Magnetic Permeability Calculator

Background.

Put the same coil, carrying the same current, around air and then around an iron bar, and the magnetic field inside changes enormously. The current sets the magnetising field H, measured in amperes per metre; the material decides how much magnetic flux density B, in tesla, that field produces. Permeability is the ratio between them, μ = B / H, and it is the number that separates a transformer core from a block of wood.

Empty space sets the reference. Its permeability, μ₀, is about 1.25663706 × 10⁻⁶ henries per metre, and dividing any material's permeability by μ₀ gives its relative permeability. Most substances sit extremely close to 1: diamagnetic materials such as water and copper fall a whisker below it, paramagnetic ones such as aluminium a whisker above. Ferromagnetic materials, iron, nickel, cobalt and their alloys, reach into the hundreds or thousands, which is why they are used to concentrate flux in motors, inductors and shielding.

For those ferromagnetic materials the ratio is not a fixed property. Their B–H curve bends as the material approaches saturation, and the flux density depends on the material's magnetic history as well as the present field. A single B and H pair therefore gives the secant permeability at that one operating point: the slope of a straight line from the origin to the point on the curve. Change the field and the answer changes with it, so record the operating point alongside the result.

What is magnetic permeability calculator?

Magnetic permeability is the ratio of magnetic flux density B to magnetic field strength H in a material, measured in henries per metre (equivalently tesla-metres per ampere). It describes how readily a material supports magnetic flux when a magnetising field is applied. The value computed here is the secant, or amplitude, permeability: B divided by H at one point, as opposed to the slope of the curve at that point or the limiting value at very low field.

How to use this calculator.

1. 01Enter the flux density B in tesla. Hall-effect probes and gaussmeters often read in gauss or millitesla: 1 gauss is 0.0001 T and 1 mT is 0.001 T.
2. 02Enter the field strength H in amperes per metre. For a long solenoid or a toroid it is the number of turns times the current divided by the magnetic path length; 400 turns carrying 1 A over 0.25 m gives 1,600 A/m.
3. 03Make sure B and H describe the same point in the same sample at the same moment; for a hysteretic material, note whether the field was rising or falling.
4. 04Read μ in henries per metre, and divide by about 1.2566 × 10⁻⁶ if you want the relative permeability.

The formula.

μ = B ÷ H. B is the magnetic flux density in tesla, the quantity that produces force on moving charges and induces voltage in a coil. H is the magnetic field strength in amperes per metre, the part of the field set by the currents in the windings. Since a tesla per ampere-per-metre equals a henry per metre, the result comes out directly in H/m. The relationship is a straight division, so doubling the measured flux density at the same field doubles the permeability. Relative permeability is μ divided by μ₀, which since the 2019 revision of the SI is a measured constant rather than exactly 4π × 10⁻⁷ H/m, though the two agree to better than one part in a billion.

A worked example.

Frequently asked questions.

References& sources.

1. [1]IEC 60050-121, International Electrotechnical Vocabulary, Electromagnetism (BIBLIOGRAPHIC/PAYWALL).
2. [2]OpenStax, University Physics Volume 2, chapter 12, Sources of Magnetic Fields.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.