The magnetic permeability calculator above derives a material's absolute permeability and relative permeability from a flux density and a magnetising field, and reports the susceptibility, the magnetisation and the magnetic polarisation that follow from the same pair of numbers. It also runs backwards, giving B from a stated relative permeability and H, or H from a relative permeability and B.
Arb Digital publishes free physics calculators, and the boundary here needs stating up front. The live permeability converter rescales a permeability value between units — henries per metre, relative permeability, and separately the fluid-mechanics darcy, which it deliberately keeps apart. It converts a number you already have. This page derives the number in the first place, from a measured B and H pair. The two do different jobs and the converter is the right tool once you have a value in hand.
What This Magnetic Permeability Calculator Does
Permeability is the constant of proportionality between the magnetising field a winding imposes and the flux density that actually appears in the material. A high permeability means a modest drive produces a large flux, which is why transformer and inductor cores are made of soft magnetic materials rather than air.
The tool computes the absolute permeability as the ratio of B to H, divides by the vacuum permeability to get the dimensionless relative permeability, and then gives the three related quantities that describe the material's own contribution: the susceptibility, which is the relative permeability minus one; the magnetisation, which is the material's internal dipole density; and the polarisation, which is the material's contribution to B expressed in tesla.
It handles tesla, millitesla, gauss and kilogauss for flux density, and amperes per metre, amperes per centimetre and oersteds for field strength, so you can work from a datasheet written in either the SI or the CGS convention.
How to Use It
- Choose what you are solving for. The default derives permeability from a measured B and H pair.
- Enter the flux density in tesla, millitesla, gauss or kilogauss. This is the total field inside the material.
- Enter the magnetising field strength in amperes per metre or oersteds. This is what the winding imposes.
- For the reverse modes, enter a relative permeability from the material's own datasheet at the relevant flux level and frequency.
- Read the relative permeability, the absolute permeability, the susceptibility, the magnetisation and the polarisation.
The Formula: How Permeability Is Derived
Absolute permeability is simply the ratio of the two field quantities:
µ = B / H, in henries per metre
and relative permeability is that ratio divided by the permeability of free space:
µr = µ / µ0
The vacuum magnetic permeability is no longer an exactly defined constant. Since the 2019 revision of the SI it is a measured quantity, and the NIST CODATA value for the vacuum magnetic permeability gives it as 1.256 637 061 27 × 10⁻⁶ N A⁻² with a relative standard uncertainty of 1.6 × 10⁻¹⁰. The old exact value of 4π × 10⁻⁷ differs from it only in the tenth significant figure, so for engineering purposes either serves.
The material's own contribution is separated out through the magnetisation M and the susceptibility χ:
B = µ0(H + M) · M = χ H · χ = µr − 1 · J = µ0 M
HyperPhysics at Georgia State University sets out the same decomposition, defining the relative permeability as the scaling factor between µ and µ0 and writing B as µ0 times the sum of H and M. That form makes the physics explicit: B counts both the imposed field and the aligned dipoles the material contributes, while H counts only the imposed part.
Work the defaults through by hand. A flux density of 1.2 T at a magnetising field of 800 A/m. The absolute permeability is 1.2 / 800 = 1.5 × 10⁻³ H/m. Dividing by µ0 gives a relative permeability of 1.5 × 10⁻³ / 1.256 637 × 10⁻⁶ = 1,193.7, so the susceptibility is 1,192.7. The magnetisation is 1,192.7 × 800 = 954,130 A/m, and the polarisation is µ0 times that, which comes to 1.199 T. Notice how close the polarisation is to B itself: the imposed field contributes only µ0 × 800 = 1.005 mT of the total 1.2 T. In a good soft magnetic material almost all of the flux comes from the material, not from the winding.
Why a Single Permeability Figure Is Nearly Always Incomplete
This is the section that matters most in practice. For a diamagnetic or paramagnetic material, permeability really is close to constant and extremely close to one. For the ferromagnetic materials that cores are actually made of, it is not a constant at all — it is the local slope of a curved, hysteretic B–H loop, and it changes with almost everything.
It varies with flux level. Permeability rises from an initial value at low excitation, peaks somewhere in the middle of the curve, then collapses towards that of free space as the material saturates and every domain is already aligned. A figure quoted at 0.1 T can be several times the figure at 1.5 T for the same material.
It depends on which slope you mean. Amplitude permeability is B divided by H at a point, which is what this page computes. Differential permeability is the local gradient dB/dH, which is smaller on the flanks. Incremental permeability is that gradient under a small alternating excitation superimposed on a DC bias, which is what governs an inductor carrying DC. They are different numbers for the same material at the same point.
It varies with temperature and falls to one at the Curie point. Above that temperature the material is simply paramagnetic. Many soft ferrites shift substantially over an ordinary operating range well below it.
It varies with frequency. Domain wall motion cannot follow arbitrarily fast excitation, so permeability rolls off and acquires a loss component, which is why datasheets publish complex permeability against frequency rather than one number.
It depends on history. Hysteresis means the material remembers, so B for a given H differs depending on whether you arrived from above or below.
Effective Permeability and the Air Gap
A practical magnetic circuit rarely sees the material's own permeability. Introduce a gap and the picture changes entirely, because the gap has a permeability of essentially one but sits in series with the core in the magnetic circuit.
A gap of a fraction of a millimetre in a core with a path length of tens of millimetres can dominate the total reluctance, dropping the effective permeability of the assembly by an order of magnitude or more. That is a deliberate design choice, not a defect: gapping linearises the inductance, makes it far less sensitive to the material's own variation, and raises the DC current the part can carry before saturating. The price is a lower inductance for the same turns, and fringing flux around the gap that can couple into nearby conductors.
The practical consequence for this page: if you measure B and H on a gapped assembly, what you get is the effective permeability of the whole magnetic circuit, not the intrinsic permeability of the material. Both are useful numbers, but they answer different questions and should not be quoted interchangeably.
Where This Sits Next to the Other Magnetism Tools
The live permeability converter rescales a permeability value you already have between units, and keeps magnetic permeability strictly separate from the unrelated fluid-mechanics quantity of the same name. The live magnetic field converter handles the field quantities themselves — flux density in tesla and gauss, field strength in amperes per metre and oersteds, and flux in webers and maxwells — which is exactly the conversion you need before entering a CGS datasheet figure here.
For the fields those quantities describe, the solenoid magnetic field calculator gives B inside a coil, the magnetic field of a wire calculator gives B around a conductor, and the magnetic force on a wire calculator and magnetic dipole moment calculator cover what those fields do mechanically. The inductance converter and Faraday's law calculator cover the circuit side.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Treating permeability as a material constant — for a ferromagnetic material it varies with flux level, temperature, frequency and magnetic history, so a single figure describes only one operating point.
- Mixing SI and CGS without converting — gauss and oersted are numerically equal in free space in the CGS system, which hides errors that only appear once you move to tesla and amperes per metre.
- Confusing B and H — H is what the winding imposes and depends only on turns, current and path length; B is what results and depends on the material.
- Quoting an effective permeability as a material property — a measurement on a gapped assembly describes the whole magnetic circuit, and the gap usually dominates it.
- Using amplitude permeability where incremental permeability applies — an inductor with DC bias is governed by the local slope under small signal, which is a smaller and quite different number.
Related Free Tools From Arb Digital
Use the live permeability converter to rescale a value between units and the live magnetic field converter to move B, H and flux between SI and CGS before entering them here. The solenoid magnetic field calculator and magnetic field of a wire calculator give the fields, the magnetic force on a wire calculator and magnetic dipole moment calculator give the mechanical effects, and the inductance converter and Faraday's law calculator cover the circuit side. Everything Arb Digital publishes sits on the free online tools hub.
Frequently Asked Questions
It is the ratio of the magnetic flux density inside a material to the magnetising field strength applied to it, with units of henries per metre. Divided by the permeability of free space it becomes the dimensionless relative permeability, which is 1 for vacuum, very slightly below 1 for diamagnets, slightly above for paramagnets and in the hundreds or thousands for soft magnetic materials.
H is the magnetising field the winding imposes, set by turns, current and magnetic path length, and it does not depend on what the core is made of. B is the flux density that results, which includes the material's own aligned dipoles. Permeability is the ratio between them, and in a good core material almost all of B comes from the material rather than from H.
Because a ferromagnetic material magnetises by aligning domains, and that process is not linear. Permeability starts at an initial value, rises to a peak partway up the curve as domain walls move easily, then falls towards the free-space value as the material saturates and no unaligned domains remain. The same material can differ by a factor of several between low and high excitation.
A converter rescales a permeability value you already have between units. This page derives the value in the first place from a measured flux density and field strength pair. Use this one when you have a B and H measurement or a point off a datasheet curve, and the converter when you have a permeability figure that needs to be in different units.
It is the relative permeability minus one, so it measures only the material's own contribution rather than the total response. It is negative for diamagnetic materials, which slightly oppose an applied field, small and positive for paramagnetic ones, and very large and positive for ferromagnetic ones. Susceptibility, magnetisation and polarisation all describe the same contribution in different units.
Because the 2019 revision of the SI redefined the ampere in terms of the elementary charge rather than the force between two current-carrying wires. That made the vacuum magnetic permeability a measured quantity with an uncertainty rather than a defined constant. Its value differs from the old exact figure of four pi times ten to the minus seven only in about the tenth significant figure.
It does not change the material's permeability, but it changes the effective permeability of the assembly, usually a great deal. A gap of a fraction of a millimetre sits in series with the core in the magnetic circuit and can dominate the total reluctance. That is normally a deliberate design choice, because gapping linearises inductance and raises the DC current a part can carry before saturating.
It depends on the part, and datasheets should say. Initial permeability describes the low-excitation limit, amplitude permeability is B divided by H at a stated point, effective permeability describes a specific gapped core assembly, and complex permeability describes the response and the loss at a stated frequency. They are different numbers and are not interchangeable.
This tool is provided for educational and preliminary engineering use only. It evaluates published relations from figures you supply and publishes no material property data. Permeability is not a constant for ferromagnetic materials: it varies with flux level, temperature, frequency and magnetic history, and it falls to that of free space at saturation and above the Curie temperature. Use the manufacturer's own data at your operating point, and confirm any magnetic design with a qualified engineer.
