The creation of the metric system following the complete destruction of the traditional-imperial French unit system marks the beginning of a series of events that eventually lead to the currently accepted International System of Units. The great German mathematician Carl Friedrich Gauss (1777–1855) was the first to promote the idea of combining metric units with the second to form a complete and consistent unit system for mechanics. With the help of the German physicist Wilhelm Weber (1804–1891), he was able to extend this concept to include the units for electricity and magnetism. What came to be known as the Gaussian System of Units arose from this proposal. Its organization served as a model for the International System.
The International System of Units (called Le Système international d'unités in French and abbreviated as SI by international convention) arose during the Eleventh General Conference of Weights and Measures (Conférence générale des poids et mesures or CGPM) conducted by the International Bureau of Weights and Measures (Bureau international des poids et mesures or BIPM) in Paris in 1960. The SI model has four major components.
- Seven defining constants (or reference constants) with exact values. These constants may be universal constants that arise from fundamental physical laws (Planck constant h, Boltzmann constant k, speed of light c), be connected to natural phenomena (hyperfine transition of a cesium atom ΔνCs, charge on a proton e), or have evolved from previous definitions of the base units (Avogadro constant NA, luminous efficacy of green light Kcd).
- Seven dimensionally independent base units (or fundamental units) defined in terms of the defining constants that are assumed irreducible by convention (second, meter, kilogram, coulomb, kelvin, mole, and candela).
- A large number of derived units formed by combining base units according to the algebraic relations of the corresponding quantities, some of which are assigned special names and symbols and which themselves can be further combined to form even more derived units.
- The derived units are coherent in the sense that they are all mutually related only by the rules of multiplication and division with no numerical factor other than 1 needed.
- The derived units are also complete in the sense that one and only one unit exists for every defined physical quantity. Although it is possible to express many units in more than one way, they are all equivalent. (The converse statement is not necessarily true, however. Some derived units are used for more than one physical quantity.)
- Twenty-four currently agreed upon prefixes that can be attached to any of the base units or derived units with special names creating multiples and division as needed. (The exception to this rule is the kilogram, which is already itself a multiple of the gram. In this case, prefixes should be added to the word gram.)
- The first three named multiples are the first three powers of ten (101, 102, 103).
Subsequent named multiples are larger than the previous named multiple by three orders of magnitude (106, 109, 1012, … ).
- The first three named divisions are the first three negative powers of ten (10−1, 10−2, 10−3).
Subsequent named divisions are smaller than the previous named division by three orders of magnitude (10−6, 10−9, 10−12, … ).
- The first three named multiples are the first three powers of ten (101, 102, 103).
Each of these components are described in greater detail below.
The defining constants listed below are used to build up the base units of the International System.
- hyperfine transition
- The unperturbed ground state hyperfine transition frequency of the cesium 133 atom (ΔνCs) is defined to be exactly9 192 631 770hertz. This definition can be used to to define the second [s] since the hertz [Hz] is an inverse second [Hz = 1/s]. The second is the base unit of time in the SI. Read the symbol ΔνCs as "delta nu cesium". This book avoids the use of the Greek letter ν (nu) as it is hard to distinguish from the Latin letter v (vee).
- speed of light
- The speed of electromagnetic radiation in vacuum (c) is defined to be exactly299 792 458meters per second [m/s]. This definition combined with the definition of the second can be used to define the meter [m], which is the base unit of length or distance in the SI.
- Planck's constant
- Planck's constant (h) is defined to be exactly6.626 070 15
× 10−34 joule seconds, where the joule [J] is a unit of energy equivalent to a kilogram meter squared per second squared [kg m2/s2]. Planck's constant is therefore a universal constant that relates mass, length, and time. This definition combined with the definitions of the second and the meter can be used to define the kilogram [kg], which is the base unit of mass in the SI.
- elementary charge
- The elementary charge (e) is defined to be exactly1.602 176 634
× 10−19 coulombs. The elementary charge is the magnitude of the charge on many subatomic particles. For example, the charge on a proton is +1 e and the charge on an electron is −1 e. This definition combined with the definition of a second can be used to define the ampère [A] since a coulomb [C] is the amount of charge transfered by one ampère of current in one second [C = A s]. The ampère is the base unit of electric current in the SI.
- Boltzmann's constant
- Boltzmann's constant (k) is defined to be exactly1.380 649
× 10−23 joule per kelvin. Boltzmann's constant is a universal constant that relates energy to temperature. Since the joule is related to the SI units of mass, length, and time [J = kg m2/s2] Boltzmann's constant can be used to define the kelvin [K], which is the base unit of temperature in the SI.
- Avogadro's constant
- Avogadro's constant (NA) is defined to be exactly6.022 140 76 × 1023 particles per mole. Avogadro's constant evolved over the history of chemistry and is not a universal constant. (I would call it a cultural constant.) It is a way to count the microscopic particles of chemistry and statistical thermodynamics (particles being things like molecules, atoms, ions, electrons, etc.). The unit that evolved out of this is the mole [mol], which is the base unit of number of particles in the SI.
- luminous efficacy
- The luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz (Kcd) is defined to be exactly 683 lumen per watt. The luminous efficacy is another example of a constant that evolved out of the cultural practice of humans doing science. It is a way to quantify how well a light source produces visible light — a concept similar to efficiency. Efficiency is for machines using energy to do work. Efficacy is for light sources using energy to make light. The watt [W] is a unit of power derived from the kilogram, meter, and second [W = kg m2/s3]. The lumen is a unit of luminous flux derived from the candela and the steradian [lm = cd sr]. A steradian is a unit of solid angle that is a unitless ratio of areas [m2/m2]. This simplified but still tortuous chain of logic can be used to define the candela [cd], which is the base unit of luminous intensity in the SI.
|ΔνCs||9 192 631 770 Hz||the unperturbed ground state hyperfine transition frequency of the cesium 133 atom|
|c||299 792 458 m/s||the speed of light in vacuum|
|h||6.626 070 15 × 10−34 J s||Planck's constant|
|e||1.602 176 634 × 10−19 C||the elementary charge|
|k||1.380 649 × 10−23 J/K||Boltzmann's constant|
|NA||6.022 140 76 × 1023 1/mol||Avogadro's constant|
|Kcd||683 lm/W||the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz|
While it is possible for many units to serve as the building blocks of a system, the following seven were chosen for historical and practical reasons to serve as the base units of the International System of Units.
The SI base unit of time is the second.
It's natural to think of the Sun as the timekeeper of our lives, but when compared to contemporary wristwatches and wall clocks, the Sun isn't the world's best clock. Variations in the speed of the Earth as it orbits the Sun are great enough that a sundial can disagree with a more conventional clock by as much as 16½ minutes and would only be considered accurate four times a year. To correct for this deviation from the ideal, astronomers imagine a fictitious sun that moves across the sky at a constant rate, makes the same number of transits across the sky as the real sun, and is in the same place as the real sun once a year. The time it takes this fictitious sun to make one circuit around the Earth is called a mean solar day. Since there are 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a mean solar day, the original definition of the second was 186,400 of a mean solar day (60 × 60 × 24 = 86,400).
Natural irregularities in the rotation of the Earth limited the accuracy achievable through this definition, however. The rotation of the Earth varies by about 1 part in 108 between May (when it is slowest) and September (when it is fastest). Furthermore, the rate of the Earth's rotation typically decreases by about 1 part in 109 per year as energy is drained away by tidal interactions between the Earth and the Moon.
Greater accuracies could be achieved from the motion of the Earth around the Sun, which is less susceptible to change over the years. The time it takes the Sun to make one complete circuit across the sky from summer solstice to summer solstice is known as a tropical year. In 1960, the CGPM on advice from the International Astronomical Union declared the second to be 131 556 925.974 7 of the tropical year. This meant that a day, which formerly defined the second, was now defined by it. A unit day is now 86,400 seconds long by definition.
The current standard, adopted in 1967, utilizes the incredible regularity at which certain atomic transitions take place. Every elementary particle can be thought of as a tiny bar magnet with a north and south pole. The hyperfine transition of the ground state occurs when the outermost electron of an atom changes its magnetic orientation relative to the nucleus from parallel (pointing in the same direction) to antiparallel (pointing in the opposite direction). A second [s] is now defined as9 192 631 770periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.
Despite the changes made to the International System in 2018, this last definition is now worded differently, but is still practically true.
A cesium oscillator, commonly referred to as an atomic clock, is a device that counts the hyperfine transitions made by a collection of cesium atoms. The most accurate clock of this type is the NIST-F2 located in the Boulder, Colorado laboratories of the National Institute of Standards and Technology. (A duplicate of NIST-F2 called ITCsF2 can be found at the Istituto Nazionale di Ricerca Metrologica in Torino, Italy.) With a fractional uncertainty of less than 0.44 × 10−15, NIST-F2 could run for 72 million years and still tell time correctly to the nearest second.
Although NIST-F2 may be the best cesium atomic clock in the world (and just may well hold this record forever) it is not the best clock in the world. New technologies are in development that use optical frequency standards rather than microwave frequency standards. (ΔνCs = 9.192 631 770 GHz is in the part of the electromagnetic spectrum used for commercial wireless data networks.) Going from roughly 10 gigahertz (1010 Hz) to say 100 terahertz (1014 Hz) is an additional four orders of magnitude or 10,000 times greater precision. When we divide things this finely, it is no longer possible to ignore the effects of local variations in gravity on time. In its next update, the SI definition of a second may look more like a unit of space-time than a unit of time alone.
The meter is the base unit of length in the International System of Units. It was originally defined in 1799 as one ten millionth of the distance from the equator to the north pole as measured on a line of longitude passing through Paris. This would then fix the circumference of the Earth at the reasonably convenient value of 40 million meters. There are two problems with this statement, however. First, although nearly spherical to the eye, the Earth is slightly flattened at the poles and deviates from the geometric ideal by about one part in three hundred. A trip around the equator is 134 km longer that a trip around the poles. Second, a small mistake was made when measuring the equator to pole distance through Paris. Thus the length of the actual meter differs from its expected length by about one part in four thousand.
In 1889, the BIPM in Paris constructed a precisely machined, platinum-iridium bar with two lines etched at opposite ends. The meter was then defined as the distance between these two lines when stored under specified conditions. Regionally authorized metrologists (measurement technicians) would travel to BIPM headquarters and copy the etched lines of the international prototype on to their own platinum-iridium bars to create regional prototypes. These would then be used to create local prototypes, which would then be used to create individual prototypes, which would then be used to calibrate practical measuring devices. Access to the international prototype was strictly controlled so as to keep the wear and tear on it to a minimum and to lessen the possibility of a catastrophic handling error.
Given the inaccessibility of the prototype and the possibility of its accidental or intentional destruction a new kind of standard was called for. The definition of the meter based on the international prototype was replaced by a series of definitions based on experiment. The meter is now effectively indestructible as it is possible to reproduced its experimental definition anywhere in the universe at any time. The original international prototype meter is still enshrined at the BIPM under conditions specified in 1889, but it is unlikely that it will ever have any official duties again.
From 1960 to 1983, the meter was defined as the length equal to 1 650 763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the krypton 86 atom.
In 1983, the definition was changed so that the meter [m] became the length of the path traveled by light in vacuum during a time interval of 1299 792 458 of a second. The change meant that lengths could now be measured as accurately as times are measured — time being the most accurately measured of all physical quantities. This had the additional effect of fixing the speed of light in a vacuum at exactly299 792 458 m/s.
Despite the changes made to the International System in 2018, this last definition is now worded differently, but is still practically true.
gram vs. grave
The gram was the base unit of mass in the metric system, but it is the kilogram (one thousand grams) that plays this role in the International System. A gram was deemed too small to be of much practical use, I suspect. The kilogram was originally defined in 1799 as the mass of one liter (one thousand cubic centimeters) of pure liquid water at 0 °C.
a mass equal to the mass of one liter of pure water under atmospheric pressure and at the temperature of its maximum density, which is approximately 4 °C.
In 1889, the BIPM in Paris constructed a precisely machined, platinum-iridium cylinder from the 1799 definition to serve as the international prototype of the kilogram. Like the international prototype meter (which was constructed at the same time) access to the international prototype kilogram is strictly controlled to reduce wear and tear from normal use and to prevent its accidental or intentional destruction. A number of secondary prototypes exist, scattered around the world in regional standards offices. They differ from the international prototype by no more than one part in 109 and could be pressed into service should something happen to the original.
Despite all the best efforts of the BIPM, the mass of the international prototype kilogram increases by approximately 1 part in 109 per year due to the inevitable accumulation of contaminants on its surface. This error is just at the limit of uncertainty in the measurement used to calibrate the secondary prototypes and is therefore of some significance. For this reason, the international prototype had a mass of one kilogram only after it has been cleaned and washed by a specified method.
The SI unit of electric current is the ampère [A].
QUOTE: Electric units, called "international units", for current and resistance, were introduced by the International Electrical Congress held in Chicago in 1893, and definitions of the "international ampere" and "international ohm" were confirmed by the International Conference in London in 1908. Although it was already obvious on the occasion of the 8th CGPM (1933) that there was a unanimous desire to replace those "international units" by so-called "absolute units", the official decision to abolish them was only taken by the 9th CGPM (1948), which adopted the ampere for the unit of electric current, following a definition proposed by the CIPM (1946, Resolution 2; PV, 20, 129-137): The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 × 10−7 newton per meter of length. It follows that the magnetic constant, μ0, also known as the permeability of free space, is exactly 4π × 10−7 henries per meter, μ0 = 4π × 10−7 H/m. The expression "MKS unit of force" which occurs in the original text of 1946 has been replaced here by "newton", a name adopted for this unit by the 9th CGPM (1948, Resolution 7; CR, 70).
The SI unit of thermodynamic temperature is the kelvin [K].
OLD: The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
DEFINITION OF WATER: This definition refers to water having the isotopic composition defined exactly by the following amount of substance ratios: 0.000 155 76 mole of 2H per mole of 1H, 0.000 379 9 mole of 17O per mole of 16O, and 0.002 005 2 mole of 18O per mole of 16O.
CELSIUS: Because of the manner in which temperature scales used to be defined, it remains common practice to express a thermodynamic temperature, symbol T, in terms of its difference from the reference temperature T0 = 273.15 K, the ice point. This difference is called the Celsius temperature, symbol t, which is defined by the quantity equation: t = T − T0. The unit of Celsius temperature is the degree Celsius, symbol °C, which is by definition equal in magnitude to the kelvin. A difference or interval of temperature may be expressed in kelvins or in degrees Celsius (13th CGPM, 1967/68, Resolution 3, mentioned above), the numerical value of the temperature difference being the same. However, the numerical value of a Celsius temperature expressed in degrees Celsius is related to the numerical value of the thermodynamic temperature expressed in kelvins by the relation t/°C = T/K − 273.15.
amount of substance
The SI unit of amount of substance is the mole [mol]. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles or specified groups of such particles.
HISTORY: Following the discovery of the fundamental laws of chemistry, units called, for example, "gram-atom" and "gram-molecule", were used to specify amounts of chemical elements or compounds. These units had a direct connection with "atomic weights" and "molecular weights", which are in fact relative masses. "Atomic weights" were originally referred to the atomic weight of oxygen, by general agreement taken as 16. But whereas physicists separated the isotopes in a mass spectrometer and attributed the value 16 to one of the isotopes of oxygen, chemists attributed the same value to the (slightly variable) mixture of isotopes 16, 17 and 18, which was for them the naturally occurring element oxygen. Finally an agreement between the International Union of Pure and Applied Physics (IUPAP) and the International Union of Pure and Applied Chemistry (IUPAC) brought this duality to an end in 1959/60. Physicists and chemists have ever since agreed to assign the value 12, exactly, to the so-called atomic weight of the isotope of carbon with mass number 12 (carbon 12, 12C), correctly called the relative atomic mass Ar(12C). The unified scale thus obtained gives the relative atomic and molecular masses, also known as the atomic and molecular weights, respectively. Following the discovery of the fundamental laws of chemistry, units called, for example, "gram-atom" and "gram-molecule", were used to specify amounts of chemical elements or compounds. These units had a direct connection with "atomic weights" and "molecular weights", which are in fact relative masses. "Atomic weights" were originally referred to the atomic weight of oxygen, by general agreement taken as 16. But whereas physicists separated the isotopes in a mass spectrometer and attributed the value 16 to one of the isotopes of oxygen, chemists attributed the same value to the (slightly variable) mixture of isotopes 16, 17 and 18, which was for them the naturally occurring element oxygen. Finally an agreement between the International Union of Pure and Applied Physics (IUPAP) and the International Union of Pure and Applied Chemistry (IUPAC) brought this duality to an end in 1959/60. Physicists and chemists have ever since agreed to assign the value 12, exactly, to the so-called atomic weight of the isotope of carbon with mass number 12 (carbon 12, 12C), correctly called the relative atomic mass Ar(12C). The unified scale thus obtained gives the relative atomic and molecular masses, also known as the atomic and molecular weights, respectively. The quantity used by chemists to specify proportionality constant being a universal constant which is the same for all samples. The unit of amount of substance is called the mole, symbol mol, and the mole is defined by specifying the mass of carbon 12 that constitutes one mole of carbon 12 atoms. By international agreement this was fixed at 0.012 kg, i.e. 12 g.
HISTORY: The units of luminous intensity based on flame or incandescent filament standards in use in various countries before 1948 were replaced initially by the "new candle" based on the luminance of a Planck radiator (a black body) at the temperature of freezing platinum. This modification had been prepared by the International Commission on Illumination (CIE) and by the CIPM before 1937, and the decision was promulgated by the CIPM in 1946. It was then ratified in 1948 by the 9th CGPM which adopted a new international name for this unit, the candela, symbol cd; in 1967.
OLD: The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.
The SI unit of luminous intensity is the candela [cd]. The lumen [lm] is a unit of luminous flux, which is a measure of the total visible light output from some source over all directions. This is the hard part. How much that light is concentrated in any particular direction is called luminous intensity. The amount of spreading is measured using the solid angle unit called the steradian [sr], which is a measure of the fraction of a sphere cut out by a solid angle. A steradian is then a ratio of areas with units that cancel out [m2/m2]. One lumen spread out over one steradian is called a candela [cd = lm/sr], which is the base unit of luminous intensity in the SI.
A large number of derived units formed by combining base units according to the algebraic relations of the corresponding quantities, some of which are assigned special names and symbols and which themselves can be further combined to form even more derived units.
The derived units are coherent in the sense that they are all mutually related only by the rules of multiplication and division with no numerical factor other than 1 needed.
The derived units are also complete in the sense that one and only one unit exists for every defined physical quantity. Although it is possible to express many units in more than one way, they are all equivalent. The converse statement is not necessarily true, however. Some units are used for more than one physical quantity.
- N m is used for energy (where it is called a joule) and torque (where it is called a newton meter)
- 1/s is used for frequency (cycles per second or hertz), angular frequency (radian per second), and becquerel (decays per second)
- J/kg is used in radiology for absorbed dose (gray) and equivalent dose (sievert)
|in terms of…|
|quantity||name||symbol||other units||base units|
|pressure, stress||pascal||Pa||N/m2||kg/m s2|
|energy, work, heat||joule||J||N m||kg m2/s2|
|power, heat flow||watt||W||J/s||kg m2/s3|
|electric charge||coulomb||C||A s|
|electric potential||volt||V||W/A||kg m2/A s3|
|capacitance||farad||F||C/V||A2 s4/kg m2|
|resistance||ohm||Ω||V/A||kg m2/A2 s3|
|conductance||siemens||S||A/V||A2 s3/kg m2|
|magnetic flux||weber||Wb||V s||kg m2/A s2|
|magnetic flux density||tesla||T||Wb/m2||kg/A s2|
|inductance||henry||H||Wb/A||kg m2/A2 s2|
|Celsius temperature||degree Celsius||°C||K|
|luminous flux||lumen||lm||cd sr||cd m2/m2|
Twenty-four currently agreed upon prefixes that can be attached to any of the base units or derived units with special names creating multiples and division as needed. (The exception to this rule is the kilogram, which is already itself a multiple of the gram. In this case, prefixes should be added to the word gram.)
The first three named mutiples are the first three powers of ten (101, 102, 103). Subsequent named multiples are larger than the previous named multiple by three orders of magnitude (106, 109, 1012, … ).
The first three named divisions are the first three negative powers of ten (10−1, 10−2, 10−3). Subsequent named divisions are smaller than the previous named division by three orders of magnitude (10−6, 10−9, 10−12, … ).
Whether one pronounces giga with hard or soft g depends on what one thinks the proper pronunciation is. A shift occurred in the US sometime in the mid 1990s (at about the time the internet took off in popular culture) when "gig" replaced "jig" as the preferred pronunciation. BIPM guidelines don't care how any SI terms are pronounced as long as they're always represented with the proper symbol.
|10−1||deci||d||Latin: ten (decem)|
|10−2||centi||c||Latin: hundred (centum)|
|10−3||milli||m||Latin: thousand (mille)||1000−1|
|10−6||micro||µ||Greek: small (μικρος, mikros)||1000−2|
|10−9||nano||n||Greek: dwarf (νανος, nanos)||1000−3|
|10−12||pico||p||Spanish: small (pico)||1000−4|
|10−15||femto||f||Danish*: fifteen (femten)||1000−5|
|10−18||atto||a||Danish*: eighteen (atten)||1000−6|
|10−21||zepto||z||Latin: seven (septem)||1000−7|
|10−24||yocto||y||Latin: eight (octo)||1000−8|
|10−27||ronto||r||Latin: nine (novem)||1000−9|
|10−30||quecto||q||Latin: ten (decem)||1000−10|
|101||deca†||da||Greek: ten (δεκα, deka)|
|102||hecto||h||Greek: hundred (εκατο, ekato)|
|103||kilo||k||Greek: thousand (χιλια, khilia)||10001|
|106||mega||M||Greek: big (μεγαλος, megalos)||10002|
|109||giga||G||Greek: giant (γιγας, gigas)||10003|
|1012||tera||T||Greek: fourth (τετρατος, tetratos)‡||10004|
|1015||peta||P||Greek: five (πεντε, pente)||10005|
|1018||exa||E||Greek: six (εξι, exi)||10006|
|1021||zetta||Z||Greek: seven (επτα, epta)||10007|
|1024||yotta||Y||Greek: eight (οκτω, okto)||10008|
|1027||ronna||R||Greek: nine (εννεα, ennea)||10009|
|1030||quetta||Q||Greek: ten (δεκα, deka)||100010|
Other scientific, traditional, and practical units and unit systems are still in use and are still useful. The Anglo-American system of units still in official use in the United States today is actually just an extension of the International System. Many of the units that are unique to the system now have definitions that refer to their SI counterparts. An inch of distance is exactly 0.0254 m, for example, and a pound of mass is exactly 0.453 592 37 kg.
|hour||h||3 600 s|
|distance||astronomical unit||au||149 597 870 700 m|
|plane and phase angle||degree||°||(π/180) rad|
|minute||'||(π/10 800) rad|
|unified atomic mass unit†||u||1.660 539 066 60 × 10−27 kg|
|energy||electron volt||eV||1.602 176 634 × 10−19 J|
|natural logarithmic ratio||bel||B||log(x/x0)|
|decadic logarithmic ratio||neper||Np||ln(x/x0)|
Units named after people…
|bel*||B||Alexander Graham Bell||level|
|coulomb||C||Charles-Augustin Coulomb||electric charge|
|degree Celsius||°C||Anders Celsius||celsius temperature|
|gray||Gy||Louis Gray||absorbed dose|
|joule||J||James Joule||energy, work, heat|
|kelvin||K||William Thomson, Lord Kelvin||temperature|
|pascal||Pa||Blaise Pascal||pressure, stress|
|siemens||S||Werner von Siemens||conductance|
|sievert||Sv||Rolf Sievert||equivalent dose|
|tesla||T||Nikola Tesla||magnetic field|
|volt||V||Alessandro Volta||electric potential|
|watt||W||James Watt||power, heat flow|
|weber||Wb||Wilhelm Weber||magnetic flux|