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

[Galileo]

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 couldn't find anything about this...

 

[quantumdude]

It is incorporated into the speed of light.

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

 

[dextercioby]

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.

 

[quasar987]

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

This is what Griffiths says.

 

[dextercioby]

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.

 

[rbj]

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

 

[dextercioby]

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

Daniel.

 

[rbj]

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.

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.

 

[quasar987]

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.

 

[Galileo]

Thanks for the link rbj.

 

[rbj]

Thanks for the link rbj.

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