Magnetic Permeability Calculator - Solve for B, H, or Mu

A magnetic permeability calculator converts the magnetizing field H applied to a material and the resulting magnetic flux density B into the absolute permeability mu, plus the relative permeability mu_r and the magnetic susceptibility chi_m that physicists and engineers read off datasheets. Magnetic permeability is the constant of proportionality in the SI relation B = mu H.

• • Physics and electrical-engineering coursework: Plug in B and H from a lab experiment to recover the material mu, then read mu_r and chi_m directly.
• • Magnetic-core design review: Compare ferrite, permalloy, and iron relative permeability values to choose a core material that meets the inductance target of a transformer or inductor design.
• • Material identification: A measured B and H pair that returns a mu_r under one suggests a diamagnetic material such as water or copper, while a mu_r in the thousands points to iron or permalloy.
• • Cross-checking textbook constants: Verify that a B of 1.256637e-6 T at H = 1 A/m reproduces the CODATA vacuum permeability mu0 before quoting the value.

When a permeability value is given in Gaussian CGS rather than SI henries per meter, the CGS system of units calculator converts between SI and CGS electromagnetic units across mechanics, electricity, and magnetism so the same constant reads the same in either system.

The calculator reads the three quantities B, H, and mu, applies the chosen solve direction, divides by the CODATA 2022 vacuum permeability mu0 to produce relative permeability mu_r, and subtracts one to report magnetic susceptibility chi_m with a plain-language classification.

• B (magnetic flux density): Magnetic flux density in tesla (T), equivalent to V*s/m^2. Reported by Gauss meters and Hall probes.
• H (magnetizing field strength): Magnetic field strength in amperes per meter (A/m). Generated by free currents.
• mu (absolute permeability): Material permeability in henries per meter (H/m). For vacuum, mu equals mu0 to twelve significant figures.
• mu_r (relative permeability): Dimensionless ratio mu / mu0. Equals 1 in vacuum, drops below 1 for diamagnets, and grows into the thousands for ferromagnets.

The three modes invert the same B = mu H relation. In solve-mu the calculator divides B by H, in solve-B it multiplies mu by H, and in solve-H it divides B by mu, then reports mu, mu_r, and chi_m on one screen.

As published by Wikipedia Permeability (electromagnetism), the relative permeability mu_r is the ratio mu / mu0 and the magnetic susceptibility is chi_m = mu_r - 1, with diamagnetic materials having mu_r less than 1, paramagnetic materials having mu_r greater than 1, and ferromagnetic materials having mu_r orders of magnitude larger.

Once the B field is known, the next step in most electromagnetic problems is to compute the force on a moving charge, and the Lorentz force calculator solves F = q(E + v x B) for any charge, velocity, and B field, which is the natural downstream calculation from this permeability result.

Four ideas cover every number the magnetic permeability calculator reports and every classification label it produces.

When the classification is ferromagnetic and the question becomes how strong an atomic magnetic moment can be, the magnetic moment calculator returns mu in Bohr magnetons and J/T from spin, orbital, or LS-coupled quantum inputs, which is the microscopic companion to this macroscopic mu answer.

Five short steps move from a known pair of B, H, or mu values to a complete permeability report and a material classification.

1. 1 Pick the solve direction: Use the Solve For dropdown to choose between solving for mu (default), B, or H.
2. 2 Enter the two known quantities: Type the known B value in tesla, the known H value in amperes per meter, and the known mu value in henries per meter. Leave the solve target at zero.
3. 3 Optionally pick a material preset: Use the Material Preset dropdown to load a representative mu for vacuum, water, copper, aluminum, iron, ferrite, permalloy, neodymium, or a superconductor.
4. 4 Read the four derived values: The result panel shows absolute mu in H/m, the relative permeability mu_r, the magnetic susceptibility chi_m, and the CODATA 2022 vacuum permeability mu0 alongside the material classification.
5. 5 Use the classification to sanity-check the material: A mu_r under 1 says diamagnetic, a mu_r just over 1 says weakly paramagnetic, a mu_r from 1 to 5 says paramagnetic, and a mu_r above 5 says ferromagnetic.

Measuring a ferrite toroid returns an inductance that implies a B of about 0.01 T at an H of 8 A/m, so you leave the B and H inputs at those values, pick Ferrite in Material Preset, and the calculator returns mu = 0.00125 H/m, mu_r = 994, chi_m = 993, with a Ferromagnetic classification.

If the same measurement was taken outdoors with a handheld compass rather than a laboratory probe, the B field is the local geomagnetic field and the right correction for the compass heading is the declination, which the magnetic declination calculator converts between magnetic and true map bearings using the east-west sign convention.

A dedicated magnetic permeability calculator removes the unit-conversion mistakes and the missing-factor errors that show up when the B = mu H relation is done by hand.

• • Three solve directions in one tool: The same screen solves for mu, B, or H by selecting the Solve For dropdown, so a single calculation handles lab analysis, inductor design, and field estimation.
• • Reports mu, mu_r, and chi_m together: Absolute permeability, relative permeability, and magnetic susceptibility appear on the same row so the values can be cross-checked against a datasheet.
• • Uses the CODATA 2022 mu0 value: The CODATA 2022 vacuum permeability of 1.25663706127 x 10^-6 H/m is baked in, so mu_r and chi_m track the current international standard.
• • Material preset shortens lookup work: The Material Preset dropdown fills the mu input from a representative value for vacuum, water, copper, aluminum, iron, ferrite, permalloy, neodymium, and superconductor.
• • Flags the diamagnetic and superconductor cases: A mu_r slightly below 1 is labeled diamagnetic and a mu of zero is recognized as a superconductor, so the user sees when their input is in a regime where the B = mu H relation is approximate.

When the permeability feeds a plasma-physics calculation, the next quantity of interest is the speed of an Alfven wave in the medium, and the Alfven velocity calculator solves v_A = B / sqrt(mu0 * mu_r * rho) for the wave speed given the magnetic field, density, and relative permeability that this calculator returns.

Three material inputs determine the answer, and two caveats tell you when the assumed linear B = mu H relation breaks down.

• • The B = mu H relation assumes a linear, isotropic material with a single scalar mu, so highly anisotropic materials such as grain-oriented electrical steel or rare-earth magnets with a preferred axis need a tensor permeability model.
• • The CODATA 2022 mu0 of 1.25663706127 x 10^-6 H/m has a relative standard uncertainty of 1.6 x 10^-10, so any answer that needs more than nine significant figures should be quoted with that uncertainty.

According to Engineering Toolbox Permeability Table, annealed permalloy has a relative permeability of about 8000, pure iron annealed in hydrogen reaches about 200000, ferrite nickel zinc ranges from 16 to 640, copper sits just under 1 at 0.999994, water sits at 0.999992, and a neodymium magnet has a relative permeability of about 1.05.

According to NIST CODATA 2022 vacuum magnetic permeability, the vacuum magnetic permeability mu0 equals 1.25663706127 x 10^-6 N/A^2 with a standard uncertainty of 0.00000000020 x 10^-6 and a relative standard uncertainty of 1.6 x 10^-10.

For the related problem of how strongly a coil or a permanent magnet responds to a field once mu is known, the magnetic dipole moment calculator returns the torque and energy of a magnetic dipole with mu equals NIA from the number of turns, current, and loop area, which is the next quantity used in electromagnet and motor design.