# Vacuum permittivity vs 1/(36π) nF/m

Vacuum permittivity is $$ϵ_0\approx8.8541878128×10^{−12}\,\mathrm{F⋅m^{−1}}$$. Therefore, within 0.14%, $$ϵ_0\approx\displaystyle\frac 1 {36\,π}\,\mathrm{nF⋅m^{−1}}$$, thus $$\displaystyle4\,π\,ϵ_0\approx \frac 1 9 \mathrm{nF}⋅\mathrm m^{−1}$$. That comes handy in electronics engineering (I've even seen equality stated).

Is that purely accidental, or is there some reason for it, like perhaps the historical definition of the Farad or Volt?

• In cgs units, charge is measured in esu and potential in esu/cm. The unit of capacitance is thus just the centimetre! 1 cm is equal 1.11 pF. Commented Jan 28, 2022 at 11:38

## 1 Answer

The old definition of the constant for the vacuum permeability is $$\mu_0 = 4\pi \cdot 10^{-7} H/m$$ (since the redefiniton of SI units in 2019, the experimental value is still close). Then your observation follows from the speed of light in vacuum and solving for vacuum permittivity: $$\epsilon_0 = \frac{1}{c^2\mu_0}$$, when approximating $$c \approx 3 \cdot 10^8 m/s$$.

• Good answer. To answer the OP's final question, it's "accidental" in the sense that the speed of light is pretty close to a round number in SI units. But it does stem from historical definitions as well. Commented Jan 28, 2022 at 12:56