Calculate magnetic permeability, flux density, and core amplification. Compare diamagnetic, paramagnetic, and ferromagnetic materials.
Magnetic permeability measures how easily a material allows magnetic flux to pass through it. It is the magnetic equivalent of electrical conductivity — materials with high permeability concentrate and amplify magnetic fields, while those with low permeability are essentially transparent to magnetic flux. Understanding permeability is essential for designing transformers, inductors, electromagnets, magnetic shielding, and any device that relies on controlling magnetic flux.
The relationship B = µH (where B is flux density, µ is permeability, and H is field intensity) is the constitutive equation of magnetism. In vacuum, µ = µ₀ = 4π × 10⁻⁷ H/m. In real materials, µ = µ₀ × µᵣ, where µᵣ (relative permeability) ranges from slightly below 1 for diamagnetic materials to over 1,000,000 for specialized ferromagnetic alloys.
This calculator computes the absolute permeability, flux density, magnetization, and susceptibility for any material. It also shows the dramatic effect of adding a magnetic core to a solenoid — an iron core (µᵣ ≈ 5000) boosts the magnetic field by a factor of 5000 compared to an air-core coil. The material reference table covers 17 common materials from diamagnetic copper to ultra-high-permeability supermalloy.
Selecting the right core material for a transformer, inductor, or electromagnet requires understanding how permeability affects the B-H relationship and the resulting field amplification. This calculator makes it easy to compare different materials and see the quantitative impact on your design.
The solenoid comparison feature is particularly useful for inductor and electromagnet designers who need to predict the field or inductance boost from adding a core. The comprehensive material table serves as a quick reference for the permeability spectrum from diamagnetic to ferromagnetic materials.
Magnetic Permeability Relationships: • Absolute permeability: µ = µ₀ × µᵣ • Flux density: B = µ × H = µ₀µᵣH • Magnetization: M = χ × H = (µᵣ − 1) × H • Susceptibility: χ = µᵣ − 1 • Solenoid field: B = µ₀µᵣNI/L Where µ₀ = 4π × 10⁻⁷ H/m, µᵣ = relative permeability, H = field intensity (A/m)
Result: B = 3.1416 T, solenoid B = 6.2832 T with iron core (5000× air)
With µᵣ = 5000 (iron) and H = 500 A/m: µ = 4π × 10⁻⁷ × 5000 = 6.283 × 10⁻³ H/m. B = µH = 6.283 × 10⁻³ × 500 = 3.142 T. The same H in air gives only B = 0.628 mT — 5000× weaker.
All materials respond to magnetic fields, but the response varies enormously. Diamagnetic materials (copper, silver, bismuth, water) slightly repel magnetic flux, with µᵣ very slightly below 1. Paramagnetic materials (aluminum, platinum, manganese) slightly attract flux, with µᵣ slightly above 1. Neither response is significant for engineering purposes.
Ferromagnetic materials (iron, nickel, cobalt, and alloys like steel, mu-metal, and permalloy) have µᵣ ranging from hundreds to millions. Their strong response comes from magnetic domain structure — regions where atomic magnetic moments are spontaneously aligned. An external field causes favorable domains to grow and unfavorable ones to shrink, producing a large net magnetization.
Real ferromagnetic materials do not have a single value of µᵣ. As H increases from zero, B initially rises steeply (high permeability), then levels off as the material approaches saturation. The slope dB/dH (differential permeability) varies continuously along this curve.
When the field is removed, ferromagnetic materials retain some magnetization (remanence), and it takes a reverse field (coercivity) to demagnetize them. This hysteresis loop represents energy lost per cycle and is a critical parameter for the core design of AC devices like transformers.
For power transformers (50/60 Hz), grain-oriented silicon steel offers high permeability, high saturation (~2 T), and moderate losses. For switch-mode power supplies (10 kHz – 1 MHz), ferrite cores provide adequate permeability with very low high-frequency losses. For magnetic shielding, mu-metal provides the highest static permeability. For sensors and precision instruments, amorphous and nanocrystalline alloys combine high permeability with low coercivity.
µ₀ is the permeability of free space (vacuum), a fundamental constant = 4π × 10⁻⁷ H/m. µᵣ is the relative permeability, a dimensionless number specific to each material. µ (absolute permeability) = µ₀ × µᵣ, with units of H/m.
Ferromagnetic materials (iron, nickel, cobalt, and their alloys) have internal magnetic domains that align with an applied field. This alignment greatly amplifies the total magnetic flux. Paramagnetic materials have weak atomic moments that barely align, and diamagnetic materials slightly oppose the field.
No. Ferromagnetic permeability varies strongly with the applied field strength (the B-H curve is nonlinear). The values given here are typical initial or maximum permeability. At high H values, the material saturates and µᵣ drops toward 1.
Saturation occurs when all magnetic domains are aligned and increasing H produces no further increase in M. Beyond saturation, B increases only by µ₀H (the vacuum contribution). For iron, saturation occurs around B = 2.0-2.2 T.
The inductance of a coil is directly proportional to the permeability of the core: L = µ₀µᵣN²A/l. Adding a ferrite core (µᵣ ≈ 3000) to an air-core inductor increases its inductance by a factor of 3000, enabling much smaller components for the same inductance.
Mu-metal (µᵣ ≈ 80,000-100,000) is a nickel-iron alloy used for magnetic shielding. Its extremely high permeability causes magnetic flux lines to preferentially flow through the shield rather than the protected region inside. It is used to shield sensitive electronics, CRT monitors, and MRI rooms.