# Magnetic field constant

Physical constant
Surname Magnetic field constant
Formula symbol ${\ displaystyle \ mu _ {0} \,}$
Size type Magnetic permeability
value
SI 1.256 637 062 12 (19)e-6th ${\ displaystyle \ textstyle {\ frac {\ mathrm {N}} {\ mathrm {A ^ {2}}}}}$
Uncertainty  (rel.) 1.5e-10
Relation to other constants
${\ displaystyle \ mu _ {0} = {\ frac {1} {\ varepsilon _ {0} \, c ^ {2}}}}$
Electric field constant speed of light${\ displaystyle \ varepsilon _ {0} \,}$
${\ displaystyle c \,}$
Sources and Notes
Source SI value: CODATA 2018 ( direct link )

The magnetic field constant , also magnetic constant , vacuum permeability or induction constant , is a physical constant that plays a role in the description of magnetic fields . It indicates the ratio of the magnetic flux density to the magnetic field strength in a vacuum. The reciprocal of the magnetic field constant (with a prefactor ) appears as a proportionality constant in the magnetostatic force law . In the International System of Units (SI) the magnetic field constant has the value: ${\ displaystyle \ mu _ {0}}$${\ displaystyle 4 \ pi}$

${\ displaystyle \ mu _ {0} \; = \; 1 {,} 2566 \ ldots \ cdot 10 ^ {- 6} {\ frac {\ mathrm {N}} {\ mathrm {A} ^ {2}} } \; \ approx \; 4 \ pi \ cdot 10 ^ {- 7} {\ frac {\ mathrm {N}} {\ mathrm {A} ^ {2}}}}$

with the units Newton (N) and Ampere (A).

## terminology

Historically, the constant has had different names. Until 1987 one spoke of the "magnetic permeability of the vacuum". Now in physics and electrical engineering it is called the magnetic field constant . ${\ displaystyle \ mu _ {0}}$

In denotes the magnetic field constant and the relative permeability. ${\ displaystyle \ mu = \ mu _ {r} \ mu _ {0}}$${\ displaystyle \ mu _ {0}}$${\ displaystyle \ mu _ {r}}$

## Relationship with other fundamental constants

From the Maxwell equations in the SI there is a simple relationship between the magnetic field constant, the electric field constant and the speed of light : ${\ displaystyle \ varepsilon _ {0}}$ ${\ displaystyle c}$

${\ displaystyle \ mu _ {0} \ varepsilon _ {0} \, c ^ {2} = 1}$
${\ displaystyle \ mu _ {0} = {\ frac {1} {\ varepsilon _ {0} \, c ^ {2}}}}$

## value

Until 2019, the value of the magnetic field constant was determined by the definition of the unit of measurement amperes . According to this definition, two parallel, infinitely long conductors in a vacuum, through which an electric current with a current of 1 ampere flows, exert a force of 2 · 10 −7 Newtons on each other. The exact value of the magnetic field constant of was derived from Ampère's law of force

${\ displaystyle \ mu _ {0} ^ {\ mathrm {old}} = 4 \ pi \ cdot 10 ^ {- 7} \ mathrm {\ frac {N} {A_ {old} ^ {2}}}}$.

As a result of the revision of the SI units resolved by the 26th General Conference on Weights and Measures (CGPM), the ampere has been defined on the basis of the elementary charge and the definition of the second since May 20, 2019 . As a result, the magnetic field constant is now (again) a variable that has to be determined experimentally and is subject to measurement uncertainty . ${\ displaystyle e}$

The exact relationship still applies . Therefore, the fine structure constant (which is subject to measurement uncertainty) and three further, firmly defined natural constants can be determined: ( = Planck constant ) ${\ displaystyle \ mu _ {0} \ varepsilon _ {0} \, c ^ {2} = 1}$${\ displaystyle \ mu _ {0}}$ ${\ displaystyle \ alpha}$${\ displaystyle h}$

${\ displaystyle \ mu _ {0} = {\ frac {2 \ cdot h \ cdot \ alpha} {c \ cdot e ^ {2}}}}$

When it was decided to change the SI units, the value was initially updated with the relative measurement uncertainty of 2.3 · 10 −10 . At the time of entry into force on May 20, 2019, the value was ${\ textstyle \ mu _ {0} = 4 \ pi \ cdot 10 ^ {- 7} \, \ mathrm {N / A ^ {2}}}$${\ displaystyle \ alpha}$

${\ displaystyle \ mu _ {0} \; = \; 1 {,} 256 \, 637 \, 062 \, 12 (19) \ cdot 10 ^ {- 6} {\ frac {\ mathrm {N}} { \ mathrm {A} ^ {2}}}}$

specified. With a relative measurement uncertainty of 1.5 · 10 −10 , it is currently generally recognized as the most accurate available value.

## unit

The unit of is expressed in different SI units depending on the use, e.g. B .: ${\ displaystyle \ mu _ {0}}$

${\ displaystyle \ left [\ mu _ {0} \ right] = {\ frac {\ mathrm {N}} {\ mathrm {A} ^ {2}}} = {\ frac {\ mathrm {V \, s }} {\ mathrm {A \, m}}} ​​= {\ frac {\ mathrm {kg \, m}} {\ mathrm {A ^ {2} \, s ^ ​​{2}}}}}$

## Individual evidence

1. SUNAMCO Commission: Recommended values of the fundamental physical constants . In: Symbols, Units, Nomenclature and Fundamental Constants in Physics . 1987, p. 52–61 , here p. 54 (English, metrology.files.wordpress.com [PDF; 624 kB ; accessed on August 14, 2018]).
2. ^ Leaflet: The legal units in Germany . June 2015 ( ptb.de [PDF; 1.6 MB ; accessed on August 14, 2018]). ptb.de ( Memento of the original from October 10, 2017 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.
3. 26th CGPM (2018) - Resolutions adopted / Résolutions adoptées. (PDF; 1.2 MB) Versailles 13–16 November 2018. In: bipm.org. Bureau International des Poids et Mesures, November 19, 2018, pp. 2–5 , accessed on August 12, 2019 (English, French).
4. CODATA Recommended Values. National Institute of Standards and Technology, accessed August 12, 2019 . Value for the magnetic permeability in a vacuum.