# Pondering an old quote from Maxwell

1. Feb 21, 2010

### diagopod

I'm reading an early paper of Maxwell's in which he defines c in a peculiar way that I hadn't heard before:

"the number of electrostatic units in one electromagnetic unit of electricity"​

Could someone help me get a sense of what he means by this? I'm familiar with the definition of c = 1 / sqrt epsilon-naught x mu-naught, but not sure how to translate that into the statement above, which seems more concise.

2. Feb 21, 2010

### BAnders1

During Maxwell's time, the theory of vector calculus was not well-established. Also, much of the vocabulary of his day is obsolete.

I tried reading Maxwell's "Treatise on Electricity and Magnetism" and I found that it took way too much concentration to follow his ideas (but it made me appreciate what brilliance it took to reach the conclusion of electromagnetic wave propagation without vector calculus).

Honestly, I am not sure what this quote means in a mathematical sense, since "electromagnetic unit," " electrostatic unit," and "electricity," may mean entirely different things than they do today.

3. Feb 21, 2010

### diagopod

Thanks BAnders, I see what you're saying. I am having a similar experience with his earlier paper "The Dynamical Theory of the Electromagnetic Field."

Btw, in your estimation, is there any way to picture the difference between the electric and magnetic constants? Is there any real sense in which one is a multiple, or at least proportional and dependent upon, the magnitude of the other?

4. Feb 21, 2010

### clem

The only mystery is who (and why) thought up muzero and epsilonzero.
The original units of charge and current were defined in absolute terms by the their contribution to force. The equation for the force between two equal charges in Coulomb's law, F=q^2/r^2, with F = 1 dyne and r = 1 cm (ugh, those outdated cgs), defined the electrostatic unit of charge of 1 esu (now, retroactively, sometimes called 1 statcoulomb). Somewhat later, the law for the force per unit length between a pair of long straight wires carrying equal currents, F/L=2I^2/r, with F = 1 dyne and r= 1 cm, defined the electromagnetic unit of current of 1 emu/sec (now, retroactively, sometimes called 1 abampere, with `absolute' shortened to ab to cloak its absolute definition.) . This also defined the emu unit of charge(now, retroactively, sometimes called 1 abcoulomb). At the time, electricity and magnetic were two almost unrelated phenomena. In a remarkable experiment, involving measuring the current caused by a discharging capacitor, Wilhelm Weber and Friedrich Kohlrausch correlated these two charge units finding their ratio to be about 3X10^10 cm/s. This is the source of Maxwell's statement that to some 'modern' ears is cryptic. It was also the first step toward the eventual unification of electricity and magnetism (since obscured by the use of different constants for electricity and magnetism), and the second step toward the recognition that light was an electromagenetic wave.

As with any advanced text, a good knowledge of the background material enhances the appreciation.

5. Feb 22, 2010

### diagopod

Thanks Clem. I appreciate your insight here. I see you write that, at that time the force b/w two charges was defined as "F=q^2/r^2" Since this would not be considered valid today, and I imagine they were using the same definition of force and certainly distance back then, it seems to me that what you're saying is that, at that time, they had a different definition of charge. And by changing that definition, we now require the electric constant to balance it out, and a similar argument would hold for magnetism. Am I understanding you correctly? Thanks again. I'll read up on that experiment too.

6. Feb 22, 2010

### clem

"this would not be considered valid today". You must be an engineer or still taking UG courses. F=q^2/r^2 is used by many working physicists in their research work, and published papers. Put "gaussian units" into google. The 'magnetic constant', muzero just corrects a mismatch of units, and is nothing like a constant of nature. As you probably know, muzero/4 pi is just a power of 10. The power comes from two causes. First, the difference between MKS and cgs. Making that conversion still leaves one power of 10, which comes from a change from the absolute ampere of my previous post to the SI ampere which was arbitrarily defined as 0.1 absolute ampere. I am not sure why this was done. Maybe it was so they could charge more for a 30 amp fuse instead of a 3 abamp fuse. The 'electric constant' epsilonzero is then defined, not from Coulomb's law, but so that 1/sqrt{e0m0}=c, the ratio between the abcoulomb and the statcoulomb, as stated by Maxwell, and now defined to be 299,792,458 m/s.

7. Feb 22, 2010

### diagopod

You're right, I don't know what I'm saying :) B/c all of the textbooks I read use the e-constant and m-constant I assumed the numbers and units just didn't add up without them, but it's actually a relief to learn that one can proceed, at least in some contexts, without those "constants," b/c they do feel contrived in a way to me, and I suppose they're only strictly necessary to make the SI units add up?

I'll read up on Gaussian units, which sound almost like "natural units" for electromagnetism, but again I don't know what I'm saying. I have a feeling I'm going to come back to this post several times, b/c there are several things in here that I need to read up on and return to, especially the history of how non-constants became the somewhat arbitrary looking "constants" that show up in the textbooks that I'm pouring through. Thanks again for your patience, hope to understand all of this with time...

8. Feb 24, 2010

### diagopod

Clem - not sure if you're still following this thread, but just in case I wanted to thank you for the Gaussian units angle. I studied up on them over the last couple days.

Comparing Gaussian units to SI, it appears to me that the a key difference lies in how charge is defined in terms of mass, distance and time in Gaussian units. In SI units, charge seems to remain undefined, even though not a fundamental unit for some reason (embedded inside the Ampere), and therefore more mysterious. At least conceptually speaking, I'd prefer to think of charge in terms of mass, distance and time, perhaps through Gaussian units. Do you think there is any substantive meaning to conceiving of charge as a derived unit of mass, distance and time, or is that simply an expediency that I shouldn't read anything into?

9. Feb 25, 2010

### clem

SI are units are based on the mistaken belief by engineers that electric charge is a new dimension, to be added to space, time, and mass. Thus they invented the coulomb, which is related to the abcoulomb in an earlier post. When it became more accurate to measure the ampere with two wires, the system became the MKSA,which was voted into universality at an international units convention in the 1950's. Physicists (but not engineers) now know that electric charge is not unique, because there are other forces in nature-- the strong and weak interactions, each with their own charge. If charge were a new dimension, more that electric would be needed. Charge is actually DIMENSIONLESS, and all the equations can be written in terms of a dimensionless charge. This is one of the bases of 'natural units', used by most high energy physicists.
In this set of units, e^2=1/137, and is defined as the 'fine structure constant' alpha.
If you are not yet using natural units (where hbar and c each equals 1), then
alpha=e^2/(hbar c)=1/137 in Gaussian units or divide by 4piepsilon0 for SI.
In either case, you get 1/137. There is also a strong force, for which
alpha_S~1, which is why it is called strong.

10. Feb 25, 2010

### diagopod

Thanks Clem. So I'm looking into natural units. Is this the system you're referring to: http://en.wikipedia.org/wiki/Natural_units#.22Natural_Units.22_.28Particle_Physics.29"

I would be interested to hear any further thoughts you have on the fundamental units/dimensions. Are all three conventional fundamental units - mass, length, time - also ultimately dimensionless in your opinion, or is only electric charge dimensionless? I'm not familiar with thinking of a fundamental unit of nature as a dimension, but I like that outlook b/c I can picture it, although it might be hard to picture length and time as dimensionless at first pass. In any case, do you have any further thoughts on that front? Thanks again.

Last edited by a moderator: Apr 24, 2017
11. Feb 25, 2010

### Meir Achuz

As with much of Wikipedia that page has too much confusion to be useful.
Their natural units seriously wrong. There is no sensible system of units in which e=1.
e cannot equal 1.
In every lsystem of units I know, (about 5), alpha=1/137 and is dimensionless.
In natural units (Gaussian based) alpha=e^2. If SI based, then alpha=e^2/4piepsilon0.

The only possible fundamental constants of nature are dimensionless.
Any other 'constant' is just a connection between different arbitrary dimensions.

mass, length, and time need dimensions to describe them. In natural units,
L~t~1/M in terms of a common unit, which can be chosen to be any convenient unit.
This is with hbar and c=1.

12. Feb 25, 2010

### LucasGB

The reason for the existence of ε0 and μ0 are unknown to most students, but nevertheless countless threads have been started in this forum where people looked for answers to these important, but neglected, questions. After a lot of reading, I think I came up with a good understanding of why these things exist. But I still have some questions, and I would like to ask clem to try and answer them, since he seems to know so much about this subject.

- Coulomb formulated Coulomb's Law, which gives the force between to charges under certain conditions, and goes F = ke q²/r², where ke is a constant.
- Ampère formulated Ampère's Force Law, which gives the force between two wires under certain conditions, and goes 2 km i1i2/r, where km is a constant.
- If one arbitrarily defines ke, then one is automatically defining a charge unit which makes Coulomb's Law true. Therefore, since the unit of charge and current are related (q=it), he is also defining the unit of current, and therefore defining km. The opposite is also true: if one abritrarily defines km, then he is ultimately defining ke.
- Physicists chose to define km and derive ke from km.
- First question: why did physicists choose to define km instead of ke? I read somewhere that it was because they wanted to leave the unit of current practically unchanged from the one that was currently being used. Could you elaborate on this?
- After defining km, physicists derived ke from km.
- When developing Maxwell's Equations from experimental facts, one finds that the term 4π appears in the numerator of both the first and the fourth equation. Therefore, physicists found it would be useful to express ke and km as a certain number divided by 4π, so it would cancel out in Maxwell's Equations and simplify them.
- So they said ke = (a certain number)/4π and km = (another number)/4π.
- Since the values of ke, km and 4π were known, they derived the values of both these numbers, which today we call ε0 and μ0, respectively. ε0 and μ0 also have units in order to make the units in Coulomb's and Ampère's Law match.
- Second question: physicists actually defined km to be μ0/4π, but they defined ke to be 1/4πε0! Why didn't they define ke to be ε0/4π (and changed the value of ε0, of course)? I have asked this question here before, but I want to see your take on it.

Thank you, and sorry for the madness.

13. Feb 26, 2010

### diagopod

Meir - Thanks for clarifying this for me. Btw, do you have a link to a good resource on the natural units you're describing above by any chance? Would like to learn more. Thanks again.

14. Feb 26, 2010

### clem

Lucas G B:
Things happened in this order:
1. Coulomb's law was formulated with ke=1, and was used to define the esu unit of charge.
2. Ampere's law was formulated with km=1, and was use to define the emu unit of current, which also defined the emu charge as 1 emu of current=1 emu of charge/second.
3. There were two systems of units, esu for electric phenomena, and emu for magnetic phenomena. There was no ke or km.
4. W & K realized that the current from the discharge of a capacitor was the flow of emu charge, and then measured the ratio 1 emu/1 esu =3.1, which I believe Maxwell called c for constant. Incidentally Maxwell built his own apparatus and got a more accurate value for c than W&K.
5. Gauss (or someone else) suggested that, instead of using two units of charge whose ratio was c, the esu could be picked as the unit of charge if every current or velocity were divided by c, thus defining the Gaussian system of units.
6. If one (It was a Italian engineer named Georgi over a hundred years ago) doesn't understand this orderly progression, he could aribitrarily define a ke and km and invent the MKSA system of units, which were actually called 'Georgi units' until an international conference in the 1950's called the SI (in French to honor the French revolution whic started all of this with the reign of terror).
7. "Physicists chose to define km and derive ke from km". Physicists did not choose anything. They were outvoted by the engineers at the Int'l conference.
8. It does make sense to 'define km first' because muzero is not a constant of nature.
It is just an arbitrary 4pi times a change of units. The episilonzero must follow to get the factor of c that is natural in Gaussian units into place.
Actually if the emu were chosen as the unit of charge and c used to multiply the elctric equations, the system of units would be close to SI with the change from cgs to MKS.
There would still be one power of ten because the definition of the Ampere was changed by a factor of 10 in 1881. That is, the SI system is really the emu system, except for the new Ampere. With the original absolute ampere, mu0over4pi would equal 1.
10. The 1/4pi to get so called 'rationalized units' is just one more factor to make EM more difficult for the beginning student, and add one more level to their confusion.
If physics were too easy, who would need PhDs to teach it? Working physicists have no trouble working with 4 or pi in any combination. What I find hard in teaching advanced EM is convincing students (who have been SI indoctrinated) that 1 is an easier constant to remember than epsilonzero and muzero. How many beginning students have any idea why the 'permeability of free space' (now no longer free)
is 12.6?
8. The answer to your last question is that 1/4piepsilon0 is required to take the 4pi out of Maxwell's equation. Of course it puts in epsilon0 which is more trouble than 4pi.
Why the 4pi is different for epsilon0 and mu0 leads to a whole new can of ugly worms.

15. Feb 26, 2010

### LucasGB

clem, that's a remarkable account on the subject. I feel like I didn't know anything about this topic before. Let me ask you something. I'm neither an engineer nor a physicist. I learn physics as a hobby. Do you suggest I learn electromagnetism using SI or Gaussian units? Since I won't be doing any practical calculations, what is the unit system which best allows me to go deeper in physics, learning subjects such as quantum electrodynamics, without having to change unit systems? Which is the most natural and intuitive one?

16. Feb 26, 2010

### LucasGB

Now that I think about it, perhaps it's a good idea to learn it without any unit systems at all, employing only ke and km. That's gotta be the purest manner of all.

17. Feb 26, 2010

### LucasGB

Are these Maxwell's Equations in terms of ke and km? I "translated" them from SI equations, so I'm not sure this is right. Please check the attachment.

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18. Feb 27, 2010

### diagopod

Like Lucas, I appreciate the clarification Clem. I did a bit more reading along the lines of the history above, but still can't quite grasp the units. Just in terms of mass-time-distance units, do I understand the history correctly now that the esu charge had units of m^1/2 t^-1 d^3/2, while the emu charge had units of m^1/2 d^1/2? Thanks again for your patience.

19. Feb 27, 2010

### clem

Those units for esu and emu charge are correct, but you never ave to consider the units of charge, either for Gaussian or SI. Just keep everything in the same set of units and things will work out. The nice thing about Gaussian is that E and B have the same units.
In natural units, charge becomes dimensionless, which makes it possible to compare the strength of interactions (EM, weak, strong). I am not sure of a good reference to 'natural units' because the don't always mean the same thing. They are made natural for the particular field. In high energy physics, the natural units are c=hbar=1,
e^2=alpha=1/137. Then variables with the same units are m~e~omega~1/L~1/t.
You can use MeV, fm, sec as the common unit. If you calculate a distance and it comes out as T sec, you can convert to fm by using the conversion 1=3X10^23 fm/sec.
The other useful conversion is 1=197 MeV-fm. Then if you get an energy as
W=3 fm^-1, you can convert to Mev by 3 fm^-1 X 197 MeV-fm =600 MeV.

20. Feb 27, 2010

### clem

Lucas:
Those equations are not wrong, but they are confusing.
Using ke and km confuses every step, and means you could not use any book.
I don't know an advanced book that does that.
Also no one would bother to work out what you write.
There is nothing wrong with doing that, but there is nothing good.
It is better to introduce H and D.
Max's equations, then in Gaussian units are:
Div D=4pi rho
Curl E=-\partial_t B/c
Div B=0
Curl H=4pi j/c +\partial_t D/c.

21. Feb 27, 2010

### LucasGB

I understand. I think the main advantage of writing the equations in that way is that it's the most "fundamental" way of doing so, since those equations can be translated into any unit system by a mere substituion of ke and km for the appropriate constants.

The downside, as you pointed out, is that since no one writes them that way, if I were to write texts which employed those equations, nobody would want to go through the trouble of understanding me, and I myself would have some difficulties understanding books.

Anyway, I think I'll just learn things using the SI. Getting used to the units of the gaussian system at this point would be quite annoying and somewhat unnecessary. Nevertheless, I should say I deeply dislike the fact that people got used to calling epsilon0 and mu0 the permittivty and permeability of free space. It's just so much simpler (and more correct) to call them the electric and magnetic constants!

22. Feb 28, 2010

### diagopod

Thanks Clem. I'm spending some time getting familiar with Gaussian units, as it seems the picture of what's really going on is, at least slightly, clearer than when using SI.

Probably well ahead of my level here, but the symmetry is so striking I think I'll learn more about these as well. Is it merely an expediency, or is there any real sense in which c and h-bar can be understood as being the same?

23. Feb 28, 2010

### clem

c and hbar are not the same, except in so far as they are each a dimensioned constant.
That is their numerical values depend on the system of units. The simplest choice is units that make the dimensioned constant 1. A familiar example is giving astronomical distance in light-years.