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$$ k_{e}=\frac{ 1 }{ 4 \pi \epsilon_{0} } ~~ \leftarrow~~ \text{coulomb constant} $$

I would like to know whether the follwing "constant of permeability"

$$ \frac{ 1 }{ 4 \pi \mu_{0} } $$

has a conventional symbol or name.

I see it many times in answers in magnetic charges, magnetic moment.

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  • $\begingroup$ Maybe you are referring to the constant $\mu_0 / 4 \pi$ which is present in front of the Biot-Savart law? $\endgroup$ Jun 21, 2021 at 8:25
  • $\begingroup$ I don't know what Biot-Savart law is ... $\endgroup$ Jun 21, 2021 at 8:32
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    $\begingroup$ The Biot-Savart law is an equation describing the magnetic field generated by a constant electric current. I have never seen the constant in your original post. $\endgroup$ Jun 21, 2021 at 8:50
  • $\begingroup$ I see. Oh really. $\endgroup$ Jun 21, 2021 at 8:53

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We see the constant $k_{e}=\frac{ 1 }{ 4 \pi \epsilon_{0} } $ quite often in electrostatics. And in a huge number of cases it comes with the attached $4\pi$.
We know that $\epsilon_{0}$ is the permittivity of air and the constant $k_{e}$ is Coulombs constant.

Now coming to $\mu_0$ , by itself it is called as the permeability of air and can be extended to other mediums by multiplying with the appropriate constant called as the relative permeability. The process is very similar to electrostatics where you define $\epsilon$ for different media.

But there is one variation. The way we use $k_{e}=\frac{ 1 }{ 4 \pi \epsilon_{0} } $ everywhere in electrostatics, we use $\frac{\mu_0}{4\pi}$ everywhere but not as often as $k_e$. It is not given a special name because it doesn't have much application (I'm talking about the constant), whereas $\mu_0$ has HUGE applications.

Fact 1: As far as I know (I'm in class 12), $\frac{1}{4\pi\mu_0}$ is NEVER seen.

Fact 2 : The $\epsilon_0$ is is denominator but the $\mu_0$ is in the numerator because of the dimensions. You can see by dimensional analysis that speed $c=\frac{1}{\sqrt{\mu\hspace{0.1cm}\epsilon}}$

Fact 3 : $\frac{1}{4\pi\epsilon_0} = 9\cdot10^9$ and $\frac{\mu_0}{4\pi} = 10^{-7}$

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  • $\begingroup$ Yes, you are correct, we don't use the expression $\frac{1}{4π\mu_{0}}$, which the OP mentioned. $\endgroup$
    – Nilabja
    Jun 21, 2021 at 9:07
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    $\begingroup$ Secondly, ${\mu_0}$, is called permeability of free space or vacuum. That of air is slightly different. $\endgroup$
    – Nilabja
    Jun 21, 2021 at 9:08
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    $\begingroup$ 1.00000037 is the permeability for air while 1 is the permeability of vacuum. They are nearly the same, at least for a 12th grade student like me or the OP $\endgroup$ Jun 21, 2021 at 9:14
  • $\begingroup$ That's true. Even I am a 12th grader. We take those to be equal. $\endgroup$
    – Nilabja
    Jun 21, 2021 at 9:15

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