What is Magnetic Permeability?

Use this calculator to compute Magnetic Permeability values with step-by-step solutions.

How do I use this calculator?

Enter the required values; results and step-by-step derivation update automatically.

Where is Magnetic Permeability used?

Electromagnetism, engineering, research, and related scientific disciplines.

What formulas does this use?

All standard magnetic permeability formulas are applied and shown step-by-step in the results.

Can I get step-by-step solutions?

Yes. The calculator shows a full step-by-step derivation when results are displayed.

How accurate are the results?

Results use standard floating-point arithmetic (~15 significant digits). Suitable for education and professional use.

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ELECTROMAGNETISMElectromagnetismPhysics Calculator

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Magnetic Permeability

Magnetic permeability quantifies how easily a material magnetizes in response to an applied magnetic field, relating flux density B to field strength H.

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Mu-metal achieves μᵣ up to 100,000 for magnetic shielding. Superconductors exhibit perfect diamagnetism (μᵣ = 0). B-H curves show hysteresis in ferromagnetic materials. Silicon steel (μᵣ ≈ 4000) reduces eddy currents in transformers.

Key quantities

4π×10⁻⁷ H/m

Vacuum μ₀

Key relation

B = μH

B-H Relation

Key relation

χm = μᵣ − 1

Susceptibility

Key relation

L = μN²A/l

Inductance

Key relation

Ready to run the numbers?

Why: Permeability determines transformer efficiency, motor design, and magnetic shielding performance.

How: Materials are classified by μᵣ: ferromagnetic (μᵣ >> 1), paramagnetic (μᵣ > 1), diamagnetic (μᵣ < 1).

Mu-metal achieves μᵣ up to 100,000 for magnetic shielding.Superconductors exhibit perfect diamagnetism (μᵣ = 0).

Sources:NIST Physical Measurement LaboratoryHyperPhysics - Magnetic Properties

Run the calculator when you are ready.

Solve the EquationCalculate permeability, B-H relationships, and material classification

🔌 Iron Core Transformer

Power transformer with iron core, typical μᵣ = 200-5000

📡 Ferrite Core Inductor

High-frequency inductor with ferrite core

💨 Air Gap Magnetic Core

Magnetic circuit with air gap to prevent saturation

🛡️ Magnetic Shielding (Mu-Metal)

Magnetic shielding using high-permeability material

🧲 Paramagnetic Material (Aluminum)

Weakly paramagnetic material analysis

💧 Diamagnetic Material (Water)

Weakly diamagnetic material analysis

⚡ Superconductor (Perfect Diamagnetism)

Superconductor with perfect diamagnetic response

Material Database

Select a material from the database to automatically load its permeability properties. Click on any material card to use its values.

Vacuum/Air

Reference material with no magnetic properties

μᵣ: 1.000000

χm: 0

Type: non-magnetic

Applications: Baseline reference, Free space calculations

Iron (Pure)

Strongly ferromagnetic material, highly responsive to magnetic fields

μᵣ: 200.000000

χm: 199.000000

Type: ferromagnetic

Applications: Electromagnets, Transformers, Motors, Generators

Iron (99.8% Pure)

Very high permeability, excellent for magnetic cores

μᵣ: 5000.000000

χm: 4999.000000

Type: ferromagnetic

Applications: Power transformers, Induction motors, Magnetic cores

Nickel

Ferromagnetic metal with moderate permeability

μᵣ: 100.000000

χm: 99.000000

Type: ferromagnetic

Applications: Magnetic alloys, Electronics, Batteries

Cobalt

Ferromagnetic element with high Curie temperature

μᵣ: 70.000000

χm: 69.000000

Type: ferromagnetic

Applications: Permanent magnets, Magnetic recording, Alloys

Permalloy (80% Ni, 20% Fe)

Extremely high permeability soft magnetic alloy

μᵣ: 100000.000000

χm: 99999.000000

Type: ferromagnetic

Applications: Magnetic shielding, Sensors, Transformers

Mu-Metal

Nickel-iron alloy with extremely high permeability

μᵣ: 100000.000000

χm: 99999.000000

Type: ferromagnetic

Applications: Magnetic shielding, Sensitive instruments, EMI protection

Ferrite (MnZn)

Ceramic ferrimagnetic material with high resistivity

μᵣ: 1500.000000

χm: 1499.000000

Type: ferrimagnetic

Applications: High-frequency transformers, Inductors, RF circuits

Ferrite (NiZn)

Nickel-zinc ferrite with lower permeability but higher frequency range

μᵣ: 100.000000

χm: 99.000000

Type: ferrimagnetic

Applications: RF inductors, EMI suppression, Antennas

Silicon Steel

Electrical steel with silicon addition to reduce eddy currents

μᵣ: 4000.000000

χm: 3999.000000

Type: ferromagnetic

Applications: Power transformers, Electric motors, Generators

Aluminum

Weakly paramagnetic metal

μᵣ: 1.000022

χm: 2.200000e-5

Type: paramagnetic

Applications: Non-magnetic applications, Conductors

Copper

Weakly diamagnetic metal

μᵣ: 0.999994

χm: -6.000000e-6

Type: diamagnetic

Applications: Electrical conductors, Non-magnetic applications

Water

Weakly diamagnetic liquid

μᵣ: 0.999992

χm: -8.000000e-6

Type: diamagnetic

Applications: Biological systems, Reference material

Superconductor

Perfect diamagnetism (Meissner effect)

μᵣ: 0

χm: -1.000000

Type: diamagnetic

Applications: MRI magnets, Particle accelerators, Quantum computing

Input Parameters

Calculation Mode

Relative Permeability (μᵣ)

Absolute Permeability (μ) - H/m

Magnetic Flux Density (B) - T

Magnetic Field Strength (H) - A/m

Magnetic Susceptibility (χm)

Vacuum Permeability (μ₀) - H/m

Optional: Inductance Calculation Parameters

Number of Turns (N)

Cross-Sectional Area (A) - m²

Length (l) - m

Frequency (f) - Hz

Temperature - °C

Please provide at least one of: Relative Permeability (μᵣ), Absolute Permeability (μ), B and H, or Magnetic Susceptibility (χm)

For educational and informational purposes only. Verify with a qualified professional.

🔬 Physics Facts

🧲

Mu-metal has relative permeability up to 100,000 for magnetic shielding.

— NIST

⚡

Vacuum permeability μ₀ = 4π×10⁻⁷ H/m appears in Maxwell's equations.

— HyperPhysics

📊

B-H curves show hysteresis loops; area inside represents energy loss per cycle.

— IEEE

🔬

Superconductors exhibit perfect diamagnetism (μᵣ = 0) via the Meissner effect.

— Physics LibreTexts

📋 Key Takeaways

• Magnetic permeability (μ) quantifies how easily a material magnetizes in response to an applied magnetic field
• Relative permeability (μᵣ) compares material to vacuum: μᵣ = μ/μ₀, where μ₀ = 4π × 10⁻⁷ H/m
• Ferromagnetic materials (iron, nickel) have μᵣ >> 1, while paramagnetic materials have μᵣ slightly > 1, and diamagnetic materials have μᵣ slightly < 1
• The B-H relationship B = μH is fundamental to magnetic circuit design and determines flux density for a given field strength

💡 Did You Know?

🧲Mu-metal has a relative permeability of up to 100,000 — making it one of the most permeable materials known, essential for magnetic shieldingSource: NIST

⚡The permeability of vacuum (μ₀ = 4π × 10⁻⁷ H/m) is a fundamental constant that appears in Maxwell's equations and defines the relationship between electric and magnetic fieldsSource: HyperPhysics

📊B-H curves show hysteresis loops for ferromagnetic materials — the area inside the loop represents energy loss per cycle due to domain wall motionSource: IEEE Magnetics

🔬Superconductors exhibit perfect diamagnetism (μᵣ = 0, χm = -1) through the Meissner effect, completely expelling magnetic fieldsSource: Physics LibreTexts

🏭Silicon steel used in transformers has μᵣ ≈ 4000, reducing eddy current losses while maintaining high permeability for efficient power transferSource: Engineering Toolbox

🌊Water is weakly diamagnetic (μᵣ ≈ 0.999992), which is why strong magnets can levitate water dropletsSource: MIT OCW

💎Permalloy (80% Ni, 20% Fe) achieves μᵣ up to 100,000, making it ideal for sensitive magnetic sensors and recording headsSource: NIST

🔋Ferrite cores in inductors have μᵣ ≈ 1500-2000 with high electrical resistivity, preventing eddy currents at high frequenciesSource: Engineering Toolbox

📖 How Magnetic Permeability Works

Magnetic permeability describes how a material responds to an applied magnetic field. When a magnetic field H is applied, the material develops an internal magnetization M, creating a total magnetic flux density B = μ₀(H + M) = μH.

Ferromagnetic Materials

Ferromagnetic materials (iron, nickel, cobalt) have unpaired electron spins that align parallel in domains. When an external field is applied, domains align, dramatically increasing B. Their μᵣ ranges from hundreds to hundreds of thousands, but permeability decreases with increasing H due to saturation.

Paramagnetic Materials

Paramagnetic materials (aluminum, oxygen) have unpaired electrons that weakly align with the field. Their μᵣ is slightly greater than 1 (typically 1.00001 to 1.001), and permeability is temperature-dependent, decreasing as temperature increases.

Diamagnetic Materials

Diamagnetic materials (copper, water) have all electrons paired. The applied field induces opposing magnetic moments, giving μᵣ slightly less than 1 (typically 0.99999). Diamagnetism is universal but weak, and independent of temperature.

B-H Curves and Hysteresis

For ferromagnetic materials, the B-H relationship is nonlinear and shows hysteresis. The initial permeability (slope at low H) is high, but as H increases, the material saturates and μ decreases. The hysteresis loop shows energy loss per cycle, important for transformer and motor design.

🎯 Expert Tips for Magnetic Circuit Design

💡 High Permeability = Higher Inductance

Using high-permeability cores (iron, ferrite) dramatically increases inductance L = (μN²A)/l. However, watch for saturation at high currents — once B reaches saturation flux density, μ drops and inductance decreases.

💡 Air Gaps Prevent Saturation

Adding small air gaps in magnetic cores reduces effective permeability but prevents saturation. The effective permeability μ_eff = μᵣ/(1 + μᵣ × gap/length) allows higher H before saturation, crucial for power inductors.

💡 Frequency Affects Permeability

At high frequencies, ferromagnetic materials show complex permeability with real and imaginary parts. Eddy currents and domain wall motion cause losses, reducing effective permeability. Use ferrites (high resistivity) for RF applications.

💡 Temperature Dependence

Ferromagnetic materials lose their magnetic properties above the Curie temperature (1043 K for iron). Permeability decreases with temperature, so design for worst-case operating temperatures in motors and transformers.

⚖️ Material Permeability Comparison

Material	μᵣ	χm	Classification	Typical Applications
Vacuum/Air	1.0	0	Non-magnetic	Reference, free space
Copper	0.999994	-0.000006	Diamagnetic	Electrical conductors
Water	0.999992	-0.000008	Diamagnetic	Biological systems
Aluminum	1.000022	0.000022	Paramagnetic	Non-magnetic structures
Iron (Pure)	200	199	Ferromagnetic	Electromagnets, cores
Iron (99.8%)	5000	4999	Ferromagnetic	Power transformers
Silicon Steel	4000	3999	Ferromagnetic	Transformers, motors
Ferrite (MnZn)	1500	1499	Ferrimagnetic	RF inductors, transformers
Permalloy	100000	99999	Ferromagnetic	Magnetic shielding, sensors
Mu-Metal	100000	99999	Ferromagnetic	EMI shielding, instruments
Superconductor	0	-1	Perfect diamagnetic	MRI, particle accelerators

❓ Frequently Asked Questions

What is the difference between absolute and relative permeability?

Absolute permeability (μ) has units of H/m and represents the actual permeability of a material. Relative permeability (μᵣ) is dimensionless and compares the material to vacuum: μᵣ = μ/μ₀. For vacuum, μᵣ = 1. Most materials have μᵣ close to 1, but ferromagnetic materials can have μᵣ ranging from hundreds to hundreds of thousands.

How does magnetic permeability affect inductance?

Inductance L = (μN²A)/l is directly proportional to permeability. Higher permeability materials (like iron cores) dramatically increase inductance compared to air-core inductors. However, ferromagnetic cores can saturate at high currents, reducing effective permeability and inductance.

What causes magnetic saturation in ferromagnetic materials?

Saturation occurs when all magnetic domains align with the applied field. Beyond the saturation point, increasing H produces minimal increase in B, and permeability drops. Saturation flux density B_sat is typically 1.5-2.0 T for iron and silicon steel. Air gaps help prevent saturation by reducing effective permeability.

Why do transformers use high-permeability cores?

High-permeability cores (μᵣ ≈ 2000-5000) concentrate magnetic flux, increasing mutual inductance between windings. This improves coupling efficiency, reduces magnetizing current, and minimizes core losses. Silicon steel is preferred for its high permeability and low eddy current losses due to added silicon.

What is magnetic susceptibility and how does it relate to permeability?

Magnetic susceptibility (χm) measures how much a material becomes magnetized per unit applied field: M = χmH. It relates to relative permeability as χm = μᵣ - 1. Positive χm indicates paramagnetic/ferromagnetic materials, negative χm indicates diamagnetic materials, and χm = 0 for vacuum.

How does frequency affect magnetic permeability?

At high frequencies, ferromagnetic materials exhibit complex permeability μ = μ' - jμ'', where the real part μ' decreases due to eddy currents and domain wall motion, and the imaginary part μ'' represents losses. Ferrites (ceramic ferrimagnetic materials) maintain high permeability at RF frequencies due to their high electrical resistivity.

What is the B-H curve and why is it important?

The B-H curve plots magnetic flux density B versus magnetic field strength H. For linear materials, B = μH is a straight line. Ferromagnetic materials show nonlinear B-H curves with hysteresis loops. The initial slope gives initial permeability, the maximum B is saturation flux density, and the loop area represents energy loss per cycle.

Can permeability be negative?

No, absolute permeability μ cannot be negative — it represents a physical property. However, relative permeability μᵣ can be slightly less than 1 for diamagnetic materials (e.g., μᵣ = 0.999992 for water). Superconductors effectively have μᵣ = 0 (perfect diamagnetism) due to the Meissner effect, expelling all magnetic fields.