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

#### LucasGB

Epsilon0, the electric constant, and Mu0, the magnetic constant, 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: 1/4 pi Epsilon0
and Ampere's Constant is: Mu0/4 pi

Why is it that these "correcting factors" (Epsilon0 and Mu0) were introduced in the denominator in one constant, and in the numerator in the other constant? For example, they could have just made Ampere's Constant 1/4 pi Mu0, in order to resemble Ampere's Constant. Considering the values of Epsilon0 and Mu0 are defined, they were free to put them wherever they wanted. They could have just said Epsilon0 was 8.8 x 10^12, as opposed to 8.8 x 10^-12 and written Coulomb's Constant as Epsilon0/4 pi, in order to resemble Ampere's Constant.

So, what gives?

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#### tiny-tim

Homework Helper
Hi LucasGB! Merry Christmas!

(have an epsilon: ε and a mu: µ and a pi: π )

I'll guess it's because E = (1/ε0)D

can be rewritten E = µ0c2D,

which makes it look like B = µ0H

#### LucasGB

Merry Christmas! Wow, thanks for the reply, but I didn't really understand anything, I don't think I'm on that level yet. Could you dumb that down for me a little bit please?

#### Pengwuino

Gold Member
The first equation relates the electric field to what is known as the Displacement Field. In this case, it relates the electric field to that field in linear materials (that is, a material who's D field is linear in E). The second you just need to know that the speed of light squared$$c^2 = \frac{1}{\mu_0 \epsilon_0}$$. The third is like the first, the magnetic field (which is actually H in most books I've read) is linear in materials vs. B which is called the magnetic induction.

My grad E/M professor actually talked about why the constants became the way they were. The $$4 \pi$$ is a matter of just how we integrate over volumes and surfaces and the usual but the actual constants is another matter. $$\mu_0$$ is defined as exact. He couldn't remember the exact details but he said something about it being one of the easiest properties to measure to the highest of precisions. The speed of light has also been defined as exact. Thus, $$\epsilon_0$$ is determined through those two.

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Staff Emeritus
$$\mu_0$$ is defined as exact. He couldn't remember the exact details but he said something about it being one of the easiest properties to measure to the highest of precisions
It's exact because it is how the ampere is defined.

#### tiny-tim

Homework Helper
µ0 is defined as exactly 4π 10-7 H/m

#### sokrates

µ0 is defined as exactly 4π 10-7 H/m
yes, exactly: $4 \times\pi \times 10^{-7}$ H/m

:)

#### Pengwuino

Gold Member
As exact as $$\pi$$ :)

#### tiny-tim

Homework Helper
Merry Christmas from here in London

#### Integral

Staff Emeritus
Gold Member
As exact as $$\pi$$ :)
Which is exactly as exact as you need it to be.

#### LucasGB

Guys, guys, you're missing the point. I understand where mu0 and epsilon0 came from, how they're defined, etc. I just want to know the reason mu0 is on the numerator and epsilon0 on the denominator. It didn't have to be that way, and I would like to know the historical reason for doing it that way. Thanks a lot for the help and merry christmas!

#### Pengwuino

Gold Member
Guys, guys, you're missing the point. I understand where mu0 and epsilon0 came from, how they're defined, etc. I just want to know the reason mu0 is on the numerator and epsilon0 on the denominator. It didn't have to be that way, and I would like to know the historical reason for doing it that way. Thanks a lot for the help and merry christmas!
Oh, so are you asking why, say, for example, Newton's law of Gravitation is $$F = \frac{GmM}{r^2}$$ as opposed to $$F = \frac{mM}{Gr^2}$$ with a simple inversion of G? I wanna guess that it comes from Maxwell's equations but at the same time, Maxwell's equations could have just as easily been defined the other way around. I think one thing to note would be that electricity and magnetism were separate fields of study until Maxwell came along. This might be interesting to find out the history behind it.

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#### LucasGB

Oh, so are you asking why, say, for example, Newton's law of Gravitation is $$F = \frac{GmM}{r^2}$$ as opposed to $$F = \frac{mM}{Gr^2}$$ with a simple inversion of G? I wanna guess that it comes from Maxwell's equations but at the same time, Maxwell's equations could have just as easily been defined the other way around. I think one thing to note would be that electricity and magnetism were separate fields of study until Maxwell came along. This might be interesting to find out the history behind it.
This, my friend, is EXACTLY what I want to know. I could understand if both mu0 and epsilon0 were in the numerators or in the denominators of their respective constants, but the fact that they're in different places is quite intriguing. This website gives a great insight into the origins of these constants: http://info.ee.surrey.ac.uk/Workshop/advice/coils/unit_systems/ but it doesn't say anything about this issue. Very curious, indeed.

#### LucasGB

In fact, both Ampere's and Coulomb's constants were DEFINED to have the values they have (10^-7 and 9 . 10^9, approximately), and only then physicists decided to express them with a 4pi, in order to simplify Maxwell's Equations. But they still had to keep the values at 10^-7 and 9 . 10^9. So they thought, "let's create a number and divided it by 4pi so that the whole thing equals 10^-7, and let's create another number and divide it by 4pi so that the whole thing equals 9.10^9". These numbers they created are mu0 and epsilon0. But in the case of epsilon0, they didn't DIVIDED it by 4pi, they divided 1 by 4pi times epsilon0! Why, lord, why?!

#### LucasGB

So... any thoughts?

#### diazona

Homework Helper
Interesting question! I suspect it's related to the original names for these constants, the "vacuum permittivity" $\epsilon_0$ and "vacuum permeability" $\mu_0$. As I understand it, the letters $\epsilon$ and $\mu$ were first introduced to represent the permittivity and permeability of materials. These properties were first measured in capacitors and inductors and various sorts of electromagnetic experiments - for example, when you replace the air filling a capacitor with a dielectric material, it changes the capacitance by a factor $\epsilon_\text{dielectric}/\epsilon_\text{air}$. Every material is associated with some permittivity value. So it seemed (seems) sensible to define a permittivity for the vacuum as well, which would have to be the right value to reproduce the capacitance of an ideal capacitor with no dielectric in it. $\epsilon_0 = Cd/A$ for a parallel-plate capacitor. I assume the same process took place for inductors and magnetic materials, that the vacuum permeability was defined to represent an ideal empty inductor.

Since the theory of electromagnetic fields was originally developed to describe physical devices including capacitors and inductors (well, actually pith balls and wires came first, but circuits weren't far behind), these constants would have been transferred from the formulas developed to describe circuit elements into the general EM field theory, i.e. Maxwell's equations and the Coulomb/Ampère force laws. So by the time anyone thought to express the force constants in terms of $\epsilon_0$ and $\mu_0$, there would have been a well-established pattern of using them as a permittivity and permeability, respectively - and that's why the algebra works out the way it does.

(At least, as I said, that's my impression, but I don't know/don't remember specific sources to back it up, so I could be wrong)

Incidentally, when Maxwell discovered the wave solution to his equations, he noticed that the math predicted a speed of $1/\sqrt{\epsilon_0 \mu_0}$ in vacuum (or $1/\sqrt{\epsilon \mu}$ in matter), but only by plugging in experimentally measured values was he able to realize that this actually corresponded to the speed of light.

#### LucasGB

Wow, that seems about right. Thank you very much, diazona, you never disappoint me!

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