# Why does the permeability of free space invoke pi?

1. Jun 30, 2011

### connorb1542

I'm not asking a question of what is permeability, but rather why is pi involved in its definition of 4pi x10^-7? As far as I know, pi is generally only used whenever dealing with a circle. How does the idea of magnetism relate to a circle?

2. Jun 30, 2011

### soothsayer

That value for the permeability of free space has the units N*A-2. The A being amperes. The Ampere is defined as being the current maintained in two wires a distance of 1 meter apart that would generate a force of 2x10-7N/m. Given this definition, along with Ampere's force law, F=($\mu$0I2)/(2$\pi$r), we can see that the permeability of free space is exactly 4$\pi$x10-7N*A-2. There's nothing magical about it, unfortunately, it's just the way we define the ampere that makes that value the way it is. Any constant could have any value if we just change the units and how they are defined =)

3. Jun 30, 2011

### connorb1542

True, but then why is it that 1/sqrt(με) is equal to the speed of light? How is it that two arbitrarily defined constants can be combined in this way to produce the speed of light?

4. Jun 30, 2011

### Andy Resnick

Eugene Wigner's article "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960) opens with almost the exact same question. It's an excellent article, well worth reading.

5. Jun 30, 2011

### silmaril89

Using Maxwell's equations you can derive the wave equation for an electromagnetic wave, it turns out that the velocity of that wave would be 1/sqrt(mu * epsilon). Those constants are not arbitrary, they are determined experimentally. When Maxwell first derived the wave equation for an electromagnetic wave he realized the velocity was extremely close to the value of the speed of light. He used that fact to realize that light must be an electromagnetic wave.

6. Jun 30, 2011

### connorb1542

so then if the constants are derived experimentally, and mu is not just a result of how we define the ampere, I go back to my first question: why pi? where does the pi come from?

7. Jun 30, 2011

### SteamKing

Staff Emeritus
Pi has been known to be associated with spheres as well as circles.

8. Jun 30, 2011

### Skaperen

Whenever you deal with two things that relate to each other in a quadrature relationship at right angles, where some total vector is always the quadrature sum of these two, then you are dealing with a circle. At least you'll have to deal with squares and square roots. Then if you need to work with the circumference or area, the value of Pi will kick in as the non-arbitrary relationship between vectors and the containing area. Magnetism will relate to this because it also involves electrical forces which are at right angles, and together form a circle.

I hope I explained it enough without having to stick you with the formulas that you were questioning in the first place. I know a lot of math oriented people think of the formulas themselves as the absolute foundation and never question why that formula is the way it is. For them, the formula is the way it is. I do like to question why, and explain things in a non-mathematical way. But it's hard to completely get away from at least some math.

9. Jun 30, 2011

### silmaril89

The best way I can possibly explain it (given my limited knowledge) is that it certainly has to do with circles/spheres. Take a look at the difference in the way Maxwell's equations look in different units (e.g. SI units and Gaussian units). In Gaussian units you will notice that there are these extra 4pi's. Where as they disappear when you look at Maxwell's equations in SI units.

Ampere's law in SI units (with constant electric field):

$$\vec{\nabla} x \vec{B} = \mu_0 \vec{J}$$

Ampere's law in Gaussian units (with constant electric field):

$$\vec{\nabla} x \vec{B} = \frac{4\pi}{c} \vec{J}$$

So you can see in Gaussian units that we have a 4pi whereas in SI units we don't. The 4pi gets absorbed into the constant $$\mu_0$$

Is that satisfactory for you?

10. Jun 30, 2011

### Antiphon

The free-space permeability is defined, not measured in MKS units. The free-space permittivity is measured.

11. Jun 30, 2011

### rbj

no it isn't. not since 1983 when they defined the meter to fix c to be 299792458 m/s.

$\mu_0, \epsilon_0, c, Z_0$ are all exact because of the definition of the meter and ampere.

12. Jun 30, 2011

### rbj

you should take a class in E&M. the last two Maxwell's equations together can form a wave equation and you can derive the wavespeed as $\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ from that wave equation.

it's actually $4 \pi$ and that comes from the formula for the surface area of a sphere. and what the surface area of a sphere has to do with it is about how the inverse-square law fundamentally works. check that out in Wikipedia.

also go to the NIST site for the definition of the Ampere. it's because how the Ampere got defined is why $\mu_0$ takes on the value it does.

13. Jun 30, 2011

### connorb1542

i just graduated from high school, and i took a calculus-based e&m class (AP physics c). unfortunately we didnt cover maxwell's equations together in one place with respect to electromagnetic waves, only independently with emphasis on problem-solving and not theory. so as i'm looking, i'm seeing that the pi either comes from the circular relationship between right angle vectors or from the surface area of a sphere.

if that were true then by redefining the ampere, we would redefine μ0. and by redefining μ0, using the wave equation from maxwells equations to get the wavespeed as 1/sqrt(μ0ϵ0) would yield a different value of c then what we observe, would it not? thats why ive been saying that it cant just have come from the way we define some unit of measurement, because if you redefine the ampere you redefine μ0, and by redefining μ0 you redefine the speed of an electromagnetic wave. and also, if the 4pi came from an INVERSE square law, wouldnt it be on the bottom, in the inverse part? like 1/4pi? in addition, not all equations using μ0 are inverse square laws

14. Jun 30, 2011

### silmaril89

If you redefined μ0 by changing the ampere and then proceeded to use 1/sqrt(μ0ϵ0) for the speed of light, you would not be changing the actual speed of light, you would just be changing the units that the speed of light is defined in. The 4pi absolutely comes from surface area of the volume that we are working in.

15. Jun 30, 2011

### connorb1542

ah, ok. thank you. im still not completely certain of the relation between permeability and a spherical volume

16. Jun 30, 2011

### cragar

I wonder if the pi comes from making an Ampererian loop. But then again when you use amperes law you never draw spheres, just circles or boxes. So im not sure.

17. Jul 1, 2011

### soothsayer

There is a 4$\pi$ inthe permeability of free space due to the way the Ampere is defined, yes, it does have something to do with spheres and inverse square force laws. If you change the definition of the ampere, it doesn't change the speed of light, it just changes the units you're measuring it is, as someone stated. Gotta involve amperes in the permittivity of free space too, I believe, for the units to check out, and then you probably get your SI value for c back or something.

18. Jul 1, 2011

### cragar

does the 4pi come from the Biot–Savart law and if it does, does the 4pi come from surface area of a sphere?

19. Jul 1, 2011

### silmaril89

While thinking this over myself. I do think it came from the Biot-Savart law. I think that they set up some arbitrary magnetic field that can be calculated using the Biot-Savart law. Then they measure the actual magnetic field and compare it to the magnetic field calculated from the Biot-Savart law which enables them to define the constant necessary to make the two equivalent. Since the the law has a 4pi on the bottom, the 4pi was introduced into the permeability of free space in order to cancel them. They did this because you can think of the 4pi in the Biot-Savart law as being a part of the constant of proportionality between calculated and measured magnetic fields. And yes that 4pi certainly comes from an integration over spherical coordinates.

Last edited: Jul 1, 2011
20. Jul 1, 2011

### f95toli

Almost, the Ampere is up for a re-definition quite soon (in a few years). It is likely (not absolutely certain) what will happen is that we will define the value of the electron charge, and then define the Ampere using number of charges/second (instead of the current definition, which no ones is using because it is so awkward to realize, the ampere is in reality usually realized via the Volt and the Ohm).
Now, from what I remember (this has more to do with politics than science, so I've heard various suggestions) this will in turn mean that epsilon_0 will be a measured ("inexact") value whereas mu_0 will still be defined (mainly because the pi is so convenient, there is no "deep" reason). The value of c will of course stay the same.

Note that all of this is provisional, we should now more after the next CGPM meeting.

Also, to those of you who are looking for a "physical" reason for the p: there isn't one; the current definition of the ampere was (arbitrarily) chosen so that mu_0 would have "nice" value; simply because it is convenient.

There should be some relevant links on the wiki for the SI, including info about future re-definitions.

21. Jul 1, 2011

### tiny-tim

the electric field at distance r from a stationary charge Q is E = Q/4πε0r2

the magnetic field at distance r from a charge Q with speed v (<< c) is B = µ0Qv/4πr2

this is so that the fluxes through a sphere of radius r (with area A(r) = 4πr2) are​

∫ E dA = Q/ε0

and ∫ B dA = µ0Qv = µ0J

in other words: if we define the coulomb (and the amp) so that the fluxes enclosing a charge Q or a current J are Q/ε0 and µ0J, then we need a 1/4π in the field formulas … we could avoid that by redefining the coulomb (and the amp), but then we'd get a 4π in the fluxes

22. Jul 1, 2011

### connorb1542

i think i've gotten a satisfactory response, thanks to all who responded!

23. Jul 1, 2011