Why magnetic constant is specifically 4*pi*e-7

[Ozgen Eren]

Why magnetic constant is specifically 4*pi*10^-7? I can understand other experimental values, for example it would make sense if it was 3.1415*10^-7. But 4*pi is something we can never confirm experimentally since pi has infinitely many digits. So there has to be theoretical reason for us to specify 4*pi. What is that? Only possible meaning I can think for 4*pi is the solid angle of sphere at the moment.

 

[Dale]

It is not an experimental value. It is a defined constant in the SI system of units.

 

[Ozgen Eren]

But it has experimental results, we cannot define it arbitrarily. For example when there is an infinite line carrying some current, we calculate the magnetic field using this magnetic constant. Then when there is another charge moving around, it experiences a force caused by this constant. So we can directly measure and test it. That means we cannot just define magnetic constant to be any value that we want. I can understand that we do some measurements and have something close to that value. But I don't understand why it is regarded specifically 4*pi. Even if we could defined it arbitrarily, 4*pi is a pretty odd number to choose among all.

For that reason I believe there must be a derivation or logic behind coming up with 4pi*10^-7 apart from saying it just is like that because we defined it to be.

 

[jedishrfu]

The 4*pi usually comes out for geometric reasons related to the surface of a sphere as 4*pi*R^2 and the types of problem where the magnetic constant is used.

 

[Ozgen Eren]

I also thought derivation should have something to do with geometry, that indeed makes sense. But still, since we do not get to choose magnetic field arbitrarily, how did it happen to be 4*pi*10^-7?

I am assuming there should be a justification behind. What is the constraining laws for example? As far as I know we cannot just say it is equal to 4*pi*10^-7 even with experiment since there is pi involved.

 

[jedishrfu]

Okay the Wikipedia article get into it somewhat:

http://en.wikipedia.org/wiki/Vacuum_permeability

 

[nasu]

But it has experimental results, we cannot define it arbitrarily. For example when there is an infinite line carrying some current, we calculate the magnetic field using this magnetic constant. Then when there is another charge moving around, it experiences a force caused by this constant. So we can directly measure and test it. That means we cannot just define magnetic constant to be any value that we want. I can understand that we do some measurements and have something close to that value. But I don't understand why it is regarded specifically 4*pi. Even if we could defined it arbitrarily, 4*pi is a pretty odd number to choose among all.

For that reason I believe there must be a derivation or logic behind coming up with 4pi*10^-7 apart from saying it just is like that because we defined it to be.

The values of the parameters measured experimentally depend on how you set-up your system of units. For example, speed of light is measured, for sure. In terms of the defined meters and defined seconds. But you also can define the speed (as it is done now) and find the meter in terms of defined seconds and defined speed of light.
Some units have to be defined.

 

[jedishrfu]

I think the original Wikipedia article sums it up nice:

The ampere defines vacuum permeability
The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 6993200000000000000 * 2×10−7 Newton per meter of length.

Adopted in 1948, the effect of this definition is to fix the magnetic constant (permeability of vacuum) at exactly 6993400000000000000 * 4π×10−7 H / m. To further illustrate:

Two thin, straight, stationary, parallel wires, a distance r apart in free space, each carrying a current I, will exert a force on each other. Ampère's force law states that the force per unit length is given by

The ampere is defined so that if the wires are 1 m apart and the current in each wire is 1 A, the force between the two wires is 6993200000000000000 * 2×10−7 N / m. Hence the value of μ0 is defined to be exactly

I don't know what else to say. I'm sure you can find various alternative definitions but this is the currently accepted one.

 

[Dale]

But it has experimental results, we cannot define it arbitrarily. For example when there is an infinite line carrying some current, we calculate the magnetic field using this magnetic constant. Then when there is another charge moving around, it experiences a force caused by this constant. So we can directly measure and test it. That means we cannot just define magnetic constant to be any value that we want.

We can, in fact, define it to be any value we want. Remember, an Ampere is not something that is determined by nature, it is something defined by humans. The BIPM (the committee which defines the SI units) took a vote and arbitrarily decided to define the Ampere as the amount of current such that $\mu_0=4\pi 10^{-7} N/A^2$ exactly.

This means that there is experimental uncertainty about a measurement of the number of amperes of any given current, but no uncertainty about the vacuum permeability. All the experimental uncertainty is placed in the realization of the unit.

 

[Ozgen Eren]

I see, so Current*Magnetic Constant is what we can observe, that is the thing fixed.

And we define magnetic constant indirectly choosing an appropriate unit for current. And we especially choose Amperes to make magnetic constant 4*pi*10^-7 but we could have chosen a different current unit and make magnetic constant even 1 for example.

And definition of this current unit also changes charge unit, thus the electrical constant. So those units all related if we specify just one.

 

[Dale]

we could have chosen a different current unit and make magnetic constant even 1 for example

Yes, and in fact this is often done and makes EM calculations much more convenient. Specifically, this is done in both Gaussian and Lorentz-Heaviside units as well as Planck units and probably others:
http://en.wikipedia.org/wiki/Centim...ensions_of_the_CGS_system_to_electromagnetism
http://en.wikipedia.org/wiki/Planck_units#Electromagnetism

 

[f95toli]

This might actually change when the SI is modified in a few years. The ampere will then be re-defined (this is almost certain, no one likes the current definition) by defining the charge of the electron; meaning the Ampere will loose its direct connection to magnetic fields. There are various ways to make sure the SI stays consistent after these changes, and altering the value of the vacuum permeability is one (of several) possibilities. Any possible the change would be extremely tiny (but it would change, since it would be a measured value), but the point is that it would no longer be a defined constant.

 

[Vanadium 50]

But 4*pi is something we can never confirm experimentally since pi has infinitely many digits.

The same thing can be said about 1, used in many other definitions. 1 has infinitely many digits as well. Granted, most of them are zero, but that doesn't change your argument.