Vacuum permeability, μ₀

The constant that sets the strength of magnetic forces in a vacuum, also called the permeability of free space or the magnetic constant. Until 2019 it was exactly 4π × 10⁻⁷ N/A²; now it is measured, and still very close to that.

μ₀ = 1.25663706127(20) × 10⁻⁶ N/A²

CODATA 2022, measured. Standard uncertainty 0.00000000020 × 10⁻⁶ N/A², which is 1.6 × 10⁻¹⁰ of the value.

Not a row of the physical constants table, which leaves the electromagnetic constants out because their digits moved between the 2018 and 2022 CODATA adjustments.

In other units

Measured values are in CODATA’s concise form: the digits in brackets are the standard uncertainty in the last two places, so 6.67430(15) means 6.67430 ± 0.00015.

Vacuum permeability in SI units and in other units
Quantity	Value
μ₀	1.25663706127(20) × 10⁻⁶ N/A²
μ₀	1.25663706127(20) × 10⁻⁶ H/m
μ₀ ÷ (4π × 10⁻⁷)	0.99999999987(16)

μ₀ in H/m

1 H = 1 V·s/A = 1 N·m/A², so H/m = N/A²
= 1.25663706127(20) × 10⁻⁶ H/m

The same number in henries per metre, the unit inductance calculations use.

μ₀ ÷ (4π × 10⁻⁷)

1.25663706127(20) × 10⁻⁶ ÷ (1.256637061… × 10⁻⁶)
= 0.99999999987(16)

How close the measured value sits to the one that used to be exact.

Where the value comes from

μ₀ follows exactly from the permittivity and the speed of light, μ₀ = 1/(ε₀c²), so this site derives it from ε₀ rather than storing a separate number.

Before 2019 the ampere was defined as the constant current that, flowing in two long parallel wires one metre apart in a vacuum, produces a force of 2 × 10⁻⁷ newtons per metre of length. That definition made μ₀ exactly 4π × 10⁻⁷ N/A². Defining the ampere through e instead left μ₀ to be measured.

CODATA 2022 puts it 1.3 parts in 10¹⁰ below 4π × 10⁻⁷, less than its own uncertainty of 1.6 parts in 10¹⁰. In the 2018 adjustment it sat about 5 parts in 10¹⁰ above, well outside the 1.5 parts in 10¹⁰ uncertainty it had then. So the old figure agrees with the measured value again, within its uncertainty, but it is no longer exact by definition.

μ₀ from ε₀ and c

μ₀ = 1 ÷ (ε₀c²)
= 1 ÷ (8.8541878188(14) × 10⁻¹² × 299,792,458²)
= 1.25663706127(20) × 10⁻⁶ N/A²

Which form to use

For any practical calculation 4π × 10⁻⁷ N/A², 1.2566 × 10⁻⁶, is indistinguishable from the measured value.

Inside a long solenoid the field is B = μ₀nI, where n is the number of turns per metre. A coil of 1000 turns, 0.5 m long, carrying 2 A has n = 2000 per metre and a field of 5.03 mT.

μ₀ also sets the impedance of free space, Z₀ = μ₀c, about 376.73 Ω: the ratio of the electric field to the magnetic field strength in a light wave, which matters in antenna and waveguide design.

Common mistakes

Calling 4π × 10⁻⁷ exact. It was until 2019; now it is an approximation, one that happens to be good to 10 significant figures.
Using μ₀ alone inside a magnetic core. Iron multiplies the field by its relative permeability, which can run into the thousands, so μ₀ by itself describes only an air or vacuum core.
Using the total number of turns for n in B = μ₀nI. n is turns per metre: the turn count divided by the length of the coil.

Tools that use it

Solenoid Magnetic Field Simulator computes the field inside the coil and its inductance with it.
Electromagnetic Induction Simulator computes the flux a magnet puts through each turn of a coil, and so the induced EMF.

Related constants

Vacuum permittivity, ε₀ = 8.8541878188(14) × 10⁻¹² F/m
Coulomb’s constant, kₑ = 8.9875517862(14) × 10⁹ N·m²/C²
Speed of light in vacuum, c = 299,792,458 m/s