How was Coulomb's law figured out?

[physics_head]

Coulomb's law says that the electrostatic force between two electric charges is given by the formula:

Ke*q1*q2/r^2

So, what I want to know is:

1. How did he figure out that Ke was equal to 1/(4*pi*electric constant)?
I mean, I know he used a torsion balance to perform measurements in his experiments. But there undoubtedly would have been error in his measurements, so it seems to me that it would be impossible for one to infer that pi was part of the parameter he was trying to infer the value of. I would have thought that he might conclude that it be 1(4*3.3*electric constant)...or to even infer such an exact value for a critical parameter in the formula at all given the error infested measurements of his experiments...How did he get such an exact value?

2. I don't know the exact details of how he did his experiments. But it seems to me that he probably measured the distance between two charges and the force between them to create a datapoint (distance i, electrostatic force i). and did this with particles at many different distances to create a dataset, and then perfomed a kind of statistical regression to fit a function to the data set...To me, this kind of procedure would seem to be unjustified, since the formula applies to particles of any distance apart, yet the measurements in his experiments could not have contained distances larger than a room. So that means for large distances, we are extrapolating when we use the formula, and a tenant of statistics is that would should not extrapolate. But the formula is supposed to be exactly accurate, without regaurd of the distance right?...so How is this justified? and If he used an alternative procedure different than the one I describe, how did he do it?

It seems mysterious how the structure and parameter values in the formula were figured out...

 

[physics_head]

Also, related to this post..I see this a lot in physics materials...A formula will be given, but they don't explain how the formula was figured out. They just assert that it is true.

Now, In mathematics, anytime a theorem is presented, and asserted to be true, a proof is included with it, that way if you have any doubts, you can just follow the steps of the proof, and reach the conclusion that the theorem is true...Why can't physics teaching materials do the same? if I don't believe that a formula is true, then present me with a detailed description of the experiments, so that way I can go step by step, and then come to the same conclusion that the scientist did when he concluded his formula...Does anyone know of any physics books that do this? or any sources that describe how all the physics formulas were figured out?

 

[jtbell]

1. How did he figure out that Ke was equal to 1/(4*pi*electric constant)?

Coulomb did not define that constant. He just determined the proportionality between force, charge and distance. The proportionality constant is a matter of which system of units you use for measuring force, distance and charge. In the MKS system, it's [itex]1/4\pi\epsilon_0[/itex]. In the Gaussian CGS system, it's 1. In the Heaviside-Lorentz CGS system, it's [itex]1/4\pi[/itex]. In the electromagnetic CGS system, it's [itex]1/c^2[/itex]. These were all invented after Coulomb's time, I think.

See Wikipedia for a discussion of different varieties of CGS units:

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

 

[Cleonis]

1. How did he figure out that Ke was equal to 1/(4*pi*electric constant)?

I have no doubt that what Coulomb probed was the dependency of electrostatic force on distance between the electrostatic charges. His investigations will have led to evidence that the law of electrostatic interaction is an inverse square law. Just that: identifying that it's an inverse square law.

At the time the later concept of electric constant did not exist yet, in that sense Coulomb's investigations did not probe such a thing as electric constant at all. As the concept of unit of electric constant was developed, it was found to be convenient to allow for a factor 4*pi. Presumably that way the electric constant can more readily be correlated with other units.

Whether or not to incorporate a factor pi is just for ease of use. Planck's constant comes in two versions, one with a factor pi incorporated, one without. Depending on the circumstances one version is more convenient than the other.


Cleonis

 

[Cleonis]

So that means for large distances, we are extrapolating when we use the formula, and a tenant of statistics is that would should not extrapolate.

Indeed asserting coulomb's law for arbitrary distances involves extrapolation, so I think you're raising a valid point. That kind of skepticism is part of what is keeping science healthy.

So what if the extrapolation would not have been justified? What would have happened in the history of physics if Coulomb's inverse square law would have been valid only for distances in the order of meters, but at larger distances somewhat different from an inverse square law.

When physicists started modeling electromagnetic processes on the surface of the Sun they got into distances in the order of hundreds of thousands of kilometers. Streams of electrically charged particles are emitted and affected by magnetic fields.

If the inverse square law of electromagnetic interaction would not hold at such distances the models would have run into discrepancies. The fact that those models did not run into discrepancies is supporting evidence for the case that electromagnetic interaction law is an inverse square law at distances up to that order of magnitude.


The way I figure it I think that if Coulomb's inverse square law would have been incorrect at large distances it would have taken quite a long time to flush it out. But I'm confident that at some point it would have been recognized.


Cleonis

 

[LeonhardEuler]

2. I don't know the exact details of how he did his experiments. But it seems to me that he probably measured the distance between two charges and the force between them to create a datapoint (distance i, electrostatic force i). and did this with particles at many different distances to create a dataset, and then perfomed a kind of statistical regression to fit a function to the data set...To me, this kind of procedure would seem to be unjustified, since the formula applies to particles of any distance apart, yet the measurements in his experiments could not have contained distances larger than a room. So that means for large distances, we are extrapolating when we use the formula, and a tenant of statistics is that would should not extrapolate. But the formula is supposed to be exactly accurate, without regaurd of the distance right?...so How is this justified? and If he used an alternative procedure different than the one I describe, how did he do it?

This relates to basic questions in the philosophy of science. Ultimately, the answer is that there is no such thing as a scientific proof in the logical sense of the word proof. In addition to extrapolating to larger distances, there are many other assumptions that need to be made. One of them is that a function that gives the electric force acting on one charge due to another which depends only on the magnitude of the charges and the distance between them even exists. In fact, it does not. Coulomb's law is not true when charges move and there are changing magnetic fields.

Even if he simply tried to make the statement that the force is x Newtons when the charges are q1 and q2 and the distance is y based on many experiments under those conditions, that would not be justified. It's always possible that the very next experiment will give a different result.

It just appears to be an odd property of the universe that there exist simple laws that it follows. But it is not possible to give a proof of that statement. It will always be just something that seems apparent, but can't be proven without knowing everything that has ever or will ever happen in the universe (unless it is disproven).

Scientific laws are always provisional and persist only until they can be falsified.

 

[Jedi_Sawyer]

I think actually it was Benjamin Franklin's friend Joseph Priestly that first stated that the actraction between charged objests varied according to the square of the distance.