.]
The charge carried by a single oil drop, measured one drop at a time by timing its fall under gravity and its rise against gravity in a known electric field, is always an integer multiple of one and the same unit. When a drop captures an ion from the surrounding air its charge changes — and it changes by an integer multiple of that same unit, never by a fraction of it. The unit is independent of the size of the drop, of the substance of which it is made, and of the means by which the ions were produced [Millikan:1911] [Millikan:1913]. Charge is therefore not a continuous quantity, and the elementary unit is that carried by the electron of [Thomson:1897]. Millikan's own value was \(e=1.592\times 10^{-19}\,\mathrm{C}\) [Millikan:1913]; since 2019 the elementary charge has been exact by definition, and its value fixes the SI ampere [BIPM:2019].
Derivation. Let \(W\) be the weight of the drop less the buoyancy of the air. With no field the drop reaches a terminal speed \(v_{f}\) at which the viscous drag balances \(W\); Stokes' law makes the drag proportional to the speed, so
with \(\eta\) the viscosity of the air and \(a\) the radius of the drop. Now apply a vertical field \(E\) strong enough to lift the drop, and let it rise at terminal speed \(v_{r}\); the balance is now \(qE-W=k\,v_{r}\). Adding the two balances eliminates \(W\):
The radius hidden in \(k\) is not an independent unknown: writing \(W=\tfrac{4}{3}\pi a^{3}\Delta\rho\,g\) with \(\Delta\rho\) the density of the oil less that of the air, Equation (63.6) gives \(a=\sqrt{9\eta v_{f}/2\Delta\rho g}\). Every quantity on the right of Equation (63.7) is therefore measured: two speeds obtained with a microscope scale and a stopwatch, a field obtained from a voltage and a plate separation, and tabulated material constants.
What the derivation delivers is a number for each drop; what the experiment delivers is the observation that those numbers cluster on integer multiples, and that a drop which changes its charge does so by jumping between multiples. No step of the derivation assumes or imposes this, which is what makes the quantization an observation rather than a construction. The one systematic that mattered is visible in Equation (63.6): Stokes' law fails when the drop is not large compared with the mean free path of the air molecules, and the correction — together with the value adopted for \(\eta\) — is the source of the difference between Millikan's \(e\) and the modern one.
∎Interpretation
[Reserved: charge is quantized; the elementary unit is carried by the electron of [Thomson:1897]; the viscosity systematic and its later correction as a case study in error analysis for Measurement, SI Units, and the Theory of Errors. Since 2019 the elementary charge is exact by definition [BIPM:2019], and the burden of measurement has moved to the quantities defined through it [Tiesinga:2021].]
Modern repetitions and precision
The Coulomb exponent
[Reserved: the lineage of null tests after Cavendish: Plimpton and Lawton's resonant detection inside charged concentric shells, bounding the exponent deviation at \(2\times 10^{-9}\) [Plimpton:1936], and the Williams–Faller–Hill experiment with concentric icosahedral shells and lock-in detection at \(4\,\mathrm{MHz}\), reaching \(\abs{\delta}\lesssim3\times 10^{-16}\) [Williams:1971]. The geometry of each apparatus and the conversion of a null voltage to an exponent bound are to be written here.]
Photon-mass bounds
[Reserved: the modern reading of every exponent test — a Yukawa deviation from Coulomb's law is a photon mass, the Proca theory of Generalized Classical Field Theory — and the compilation of bounds in the Review of Particle Physics, reaching \(m_{\gamma}<10^{-18}\,\mathrm{eV}/c^{2}\) from solar-wind magnetohydrodynamics [Navas:2024], with the laboratory bound of [Williams:1971] translated into the same units. The plasma physics behind the solar-wind limit lives in Plasmas and Magnetohydrodynamics.]
Charge quantization today
[Reserved: where Millikan's result stands now: the exact elementary charge of the revised SI [BIPM:2019] [Tiesinga:2021]; fractional charges of quarks, never observed free, inferred from deep inelastic scattering (Experiment: Deep Inelastic Scattering); and the fractionally charged quasiparticles of the fractional quantum Hall effect (Experiment: The Quantum Hall Effect), which quantize collective excitations rather than violate charge quantization.]