The vacuum magnetic permeability is the magnetic permeability in a classical vacuum. It is a physical constant, conventionally written as μ₀. It quantifies the strength of the magnetic field induced by an electric current. Expressed in terms of SI base units, it has the unit kg⋅m⋅s⁻²⋅A⁻².

μ₀ = 1.25663706 × 10-6 kg m s^-2 A^-2

The inverse of this constant is: μ₀^-1 = 8.0 x 10⁵ kg^-1 m^-1 s² A²

 

Vacuum permittivity, commonly denoted ε₀, is the value of the absolute dielectric permittivity of a classical vacuum. It may also be referred to as the permittivity of free space, the electric constant, or the distributed capacitance of the vacuum. It is an ideal physical constant. Expressed in terms of SI base units, it has the unit kg^-1⋅m^-3⋅s⁴⋅μ₀^-1 A².

ε₀ = 8.85418782 × 10-12 kg^-1 μ₀^-1 m^-3 s⁴ A²

The inverse of this constant is: ε₀^-1 = 1.1 x 10¹¹ kg m³ s^-4 A^-2

 

Multiplied together, the two inverses yield the speed of light squared. Expressed in terms of SI base units, the speed of light has the unit m⋅s^-1.

ε₀^-1 ⋅ μ₀^-1 = c²

For reasons that will become apparent later: • the constants on the left-hand side are renamed • constants of proportionality are divided out, where B = D = 1, and • B and D are assigned units so that ε⁰ and μ⁰ are in expressed in units of velocity:

ε₀^-1 = B ⋅ ε⁰

μ₀^-1 = D ⋅ μ⁰

 

Expressed in terms of SI base units, the units of B and D are kg-1 m-2 s³ A² and kg m² s-³ A^-2, respectively.

B ⋅ D ⋅ ε⁰ ⋅ μ⁰ = c²

 

Constants B and D are multiplied together, the units cancel out and F is dimensionless with a magnitude of 1:    

B ⋅ D = F

And:

F ⋅ ε⁰ ⋅ μ⁰ = c²

 

These values are plotted as a field below, where light has the velocity = c, and rest mass = 0.  Due to these relationships, the absolute values of the integrals between μ⁰ and c, and c and ε⁰, are equal.

This is my Friday night, Pi Day 2025 post. This is the best I can do for now. I added the πR just for fun. Happi Pi Day, everyone!