Hong Life CalculatorsUnderstanding the magnetic permeability calculator — concept and principle
This calculator converts between relative permeability, magnetic susceptibility and absolute permeability, or finds the permeability from a measured B and H, and classifies the material as diamagnetic, paramagnetic or ferromagnetic. It is used in electromagnetism courses and when choosing core materials for inductors, transformers and shields.
How it is calculated
μ = μ₀ × μ_r, with μ₀ = 4π × 10⁻⁷ H/m ≈ 1.2566 µH/m
μ_r = 1 + χ
From a measurement: μ_r = B ÷ (μ₀ H)
Field gain over vacuum = (μ_r − 1) × 100% = χ × 100%
- Results are given in H/m and µH/m.
The principle
When a material sits in a magnetic field, its atoms respond. In diamagnetic materials the response weakly opposes the field (χ slightly negative). In paramagnetic materials atomic moments line up slightly with the field (χ slightly positive). In ferromagnetic materials such as iron, groups of atoms (domains) align strongly and multiply the field thousands of times. Permeability measures this overall response: B = μH. For ferromagnets μ_r is not constant; it depends on field strength and history, which is why a measured B and H gives the value at that operating point.
Worked example
Default inputs: start from μ_r = 4,000.
- μ = 4,000 × 4π × 10⁻⁷ = 0.00502655 H/m = 5,026.55 µH/m.
- χ = 4,000 − 1 = 3,999.
- Field gain over vacuum = 3,999 × 100% = 399,900%.
- Classification: ferromagnetic.
| Relative permeability μr | Permeability μ (H/m) |
|---|---|
| 1,000 | 0.00125664 |
| 2,000 | 0.00251327 |
| 3,000 | 0.00376991 |
| 4,000 | 0.00502655 |
| 5,000 | 0.00628319 |
| 6,000 | 0.00753982 |
| 8,000 | 0.0100531 |
How to read the result
Copper and water have χ of the order of −10⁻⁵, aluminium about +10⁻⁵, and iron-based core materials can have μ_r in the thousands or more. A high μ_r concentrates magnetic flux, which is why transformer and inductor cores use ferromagnetic materials, but they saturate at high flux density.
Common mistakes
- Confusing μ_r with χ for ferromagnets (they differ by 1, which matters only for weak materials).
- Treating a ferromagnet's μ_r as a fixed constant.
- Mixing H/m and µH/m.
Frequently asked questions
What is the permeability of free space?
μ₀ ≈ 4π × 10⁻⁷ H/m, about 1.2566 µH/m.
Can μ_r be less than 1?
Yes, slightly, for diamagnetic materials.
Why does iron saturate?
Once all domains are aligned, the material cannot add more magnetisation.
How is permeability measured?
By measuring B for a known H, for example with a wound toroid sample.
Sources: OpenStax, University Physics Volume 2 (2016), section 12.7 "Magnetism in Matter"