# Why does the permeability of free space have an exact value?

1. May 21, 2004

And why is it related to pi?

2. May 21, 2004

### AKG

I don't think it does have an exact value. Do you have a reference stating that? I believe it would have been determined experimentally, and to the first 5 or 6 digits, it may work out to be the same first 5 or 6 digits of $$4\pi \mbox{ x } 10^{-7}$$, but I doubt that if they could perform some perfectly precise and accurate experiment, that they would find that $$\mu_{o} \neq 4\pi \mbox{ x } 10^{-7}$$, probably just $$\mu_{o} \approx 4\pi \mbox{ x } 10^{-7}$$.

3. May 22, 2004

### TALewis

When you see physical constants listed as exact, rather than given to an approximate number of decimals, in my experience this means that the units the constant is given in are defined in some way or another in terms of the constant itself.

For example, my text says that the speed of light in vacuum is exactly:

299792458 meters per second

How can anyone say this is the number, exactly? It's not because they've got a really good stopwatch and meterstick. They can do it because they they define the meter in the following way:

1 meter is equal to the distance traveled by light in vacuum during a time of 1/299792458 second.

Do you see how that works? That's where the exactness of the constant comes from, because the constant's units are defined in terms of the same constant!

Something similar is probably the case for the permeability constant. The units T*m/A must somehow be defined such that the permeability of free space can be given as an exact number. Pi fits in as well from the definition of the constant in some way. Clearly I don't know a whole lot about the details: I had one introductory course with a section on magnetic fields, and it was not very deep.

One last example: Why does 1 in = 2.54 cm exactly? Not because we got our meterstick and our yardstick and lined them up and determined that was the exact value. No, it's exact because we say it's so -- that's how the units are defined.

My point is, if someone tells you they have measured something and can report it to you exactly, they're lying or they're misinformed. We can measure things to high degrees of precision that are more than enough for practical uses, but never to absolute exactness. However, we can say that certain quantities are exact so long as we define the units in an appropriate way to make it so.

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Edit: Through a little more quick research it seems the exact value of the permeability of free space comes from the definition of the Ampere:

"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 2 x 10^-7 newton per meter of length."

http://physics.nist.gov/cuu/Units/ampere.html

Also, it can be shown that the magnetic force per unit length on on two parallel wires is given by:

$$\frac{F_B}{L} = \frac{\mu_0I_1I_2}{2\pi a}$$

FB/L is the force per unit length on the wires, $\mu_0$ is the permeability of free space, I1 and I2 are the currents in each wire, and a is the distance between the wires.

If you plug in the values from the definition of the ampere, you can get the exact value of $\mu_0$.

Last edited: May 22, 2004
4. May 22, 2004

### AKG

I believe the French came up with the meter by making a very large measurement and dividing it by some number, and having the result be the meter. Although it is possible that they've changed the definition of the meter.

In fact, I did some research, and that's exactly what happened.

The definition of the Ampere seems odd, perhaps "flawed". Since they can never have infinitely long wires, the Ampere isn't really standard. The forces between wires that are 100 m long and wires that are 101 m long may be different, and so the Ampere would have different values.

The definition for "second" corresponds to something that, as far as I can tell, can be counted (something to do with the number of vibrations of a certain molecule or atom). The meter seems like something that can actually be measured too, if they can in fact mark off the position of a beam of light at t = 0 and t = 1/299792458 s. Although, I would suspect that we may not actually be capable of such precision, potentially making the definition of the meter "flawed" as well.

5. May 22, 2004

### TALewis

It's the same kind of mathematical idealism that you see in something like the Pythagoream theorem. Sure, I say that a^2 + b^2 = c^2, but I could never physically construct a right triangle so perfect as to be able to say that the measurements of the sides satisfy the equation exactly. There is always uncertainty.

I find the definition of the meter particularly tickling. 1 meter is the distance light travels in 1/299792458 second. OK, how far is that? Well, no one can say, exactly, other than to define it so. To call it one meter.

But if I'm going to make a meterstick, I'm not going to shut myself in a dark room with a flashlight and a wristwatch. If I'm going to make a wristwatch I'm not going to count the vibrations of a cesium atom to tell me how long a second is. So while a physicist offers an intellectually satisfying definition, it's up to an engineer to make it practical.

6. May 22, 2004

### AKG

Well, mathematics (including geometry) are deductive axiomatic systems. Although the Pythagorean Theorem is not a definition, it's practicality is essentially irrelevant since it need only apply to mathematical objects which exist "in" mathematics, and need no correspondence to the real world. On the other hand, charge will pass through a wire, whether we define it or construct a logical system that defines it or not. However, to define the basic unit of measurement of the Ampere to something that is necessarily immeasurable is, well, pretty odd.

In other words, a mathematical theorem or definition would be "bad" if it contradicted the existing axioms and definitions in mathematics. The Pythagorean Theorem is not so. A physical definition would be "bad" if it did not correspond to something that existed or could exist in reality. The definition of Ampere is so.

7. May 22, 2004

### Integral

Staff Emeritus
The value of $$\mu_0 = 4 \pi x10^{-7} \frac H m$$ is due to the choice of constants in Maxwell's equations. This system is called the Rationalized MKS system. Do a web search on that term you should be able to find the details.

8. May 22, 2004

### HallsofIvy

I was about to protest that that was ridiculous until I checked up on it!
Yes, in 1986, the meter was redefined exactly as you say! Gosh, do I feel old.

9. May 22, 2004

### Hurkyl

Staff Emeritus
Basically, the permeability of free space is a geometric constant (like pi), not an empirical constant.

10. May 22, 2004

### krab

$$\mu_0=4\pi\times 10^{-7}\,\mbox{webers/amp-m}$$
The above makes no sense if the units, webers per amp-metre, or newtons per amp squared, are omitted. (BTW, it seems no one knows how to make a multiplication sign with tex. Please click on the tex above to find out.) This is exact, because after extrapolating from experiment that the permeability of free space is a universal constant, the unit of current (amp) was redefined to make it exact. You might ask, why not make the unit of current exactly equal to the square root of the force unit. Then we would get $\mu_0=1$, exactly, dimensionless, and $\mu_0$ would not have to appear in any formulas. This has in fact been done, and the system of units is called the emu system.

This may not be clear, so I'll draw an analogy. Before Newton, it was not understood that F=ma. What Newton actually found out was that F is proportional to mass times acceleration, or $F=Cma$. From that time onward, units were used for Force, mass and acceleration (distance and time) to make C exactly equal to 1.

Last edited: May 22, 2004
11. May 22, 2004

### H-bar None

Some of the newer physics texts offer this explanation:

The problem is the standard rod or weight. Most of the standards were held in some bureau in France. For example, the standard for mass, the kilo, was a cylidrical metal weight. I'm not as sure about length but I'm sure there was a physical rod as the standard of length.

This worked for awhile until a set of problems arose. First, most materials warp, expand, and contract over time make the standard useless. Second no one can precisely duplicate the standard.

Tired of useless, imprecise, standards of measurement. Worry no more, the UNIVERSE provides with the new, unshakeable, standard.

Light!!! :surprise:

Yes! Light travels at a constant speed. The meter is now defined in terms of light travel time. The second is also defined as 2.999792E8 meters of light travel time.

The vibration of a cesium atom is constant. A cesium atom electons are excited and gives of light. From Q.M. the energy of photon also reveals the frequency. So the second is the inverse of the frequency of cesium atom
vibrations.

12. May 22, 2004

### EL

You can't define both meter and second from the velocity of light.
As you said later on, the second is now defined from vibrations in atoms, and not from the speed of light.

13. May 22, 2004

### H-bar None

Your'e right. I was at work when I typed up the response and I had to go off the top my head.

According to the Neil Gershenfeld's The Physics of Information Technology, The velocity of light became the standard to define the meter in terms the time light travels in 1/2.9999792E8 second. For mass, the kilogram platnum-iridium cylinder is kept the Bureau of International des Poids et Measures(BIPM) in Sevres, France.

EL, this is where I got it confused. In Relativity it makes life a little easier to relate time and space in terms of c. As length can be measured in terms of time, time can be measured in terms of length. One second is 2.999E8 meters of light travel time in this case. The meter is defined as before. Ah the blessed unity of time and space.

We see this everday conversation. Detroit is 12 hrs away from Atlanta. There is no reason we couldn't say 12 hours is the distance from Detroit to Atlanta. The assumption is the speed of travel is constant.

14. May 22, 2004

### AKG

krab

One problem I have, which I pointed out, is that the definition of the Ampere relies on something physically impossible (infinitely long wires). In fact, since there will be gravitational attraction between any two wires as well, it seems the definition of the Ampere is flawed in comparison to, say, the definitions of the second (which depends on a constant and something countable) or the meter (which depends on the constant speed of light).

Second of all, it is said that any time less than the Planck time is meaningless. We can take this to suggest that, for all meaningful intents and purposes, time is quantized. Charge, too, is quantized. Current measures the rate of flow of charge with respect to time. But if current depends on $\mu_{o}$ which depends on $\pi$ which is irrational, then doesn't that contradict the fact that current is based on quantum measurements (sorry if I'm using the wrong terminology). A second must be a multiple of the Planck time, I would think, since, if, say a second was 100.5 planck times (I know that's way off) that would mean 0.5 planck times would have some meaning, but it doesn't, as far as I know. So, if an amp is charge/time, then it must be some_natural/some_natural x some units, giving some rational with units, whereas pi is irrational. Is this not contradictory?

15. May 23, 2004

### TillEulenspiegel

AKG: Although the Pythagorean Theorem is not a definition, it's practicality is essentially irrelevant since it need only apply to mathematical objects which exist "in" mathematics, and need no correspondence to the real world."

Are you saying that there is no real world correspondence between the theory and the reality?

16. May 23, 2004

### AKG

When did I say that? But anyhow, I'd like you to show me a truly two-dimensional object, a perfectly straight line, etc. We can use mathematical models to very usefully model reality and thus engineers and scientists can actually make use of the theorem. But the theorem proves something about the an object in the "world" of mathematics because it is deduced by the axioms which define the world of mathematics; these axioms do not define reality, and thus the theorem is not a truth of reality, simply mathematics. Mathematics is so great in part because it can model reality so well that we can apply mathematical concepts to reality.

17. May 23, 2004

### Integral

Staff Emeritus
Any measurement made is subject to errors, so the effects wire length is easily dealt with, as for gravitational attraction, that will be constant with and without applied current, What is measured is a CHANGE in force, thus once again easily dealt with.
Current measurements are made with quantities and times well outside of quantum limits. QM need not be considered for this measurement. Once again someone raises an irrational fear of irrational numbers. Any measurement ever made will only be good to a finite number of digits, we are able to generate as many digits of Pi as needed.

You must remember while these are physical quantities the numbers used to express magnitude are a result of the system of units used. The pi is present in rationalized MKS units but not in other systems.

18. May 30, 2004

### confuted

The Ampere was originally defined as the amount of current that would deposit a certain weight of silver in an electrolytic cell in a given amount of time. The definition was later changed to be a tad bit more precise. Anyway, even with the current (haha pun) definition, it's very possible to have wires so long that they are "close enough" to infinitely long that our equipment will never tell the difference. I mean, really, how many decimal points do you need? I believe something like 30 decimals of $$\pi$$ is sufficient to calculate the circumfrence of the universe to within the radius of a proton - surely this is sufficient.

19. May 30, 2004

### Creator

Oh yea, H-bar? Just take your cesium powered wrist watch to the moon and back and then compare it to mine and you'll see that the number of vibrations are not at all constant. :tongue2: Each successive transition is only constant in a constant gravitational field and in equivalent inertial frames.

In actuality, we DEFINE the SECOND as a set NUMBER of cycles of the cesium transition. The second is defined as exactly 9, 192,631,770 cycles of the cesium 133 atom, (not as the inverse frequency of cesium).

Creator

20. Oct 3, 2004

### rogerperkins

permeability

What if I told you that the permeability of free space was increasing due to the expansion of the universe? That would mean that the speed of light is also increasing. Which would mean that light emitted from distance stars billions of years ago would actually be travelling faster now than when it was first emitted, and therefore appear to be red shifted more to the red than it actually would be just due to the expansion of the universe alone. :surprised