Mu naught (μ₀) is a fundamental constant in physics that describes how well a vacuum supports a magnetic field. Known formally as the permeability of free space or the vacuum magnetic permeability, its value is approximately 1.257 × 10⁻⁶ newtons per ampere squared (N/A²). It shows up in nearly every equation involving magnetism and electromagnetic waves, including the relationship that defines the speed of light.
What Mu Naught Physically Represents
Think of μ₀ as a measure of how “easy” it is for a magnetic field to form in empty space. When electric current flows through a wire, it generates a magnetic field around it. The strength of that field doesn’t just depend on the current; it also depends on the medium the field is spreading through. In a perfect vacuum with nothing to help or hinder the field, μ₀ is the baseline resistance the universe offers.
This is why it’s called “permeability.” The term, coined by the physicist William Thomson (Lord Kelvin), captures the idea that space itself is permeable to magnetic influence. Every material has its own permeability, but μ₀ is the reference point. The permeability of any material is expressed as a multiple of μ₀. Iron, for instance, has a relative permeability thousands of times larger, which is why magnets stick to it. Air is so close to a vacuum magnetically that μ₀ works as a practical stand-in for everyday calculations.
Its Numerical Value
For most physics courses and engineering work, μ₀ is treated as exactly 4π × 10⁻⁷ H/m (henries per meter), which works out to roughly 1.2566 × 10⁻⁶ N/A². That value was historically exact by definition, not by measurement. The old SI definition of the ampere was built around the force between two current-carrying wires, and that definition locked μ₀ to precisely 4π × 10⁻⁷.
That changed in 2019. The international system of units was revised so the ampere is now defined by fixing the value of the elementary electric charge (the charge of a single electron) instead. As a result, μ₀ is no longer exact by definition. It must be determined experimentally. The 2022 CODATA recommended value is 1.256 637 061 27 × 10⁻⁶ N/A², with an uncertainty of about 1.6 parts in ten billion. In practice, the difference from 4π × 10⁻⁷ is so tiny that it only matters in the most precise metrology work. For homework, lab calculations, and engineering, 4π × 10⁻⁷ H/m is still the number to use.
Where You’ll See It in Equations
μ₀ appears wherever magnetism meets math. A few of the most common places:
Magnetic field of a solenoid. The field inside a coil of wire is B = μ₀nI, where n is the number of turns per unit length and I is the current. This is one of the simplest and most practical uses of μ₀. It tells you that the magnetic field inside a coil scales directly with current and coil density, with μ₀ setting the proportionality.
Inductance of a solenoid. The inductance (a measure of how much a coil resists changes in current) is L = μ₀AN²/ℓ, where A is the cross-sectional area, N is the total number of turns, and ℓ is the length. Every inductor you’ve ever used in a circuit has μ₀ baked into its behavior.
Ampère’s law and Maxwell’s equations. In the full set of Maxwell’s equations, which describe all classical electromagnetic behavior, μ₀ appears in the Ampère-Maxwell law. That equation relates the curl of a magnetic field to the electric current producing it and to any changing electric field nearby. Without μ₀ in that equation, the math wouldn’t produce the correct field strengths.
Its Connection to the Speed of Light
One of the most elegant results in physics ties μ₀ directly to how fast light travels. The speed of light in a vacuum, c, is related to both the magnetic constant (μ₀) and the electric constant (ε₀, the permittivity of free space) by:
c = 1 / √(μ₀ε₀)
This equation fell out of Maxwell’s equations in the 1860s and was one of the great unifying discoveries in science. It showed that light is an electromagnetic wave, and that its speed is entirely determined by how easily empty space supports electric and magnetic fields. Plug in μ₀ ≈ 4π × 10⁻⁷ H/m and ε₀ ≈ 8.85 × 10⁻¹² F/m, and you get almost exactly 3 × 10⁸ m/s.
Mu Naught vs. Relative Permeability
When you move from a vacuum into a real material, the relevant constant becomes μ, the absolute permeability of that material. It’s calculated by multiplying μ₀ by a dimensionless number called relative permeability (μᵣ):
μ = μ₀ × μᵣ
For a vacuum, μᵣ is exactly 1. For most everyday materials like wood, plastic, and aluminum, μᵣ is very close to 1, meaning they barely affect magnetic fields. Ferromagnetic materials like iron, nickel, and cobalt have μᵣ values in the hundreds or thousands, which is why they concentrate magnetic field lines so effectively and are used in transformer cores, electromagnets, and electric motors.
Why the Name “Mu Naught”
The symbol μ (the Greek letter mu) has long been used for permeability in physics. The subscript 0 (read as “naught” or “zero”) indicates that this is the value in free space, the baseline with no material present. You’ll also see it written as μ₀ and referred to by any of several names: the magnetic constant, vacuum permeability, or permeability of free space. They all mean the same thing.