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Foundations: Physics
Speed of light from the permeability and permittivity of free space
Maxwell’s equations give electromagnetic waves a speed fixed by two constants of free space: the magnetic permeability μ₀ and the electric permittivity ε₀.
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- FPH-60
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- v1
- Field
- Foundations: Physics
- Result
- c [m/s]
μ₀ is the permeability of free space (1.2566 × 10⁻⁶ H/m) and ε₀ its permittivity (8.854 × 10⁻¹² F/m). Maxwell found that electric and magnetic fields can travel as a wave whose speed is fixed by these two constants. In a material, use μ and ε of the material to get the wave speed in it, v = 1/√(με).
Maxwell’s equations give electromagnetic waves a speed fixed by two constants of free space: the magnetic permeability μ₀ and the electric permittivity ε₀. The result is the speed of light in a vacuum, about 3.00 × 10⁸ m/s.
Calculator
- Speed in km/s299,800 km/s
- Impedance of free space, √(μ/ε)376.7 Ω
- Wavelength of a 100 MHz wave, c/f2.998 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With the standard constants the speed is 2.997925×108 m/s, about 3 × 10⁸ m/s: the speed of light.
- Light, radio waves and X-rays are the same kind of wave; they differ only in frequency. Electrical measurements of μ₀ and ε₀ gave the speed of light before it was measured optically.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| μ0 Permeability | 1e-7 to 1e-5 H/m | 1.063e8 | 1.063e9 | 9.565e8 | +10% | −4.65 % |
| ε0 Permittivity | 1e-12 to 1e-10 F/m | 8.921e7 | 8.921e8 | 8.029e8 | +10% | −4.65 % |
Sweep of Permittivity
| Permittivity (F/m) | Wave speed (m/s) |
|---|---|
| 1e-12 | 8.921e8 |
| 1.338e-11 | 2.439e8 |
| 2.575e-11 | 1.758e8 |
| 3.813e-11 | 1.445e8 |
| 5.05e-11 | 1.255e8 |
| 6.288e-11 | 1.125e8 |
| 7.525e-11 | 1.028e8 |
| 8.763e-11 | 9.53e7 |
| 1e-10 | 8.921e7 |
Sweep one variable
Variables and how each one contributes
Wave speed
What the equation gives you. Everything else feeds it.
Permeability
The permeability of the medium, in H/m (1.2566 × 10⁻⁶ for free space). Lowers the wave speed in inverse proportion to the square root of μ. Raising it by 10 % (to 1.382×10−6) moves the result by −4.65 %, from 2.998×108 to 2.858×108 m/s.
Permittivity
The permittivity of the medium, in F/m (8.854 × 10⁻¹² for free space). Lowers the wave speed in inverse proportion to the square root of ε. Raising it by 10 % (to 9.74×10−12) moves the result by −4.65 %, from 2.998×108 to 2.858×108 m/s.
Worked example with the values above
- Product: μ₀ε₀ = 1.25664×10−6 × 8.85419×10−12 = 1.11265×10−17 s²/m².
- Square root: √(μ₀ε₀) = 3.33564×10−9 s/m.
- c = 1 / 3.33564×10−9 = 2.997925×108 m/s.
Assumptions and limits
- Free space (vacuum); air differs by about 0.03 %.
- Plane electromagnetic waves.
Common mistakes
- Entering ε₀ in pF/m or μ₀ in µH/m without converting to SI.
- Taking the square root of only one constant.
- Using the vacuum values for a wave in a cable or dielectric.
Statistics and uncertainty
The speed of light is exact by definition, and ε₀ follows from μ₀ and c, so the product μ₀ε₀ is fixed even though each constant is measured. Any error here comes from the values you enter.
Where it is used
- Showing that light is an electromagnetic wave.
- Finding wave speeds in cables and dielectrics from μ and ε.
Maths you need
- Permittivity and permeability: constants of free space
- Wave speed: v = fλ