Permeability of Free Space (μ₀): Definition, Value, Formula, and Applications

The permeability of free space is an important physical constant in electromagnetism that describes the relationship between magnetic fields and electric currents in a vacuum. Represented by the symbol μ₀ (mu naught), it is used in electromagnetic field calculations, magnetic force equations, and the study of electromagnetic waves.

Understanding the permeability of free space is essential for electrical engineering, physics, and electronics because it helps explain how magnetic fields behave in a vacuum and how electromagnetic quantities are related.

What Is the Permeability of Free Space?

The permeability of free space (μ₀), also called the vacuum magnetic permeability, is a physical constant that characterizes the magnetic properties of a vacuum.

It appears in several fundamental electromagnetic equations, including Ampère’s law, the magnetic field equations for current-carrying conductors, and the relationship between the speed of light and the electromagnetic constants of free space.

The permeability of free space is approximately:

μ₀ ≈ 1.25663706127 × 10⁻⁶ H/m

where H/m represents henries per metre.

In SI base units, it can also be expressed as newtons per ampere squared (N/A²).

Unlike the relative permeability of a material, which depends on its magnetic properties, μ₀ is the permeability constant associated with a vacuum.

Value of the Permeability of Free Space

The commonly used value of the permeability of free space is approximately 1.2566 × 10⁻⁶ H/m.

Its value and units are given below.

PropertyValue
NamePermeability of free space
Symbolμ₀
Approximate value1.25663706127 × 10⁻⁶ H/m
Equivalent unitN/A²
SI unitHenry per metre (H/m)
Alternative nameVacuum permeability
ApplicationElectromagnetic field calculations

Important note: In the modern SI system, μ₀ is no longer defined as having the exact value 4π × 10⁻⁷ H/m. That was its exact value under the previous SI definition. Following the 2019 SI revision, its value is experimentally determined and is very close to 4π × 10⁻⁷ H/m.

For most practical engineering calculations, the approximation

μ₀ ≈ 4π × 10⁻⁷ H/m

remains sufficiently accurate.

Formula for the Permeability of Free Space

The permeability of free space is related to the speed of light and the permittivity of free space through the following equation:

c=1μ0ε0c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}

Rearranging the equation gives:

μ0=1c2ε0\mu_0=\frac{1}{c^2\varepsilon_0}

Where:

  • μ₀ = permeability of free space, in H/m
  • ε₀ = permittivity of free space, in F/m
  • c = speed of light in a vacuum, approximately 299,792,458 m/s

This relationship demonstrates that the magnetic permeability and electric permittivity of free space jointly determine the speed at which electromagnetic waves propagate through a vacuum.

Dimensional Formula of μ₀

The dimensional formula can be derived using the magnetic force between two long, straight, parallel current-carrying conductors.

The force per unit length is:

FL=μ0I1I22πr\frac{F}{L}=\frac{\mu_0 I_1I_2}{2\pi r}

Rearranging the equation:

μ0=2πrFLI1I2\mu_0=\frac{2\pi rF}{LI_1I_2}

Here,2π 2\pi is dimensionless. Therefore, the dimensional formula of permeability is:

[μ0]=[L][F][L][I]2[\mu_0]=\frac{[L][F]}{[L][I]^2}

In this expression, the first ([L]) represents the dimension of separation distance, while the second represents the dimension of length used in the force-per-unit-length term. They cancel each other.

Thus,

[μ0]=[F][I]2[\mu_0]=\frac{[F]}{[I]^2}

The dimensional formula of force is:

[F]=[MLT−2][F]=[MLT^{-2}]

Substituting this into the equation gives:

[μ0]=[MLT−2][I2][\mu_0]=\frac{[MLT^{-2}]}{[I^2]}

Therefore, the dimensional formula of permeability of free space is:

[MLT−2I−2]\boxed{[MLT^{-2}I^{-2}]}

Relationship Between Permeability and Permittivity of Free Space

Permeability and permittivity are two fundamental electromagnetic constants, but they describe different properties.

  • Permeability of free space (μ₀): Appears in equations describing magnetic fields and magnetic effects produced by electric currents.
  • Permittivity of free space (ε₀): Appears in equations describing electric fields and the electric response of a vacuum.

Their relationship with the speed of light is:

c=1μ0ε0c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}

Both constants are essential for understanding electromagnetic wave propagation, including radio waves, visible light, and other forms of electromagnetic radiation.

Magnetic Energy Density and Permeability of Free Space

A magnetic field stores energy in the space where the field exists. The magnetic energy density in a vacuum is calculated using the following equation:

uB=B22μ0u_B=\frac{B^2}{2\mu_0}

Where:

  • uBu_B = magnetic energy density, in J/m³
  • B = magnetic flux density, in tesla (T)
  • μ₀ = permeability of free space, in H/m

The equation shows that magnetic energy density increases with the square of the magnetic flux density. For example, doubling the magnetic flux density increases the magnetic energy density by a factor of four, assuming the vacuum permeability remains constant.

Example Calculation

Suppose the magnetic flux density in a vacuum is 0.02 T. Calculate the magnetic energy density.

Given:

  • B = 0.02 T
  • μ₀ ≈ 1.2566 × 10⁻⁶ H/m

Using the formula:

uB=B22μ0u_B=\frac{B^2}{2\mu_0}

Substituting the values:

uB=(0.02)22(1.2566×10−6)u_B=\frac{(0.02)^2}{2(1.2566\times10^{-6})}
uB≈159.2 J/m3u_B\approx159.2\text{ J/m}^3

Therefore, the magnetic energy density is approximately 159.2 J/m³.

Magnetic Force Between Two Parallel Current-Carrying Wires

The permeability of free space is also used to calculate the magnetic force between two long, straight, parallel conductors carrying electric currents.

For conductors separated by a distance r, the magnitude of the magnetic force per unit length is:

FL=μ0I1I22πr\frac{F}{L}=\frac{\mu_0 I_1 I_2}{2\pi r}

Where:

  • F/L = magnetic force per unit length, in N/m
  • μ₀ = permeability of free space
  • I₁ = current in the first conductor, in amperes
  • I₂ = current in the second conductor, in amperes
  • r = distance between the conductors, in metres

Currents flowing in the same direction produce an attractive force, while currents flowing in opposite directions produce a repulsive force.

Example Calculation

Two long parallel conductors carry currents of 10 A and 20 A. Their separation is 0.1 m. Calculate the magnetic force per unit length, assuming the surrounding medium is a vacuum.

Given:

  • I₁ = 10 A
  • I₂ = 20 A
  • r = 0.1 m
  • μ₀ ≈ 4π × 10⁻⁷ H/m

Using the formula:

FL=μ0I1I22πr\frac{F}{L}=\frac{\mu_0 I_1 I_2}{2\pi r}

Substituting the values:

FL=(4π×10−7)(10)(20)2π(0.1)\frac{F}{L}=\frac{(4\pi\times10^{-7})(10)(20)}{2\pi(0.1)}
FL=4×10−4 N/m\frac{F}{L}=4\times10^{-4}\text{ N/m}

Therefore, the magnetic force per unit length is 0.0004 N/m. The force is attractive if the currents flow in the same direction and repulsive if they flow in opposite directions.

Applications of the Permeability of Free Space

The permeability of free space is used in several areas of electrical engineering and physics.

1. Magnetic field calculations: It helps calculate magnetic fields produced by current-carrying conductors and coils in a vacuum.

2. Electromagnetic wave propagation: Together with the permittivity of free space, it determines the speed of electromagnetic waves in a vacuum.

3. Inductance calculations: It appears in inductance equations for coils, solenoids, and other electromagnetic structures, particularly when deriving idealized vacuum-field relationships.

4. Magnetic energy calculations: It is used to determine the energy density stored in a magnetic field in a vacuum.

5. Electrical power systems: It forms part of the electromagnetic theory used to analyze conductors, busbars, transformers, motors, and other electrical equipment.

6. Antenna and radio-frequency engineering: It is used in electromagnetic field models for antennas, transmission structures, and radio-frequency systems.

7. Scientific measurements: It is an important constant in experimental electromagnetism and the study of fundamental physical relationships.

Difference Between Permeability of Free Space and Relative Permeability

The permeability of free space and relative permeability are related but different quantities.

BasisPermeability of Free SpaceRelative Permeability
Symbolμ₀μᵣ
MeaningMagnetic permeability constant associated with a vacuumRatio of a material’s permeability to μ₀
UnitH/mDimensionless
ValueApproximately 1.2566 × 10⁻⁶ H/mDepends on the material and conditions
ApplicationFundamental electromagnetic equationsComparing the magnetic properties of materials

For a linear, isotropic material, the absolute permeability is expressed as:

μ=μ0μr\mu=\mu_0\mu_r

Where μ is the material’s absolute permeability and μᵣ is its relative permeability.

Conclusion

The permeability of free space (μ₀) is a fundamental constant used to describe magnetic phenomena in a vacuum. Its approximate value is 1.2566 × 10⁻⁶ H/m, and it appears in equations for magnetic fields, forces between current-carrying conductors, and magnetic energy density.

It is also related to the permittivity of free space through the speed-of-light equation. A clear understanding of this constant helps students, physicists, and electrical engineers analyze electromagnetic systems and apply fundamental electromagnetic principles in practical calculations.

Frequently Asked Questions

Q1. What is the symbol for the permeability of free space?

The symbol is μ₀, pronounced “mu naught.” It represents the magnetic permeability associated with a vacuum.

Q2. What is the SI unit of permeability of free space?

Its SI unit is the henry per metre (H/m). It can also be expressed as newtons per ampere squared (N/A²).

Q3. Is the permeability of free space constant?

It is a fundamental physical constant in the SI system. However, its numerical value is experimentally determined under the modern SI definitions rather than fixed to the exact value 4π × 10⁻⁷ H/m.

Q4. Why is the permeability of free space important?

It connects electric currents with magnetic fields and appears in equations for magnetic forces, magnetic energy density, inductance, and electromagnetic wave propagation.

Q5. What is the relationship between μ₀ and ε₀?

The permeability of free space and permittivity of free space are related to the speed of light by: c=1μ0ε0c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}
This equation describes how these two electromagnetic constants are connected to the propagation speed of electromagnetic waves in a vacuum.

Reference:

  1. Visit Hyperphysics for more information on this constant.

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