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$ε_0$, 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?

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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.

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  • $\begingroup$ How is H easier to measure than B? We can measure B easily using Biot-Savart law, how can we measure H more easily? $\endgroup$ – Hisab Jul 6 '17 at 9:14
  • $\begingroup$ @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. $\endgroup$ – tparker Jul 6 '17 at 16:07
  • $\begingroup$ How is H measured? $\endgroup$ – Hisab Jul 6 '17 at 16:13
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According to me both these constants are connected with light. The speed of light is given by equation square root (1/mew knot×epsilon knot).In calculation of the forces the scientist found that there was always a factor of 4pie and the above constants. So they were introduced.

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