Weber
SI derived unit · quantity: magnetic flux
01 · Definition
One weber is the flux which, on vanishing within a second, induces one volt in a single turn. The definition through a change is no accident: the magnetic flux by itself creates nothing, only its change does any work. All power engineering rests on this — the generator, the transformer, the induction hob. The flux equals the induction times the area: a tesla per square metre is exactly a weber.
Hence the difference between the weber and the tesla, the one most often confused. The tesla is flux density, flux per square metre; the weber is the whole flux through a contour entire. A strong neodymium magnet gives more than one tesla at its surface, but the flux through its small cross-section is hundredths of a weber. To gather a whole weber you need either a large area or a strong field: a loop of one square metre in the field of the Earth catches only fifty microwebers.
Two loops in one and the same uniform field see the same induction in tesla but a different flux in webers: it is proportional to the area. From this follows the rule for designing generators too: the emf can be raised either by strengthening the field, or by widening the loop, or by adding turns, or by spinning faster. The first is limited by the saturation of iron at about two tesla, the second and third by the size and resistance of the winding, so industrial machines come up against the fourth and run at high speed.
02 · Conversion
Enter a flux and the passport will break it down by unit
The maxwell is the unit of flux in the CGS system, a hundred million times smaller than the weber. In the old literature on electrical machines all the calculations run in maxwells and gausses; the move to the weber and the tesla happened only in the 1960s. Engineers literally counted the lines of force on a drawing: one line, one maxwell. Hence the expression “a flux of lines”.
In a transformer the flux sets the size: the more volt-seconds the core must withstand, the larger its cross-section. Hence the rule familiar to every designer: a mains transformer at fifty hertz is twice the bulk of one of the same power at a hundred hertz, while a switching supply at a hundred kilohertz gets by with a ferrite ring the size of a fingernail. The cross-section can only be reduced by raising the frequency — otherwise the iron goes into saturation, the flux stops growing with the current, and the winding burns.
| Unit | Name | Value | Where it is met |
|---|---|---|---|
| Wb | weber, the SI unit | 1 | the unit of this sheet |
| mWb | milliweber | 1000 | the working prefix of magnetic calculations |
| µWb | microweber | 1000000 | below this begins the country of SQUIDs |
| nWb | nanoweber | 10⁹ | here a weber-meter still works confidently |
| V·s | volt-second | 1 | the same thing in other words: the weber is the volt-second |
| T·m² | tesla per m² | 1 | the notation is correct, but since 1960 the quantity has a name of its own |
| Φ₀ | flux quantum | 4.836·10¹⁴ | h/2e — a value exact by the 2019 definition |
| Mx | maxwell, CGS | 10⁸ | exactly a hundred million in a weber |
| kMx | kilomaxwell | 100000 | a handy multiple for machine calculations |
| G·cm² | gauss per cm² | 10⁸ | the same as the maxwell: a line on a square centimetre |
| line | line of force | 10⁸ | engineers counted the lines on a drawing one by one |
| Wb | weber | 1 | the unit of this sheet |
The weber is the centre of the magnetic trio: divide by area and you get the tesla, divide by current the henry, divide by time the volt
03 · Orders of magnitude
from the flux quantum to an accelerator magnetOne weber: a large machine, or a strong magnet with a large area
04 · Measuring instruments
What magnetic flux is measured with
A moving-coil frame with a large moment of inertia: during the pulse it has no time to turn, so the first throw is proportional to all the charge that has passed, and that in turn to the change of flux. The classic instrument of magnetic laboratories, displaced by electronic integrators only by the seventies.
An electronic integrator: it adds up the voltage from the search coil over time and gives volt-seconds straight away. Its chief trouble is zero drift: the integrator accumulates the amplifier’s own offset as well, so a measurement begins with a reset and is kept to a few seconds.
Turns of known area are placed in the field and sharply pulled out, or turned through a hundred and eighty degrees. The integrated voltage pulse gives the whole flux that passed through them — the most direct way of measuring the weber straight from the definition of the unit.
A superconducting ring with two Josephson junctions: the flux through it can change only in portions of two femtowebers and a little, and the instrument counts those portions. The most sensitive magnetometer in existence — it catches the magnetic field of a working brain.
05 · Writing rules
Wb — two letters, and the second one lower case
The symbol is written as one piece, and a prefix stands before the whole of it: mWb, µWb. A single capital W nearby is the watt, an entirely different quantity. The flux is denoted by the Greek letter Φ, and its change by dΦ; in the formulas of induction the minus sign is obligatory, it expresses Lenz’s rule.
Flux linkage — the flux multiplied by the number of turns — is measured in the same webers but denoted Ψ. In coil calculations Φ and Ψ must not be confused: the difference runs to hundreds of times. The Russian symbol is Вб. Writing “weber per square metre” is technically correct and equals the tesla, but it should not be used: since 1960 that quantity has had a name of its own.
06 · Neighbouring units
The weber ties magnetism to electricity: divided by area it gives the tesla, divided by current the henry, differentiated with respect to time the volt.
weber per square metre
magnetic flux
weber per ampere
Of the Simetrium passports the weber is adjoined by tesla as the density of that same flux, henry from the definition of inductance, volt s a second, whose product it is, and also maxwell and gauss — their predecessors in the CGS system.
07 · Historical section
What flux was measured in before the weber
Faraday wound two windings on an iron ring and noticed that the galvanometer needle twitched only at the moment the current was switched on and off. A steady field gives nothing — only its change does the work.
Weber, together with Gauss, built the first electromagnetic telegraph and was the first to reduce electrical measurements to mechanical ones. The unit of magnetic flux bears his name.
The International Electrotechnical Commission named the practical unit of flux the weber, a power-of-ten multiple of the maxwell. The General Conference brought it into the system together with the volt, the ampere and the rest of the electrical units.
Deaver and Fairbank at Stanford, and simultaneously Doll and Näbauer in Munich, measured the flux in a superconducting cylinder and found it to be a multiple of a fundamental portion. The two in the denominator confirmed that charge is carried by pairs of electrons.
Flux cannot be poured out in arbitrary fractions — nature measures it out in portions
Magnetic flux does not yield to direct measurement: any instrument responds not to the flux itself but to its change. So the flux is either broken off sharply by pulling the coil out, or the coil is flipped over, or the current in the winding is killed, and then the voltage pulse that arises is integrated. The absolute accuracy of such a measurement is limited by the drift of the integrator and by the knowledge of the area of the turns, and is rarely better than a tenth of a per cent. The quantisation of flux changed the picture: in a superconducting ring the flux can take only values that are multiples of Φ₀ = h/2e, and that portion is known exactly, since the Planck constant and the electron charge have been fixed by numbers since 2019. A SQUID that counts such portions distinguishes a change of flux of a millionth of a quantum — a sensitivity that makes it possible to record the magnetic field of the heart through the chest and of the brain through the skull. The quantum itself is vanishingly small: to gather one weber, almost five hundred trillion of them are needed.
Catalogue · units of measurement
A passport for every quantity
Seven SI base units, twenty-two derived ones with names of their own, and the non-SI units that neither engineering nor daily life can do without. Each gets its own sheet: definition, conversion, instruments, writing rules. In 11 languages.
SI derived units with names of their own
the weber passport is openSI base units
in terracotta — the four units the weber is built fromMagnetic flux in other systems
Passport language
11 languages. The symbol Wb is international, but the habit of measuring in maxwells holds on unevenly: in Russian and Japanese literature on magnetic materials the CGS units are still met today.