Reality → Base → Physics → Modern
In analogy to Newton’s law of gravitation, Coulomb’s law describes the electrostatic force [1] as the product of the electric charges of two bodies, divided by the square of their distance and multiplied by a constant of proportionality, called Coulomb’s constant [2] . The formula is great for understanding electric fields and related forces, but is restricted to static conditions [3] . Moving electric charges (electric currents) cause magnetic fields and related forces. This was first observed by Oersted and shortly thereafter formulated more generally by Ampere in his circuital law [4] . Faraday demonstrated by experiment the reverse: a changing magnetic field causes an electric current in a conductor (his famous law of induction formulates the relation between a changing magnetic flux and the induced electromotive force in a closed circuit) [5] . Based on Ampere’s and Faraday’s work, as well as on Gauss’ equations for electric and magnetic fields, Maxwell developed a groundbreaking electrodynamic theory that accurately describes the close interdependence of changing electric and magnetic fields, their wave nature and propagation in free space with the speed of light, leading to the discovery that light is electromagnetic radiation [6] . This is one of the miraculous achievements of physics, where theoretical work based on experimental observations and measurements led to a discovery of huge implications [7] . In quantum electrodynamics (QED), the wave properties of light (electromagnetic radiation) are explained by quantum mechanical principles attributed to photons (particles) [8] .
In 1900, Planck discovered that the measured intensity distribution of a black body’s thermal (electromagnetic) radiation could only be explained by assuming that energy occurs in small packages (quanta) rather than as a continuum [9] . Einstein then concluded that light also consists of energy quanta or elementary particles (later named photons). Planck's discovery and Einstein's interpretation established the fundamental relationship between frequency and energy of an energy particle [10] . De Broglie postulated that all moving particles, including those that build matter, possess wavelike properties [11] . Schrödinger developed an equation that combines particle and wave aspects and led to the modern atomic model [12] . To the same effect, Heisenberg created a different mathematical model and the uncertainty principle [13] . Further important contributions were made by Pauli [14] and Dirac [15] . Feynman added a new view of quantum mechanics [16] . The inscrutable rules of quantum mechanics have been astonishingly successful in pondering fundamentals of the micro and macro worlds and thereby preparing the ground for today's rapid advances in high-tech industries.
Based on the experimentally demonstrated constancy of the speed of light in vacuum and the principle that the laws of physics must be the same for reference frames that are moving relative to each other at constant velocities, Einstein deduced in his special relativity theory that time and space are not absolute physical quantities. For systems that move at velocities close to the speed of light, time is dilated and length contracted in accordance with the Lorentz transformation [17] . Special relativity is of great importance in particle accelerator experiments, where speeds are very high and the effects of relativity can be observed and measured. A consequence of special relativity is the equivalence of mass and energy as expressed in the famous formula E = m c2 [18] . The general relativity theory extends the principle of relativity (laws of physics independent from reference frame) to accelerated systems, based on the experienced equivalence of inertial and gravitational mass [19] . Einstein deduced that gravitation can be explained by curved spacetime. The theory rests heavily on mathematics [20] and explains several observed cosmic phenomena [21] .
Whereas in gravitation the force is always positive, the Coulomb law gives a negative value for attraction between positively and negatively charged bodies, and a positive value for the repulsion between bodies of same polarity.
The Coulomb constant refers to electrostatic forces in free space (vacuum) and is defined through the electric constant ε0, which in turn can be calculated from the magnetic constant and the speed of light (see Note 6). In a dielectric medium, a material constant is to be applied.
The force exerted by the static field causes a shifting of electric charges in electric conductors (see Electrostatic induction or Influenz). The electrostatic force acting between an electron and a proton is about 40 orders of magnitude stronger than the gravitational force derived from their masses.
The law defines the relation between the magnetic field (represented by circular closed lines around an electric conductor) and an electric current or, more generally, moving electric charges. The law, as one of the Maxwell equations, became an important base of classical electrodynamics (see Sheet for some formulas).
Faraday’s discovery of electromagnetic induction was crucial to the development of the electric industry (generators, motors, and transformers are induction devices) and his related physical work became a foundation of electrodynamics, a branch of theoretical physics that investigates the movement of electric charges, interaction of electric and magnetic fields, and related forces.
The essence of Maxwell’s work on electromagnetism is summarized in four beautifully brief and symmetric equations (the Maxwell equations). These differential equations can be transformed into wave equations. They contain physical parameters (ε0 and μ0) measured in lab experiments with stationary electric and magnetic fields. Mathematical solution of the wave equations yields the speed of propagation of the electromagnetic waves, i.e., the speed of light !
Hertz was first to experimentally produce electromagnetic waves (incidentally, in the VHF/UHF range) that could freely (without wire) travel through space. Maxwell's theory and Hertz's experiential proof were groundbreaking for revolutionary technical/commercial developments (see also Electromagnetic radiation, Note 2).
QED doesn’t make it easier to understand electromagnetism but provides a fascinating theory (based on probability) capable of predicting with great accuracy results verifiable by experiments.
Great reading: Richard P. Feynman (1985), QED The Strange Theory of Light and Matter.
The black body is an ideal absorber (and emitter) of heat (and other electromagnetic) radiation, used for the experimental determination of the relationship between energy, temperature, and wavelength. The observed relations do not match classical thermodynamic theory. Planck succeeded in matching experiment and theory by introducing the constant h into his radiation formula, which led to the conclusion that the observed black body radiation is not continuous but occurs in small packages (quanta). The constant h occurs in the most basic formulas of quantum mechanics (see Sheet).
The energy E of a photon is proportional to its frequency f, where the proportionality factor h is the Planck constant (E = h f). The constant h occurs very frequently in formulas of quantum physics, often in its reduced form ħ (h-bar, h divided by 2π). The formula is central to wave-particle duality.
See also Einstein and light, Note 1.
The de Broglie wavelength λ of a matter particle is inversely proportional to the particle’s momentum p, where the proportionality factor is the Planck constant h (λ = h / p). Matter waves were originally hypothesized by de Broglie for electrons, followed soon by experiments demonstrating wave-like properties of electrons, atoms, and even molecules.
The Schrödinger equation is a wave equation built on the concept of wave-particle duality (see also Schrödinger's Nobel lecture). The solutions of the Schrödinger equation are wave functions that contain complex numbers composed of real and imaginary parts. The square of a wave function is interpreted to signify the probability density of a particle’s location, e.g., showing the probabilities of how an electron is distributed over the space around an atomic nucleus.
See also Cloud of electrons, Note 1.
Heisenberg completely abandoned the notion of visualizable electron circuits and developed abstract matrix mechanics to 'explain' the atomic model (see also Nobel lecture). His model is compatible with Schrödinger's wave model (the matrices use, inter alia, parameters derived from Fourier series). Heisenberg postulated that it is principally impossible to determine the exact location and momentum of a particle at the same time. Due to the wave nature of particles, the product of the standard deviations of momentum and location is at a minimum 1/2 of ħ, as expressed in the uncertainty principle. The formula is used, inter alia, to explain quantum tunnelling (an effect that turns up in modern chip manufacture).
Pauli introduced the notion of spin and formulated the exclusion principle, an important rule that reconciles the modern atomic model with the periodic table of elements (see Sheet).
Dirac formulated an equation that unifies quantum mechanics and special relativity. His work led to the prediction of the positron and antimatter (see also decay, positron emission, and PET).
Feynman elaborated the path integral formulation, a third way of interpreting quantum mechanics (after the wave and matrix formulations). Path integrals are used in quantum electrodynamics and quantum field theory.
See also Feynman's diagrams, Nobel lecture, and book QED.
Lorentz developed the transformation formulas as part of his work on electromagnetic fields. Einstein had the insight that relativity effects (time dilatation and length contraction) always apply to an observer’s reference frame when his frame moves uniformly relative to any other frame. However, if the relative velocity is low compared to the speed of light (which is always true in our normal world), the impact is imperceptibly small (see Sheet).
The formula follows from the conservation of mass, energy, and momentum. It describes a true equivalence, not a transformation. The equivalence is extensively used in particle physics and can be observed and measured in nuclear reactions.
A person inside an accelerated enclosure in 'free space' has no means to establish whether the enclosure is being accelerated or wether it is at rest in a gravitational field.
Einstein's equations describe how spacetime is formed when matter is present. The underlying mathematical concepts include Gaussian curvature, 4-dimensional Minkowski space, Riemannian geometry, and tensor analysis. A solution to the problem of gravitational singularity was found in 1965 by Penrose when he introduced the concept of trapped surfaces (rewarded with half of the 2020 Physics Nobel prize).
Bending or deflection of light is observed in solar eclipses (Eddington’s observation of 1919 made Einstein world famous over night) and in gravitational lensing (Hubble telescope has detected ‘Einstein rings’, the ‘ Einstein cross’, and irregular mirror images of very distant quasars directly behind a galaxy). The measured deflections range from a fraction of an arc second to just a few arc seconds. Gravitational redshift has been measured by the highly sensitive Mößbauer effect (energy difference of absorbed and emitted gamma-rays of radioactive solids, ‘recoil-free’ due to huge mass difference between solid crystal and photon). Gravitational time dilatation is measured with atomic clocks in satellites. To correctly determine a GPS location, satellite clocks have to be slowed down by 38 microseconds per day (or 1 second in 72 years) compared to terrestrial clocks. Special relativity causes the satellite clock, which moves at 14,000 km/h, to go 7 μs/d slower, but general relativity (gravitational time dilatation) causes the 20,000 km distant clock to go 45 μs/d faster, hence the adjustment of 38 μs/d. The perihelion of Mercury is precessing 93 arc minutes in a century, in line with the prediction of general relativity.