# Why are the electric and magnetic constants where they are?

$ε_0$, the electric permittivity and $μ_0$, the magnetic magnetic permeability were introduced in Coulomb's Constant and Ampere's Constant in order to make units and magnitudes match, in Coulomb's Law and Ampere's Force Law, respectively.

But Coulomb's Constant is: $1/4πε_0$ while Ampere's Constant is: $μ_0/4π$.

Why is it that these "correcting factors" ($ε_0$ and $μ_0$) were introduced in the denominator in one constant, and in the numerator in the other constant?

The value of $ε_0$, the electric permittivity of free space is: $8.8\times 10^{-12}$, and the value of $μ_0$, the magnetic permeability of free space is: $4\pi\times10^{-7}$.

Both of these values are less than 1.

So, while the presence of $ε_0$ in the denominator makes the value of $E$ larger than $D$, the presence of $μ_0$ in the numerator makes the value of $B$ smaller than $H$ . (Look at the expressions: $$E=D/ε_0 \qquad \hbox{and} \qquad B=H μ_0$$ )

Why is it so?

A combination of historical accident and a quirk of experimental practice. It's easier to measure ${\bf E}$ than ${\bf D}$, and ${\bf H}$ than ${\bf B}$. So the constitutive relations for electromagnetic fields in matter are ${\bf D} = \epsilon {\bf E}$ and ${\bf B} = \mu {\bf H}$, where in both cases you multiply the field that easier to measure by a constant in order to get the field that's harder to measure. There's no good theoretical reason for it. In fact, theorists often work in Heaviside-Lorentz units where $\epsilon_0$ and $\mu_0$ are both equal to $1$, so this issue doesn't come up.
• @Hisab You can't directly measure $B$ (or anything else) with the Biot-Savart law. By "measure" I mean sticking a magnetometer somewhere and reading off the field from that. – tparker Jul 6 '17 at 16:07