# Why \mu_0 is Used & Where it Comes From in Physics

• Galileo
In summary: The magnetic constant, or permeability of vacuum \mu_0 is defined to be 4 \pi \times 10^{-7} N/A^2. In summary, the magnetic constant is an incorporated quantity that is used to define the ampere and Coulomb.
Galileo
Homework Helper
The magnetic constant, or permeability of vacuum $\mu_0$ is defined to be $4\pi \times 10^{-7} N/A^2$.

The first time this constant comes up is usually in the Biot-Savart law, however it is not an empirical constant. Why? In what quantity is it incorporated? Is it used to define the ampere (which defines the Coulomb)? Then, which came first? The Coulomb, $\epsilon_0$ or $\mu_0$?

It is incorporated into the speed of light.

$$c=\frac{1}{\sqrt{\epsilon_0\mu_0}}$$

Incidentally,in SI,the Coulomb's law contains $$\frac{1}{4\pi\epsilon_{0}}$$ and the values are meant to yield EXACTLY $c=3\cdot 10^{8} \mbox{m/s}$,however we know that

EXACTLY $$c=299,792,458 \mbox{m/s}$$

Isn't this weird?

Daniel.

Galileo said:
Is it used to define the ampere (which defines the Coulomb)?

This is what Griffiths says.

Yes,the Ampère is defined using the attraction/repulsion force between 2 infinite long parallel conductors situated in vacuum and,obviously,through which a constant current of 1 A flows...And it that force comes up this #.

Daniel.

Galileo said:
The magnetic constant, or permeability of vacuum $\mu_0$ is defined to be $4 \pi \times 10^{-7} N/A^2$.

The first time this constant comes up is usually in the Biot-Savart law, however it is not an empirical constant. Why? In what quantity is it incorporated? Is it used to define the ampere (which defines the Coulomb)? Then, which came first? The Coulomb, $\epsilon_0$ or $\mu_0$?

i think you've almost answered your own question. check out both the current definitions and the historical definitions at http://physics.nist.gov/cuu/Units/background.html

first (after the meter, kg, second) came the Ampere. it was defined to be such a current that when passed in two infinitely long and very thin parallel conductors in vacuum spaced apart by exactly 1 meter, induced a magnetic force on those conductors of exactly $2 \times 10^{-7}$ Newtons per meter. that is what set $\mu_0 = 4 \pi \times 10^{-7} N/A^2$ if $\mu_0$ was anything different, that force per unit length would come out different than the defined value. then, of course, the Coulomb comes out to be an Ampere-second. there is nothing magical about these choices of units, they're quite anthropocentric and might not be used in 200 years.

until 1983, the meter was defined to be the distance between the centers of two little scratch marks on a plantinum-iridium bar in Paris (and got its original definition as 10,000,000 meters from the North pole to the equator) and the speed of light was measured to be 299792548 meters/second with some experimental error. at that time, then $$\epsilon_0 = \frac{1}{c^2 \mu_0}$$ also had experimental error. but in 1983 they changed the definition of the meter to be the distance that light in a vacuum travels in 1/299792548 seconds. that, plus the fact that $\mu_0$ was defined, had the effect of defining $\epsilon_0$. someday reasonably soon, they may redefine the kilogram to effectively give Planck's constant $\hbar$ a defined value.

r b-j

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The meter was defined as the distance on a Pt-Ir bar till 1961.

Daniel.

rbj said:
i think you've almost answered your own question. check out both the current definitions and the historical definitions at http://physics.nist.gov/cuu/Units/background.html

first (after the meter, kg, second) came the Ampere. it was defined to be such a current that when passed in two infinitely long and very thin parallel conductors in vacuum spaced apart by exactly 1 meter, induced a magnetic force on those conductors of exactly $2 \times 10^{-7}$ Newtons per meter. that is what set $\mu_0 = 4 \pi \times 10^{-7} N/A^2$ if $\mu_0$ was anything different, that force per unit length would come out different than the defined value. then, of course, the Coulomb comes out to be an Ampere-second. there is nothing magical about these choices of units, they're quite anthropocentric and might not be used in 200 years.

until 1983, the meter was defined to be the distance between the centers of two little scratch marks on a plantinum-iridium bar in Paris (and got its original definition as 10,000,000 meters from the North pole to the equator) and the speed of light was measured to be 299792548 meters/second with some experimental error. at that time, then $$\epsilon_0 = \frac{1}{c^2 \mu_0}$$ also had experimental error. but in 1983 they changed the definition of the meter to be the distance that light in a vacuum travels in 1/299792548 seconds. that, plus the fact that $\mu_0$ was defined, had the effect of defining $\epsilon_0$. someday reasonably soon, they may redefine the kilogram to effectively give Planck's constant $\hbar$ a defined value.

i'm having great trouble getting the tex to format correctly. sometimes when stuff with math is quoted, it comes out right.

dextercioby said:
The meter was defined as the distance on a Pt-Ir bar till 1961.

you're right. it was defined as some number of wavelengths of krypton-86 radiation from then until 1983 where it was simply the distance light (of unspecified frequency) travels in 1/299792548 second.

rbj said:
first (after the meter, kg, second) came the Ampere. it was defined to be such a current that when passed in two infinitely long and very thin parallel conductors in vacuum spaced apart by exactly 1 meter, induced a magnetic force on those conductors of exactly $2 \times 10^{-7}$ Newtons per meter. that is what set $\mu_0 = 4 \pi \times 10^{-7} N/A^2$

I understand that if we consider the Biot-Savart law to be

$$\vec{B} = kI\int\frac{d\vec{l}\times \hat{r}}{r^2}$$

, we find the magnetic field produced at a distance of 1m from an infinitely long wire carrying a unit current to be

$$\vec{B} = 2k \hat{\phi}$$

But the force part is nebulous, for the TOTAL force is infinite (we have to integrate a constant from -infty to +infty). It would work if it were the total force on 1m of wire:

$$\vec{F} = I\int d\vec{l} \times \vec{B} = \int_0^1 2kdl = 2k$$

So

$$k = 10^{-7}[N/A^2] = \frac{4\pi \cdot 10^-7}{4\pi}[N/A^2]$$

Now define $\mu_0 = 4\pi \times 10^-7$ and we get Biot-Savart law as we know it.

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Galileo said:

yer welcome. that whole NIST site at http://physics.nist.gov/cuu/ is very, very useful. i wasn't so good explaining the candela but i found a website http://www.electro-optical.com/whitepapers/candela.htm that did a better job. IMO the candela has no business as a fundamental physical unit in SI (or any other system of physical units) because it ain't a physical unit. it's a perceptual one. anyway, with this exception, the NIST site is good at explaining how anything is defined.

r b-j

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## 1. Why is \mu_0 used in physics?

\mu_0, also known as the permeability of free space, is used in physics to describe the relationship between electric and magnetic fields. It is a fundamental constant that helps us understand the behavior of these fields and their interactions with each other.

## 2. What does \mu_0 represent?

\mu_0 represents the intrinsic ability of free space to allow electric and magnetic fields to propagate. It is a measure of the strength of the vacuum and is an essential component in many equations that describe the behavior of electromagnetic waves.

## 3. Where does \mu_0 come from?

\mu_0 is a constant that was first introduced by the physicist James Clerk Maxwell in his famous set of equations known as Maxwell's equations. These equations describe the fundamental principles of electromagnetism and \mu_0 was included as a necessary constant to make the equations consistent and complete.

## 4. How is \mu_0 experimentally determined?

There are several ways that scientists have determined the value of \mu_0 experimentally. One method is by measuring the force between two parallel wires carrying electric currents and using this to calculate the value of \mu_0. Another method involves measuring the speed of light in a vacuum and using this to derive the value of \mu_0.

## 5. Why is \mu_0 considered a fundamental constant?

\mu_0 is considered a fundamental constant because it is a fundamental property of the vacuum and is not dependent on any other physical quantities. It is the basis for many other important constants in physics, such as the speed of light and the permittivity of free space, and plays a crucial role in our understanding of electromagnetism.

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