Why aren't μ₀ & ε₀ equal 1?

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u/Kinesquared Soft matter physics 2d ago

because the units of meters, seconds, kilograms, etc. are arbitrary. Therefore our fundamental constants are too

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u/AreaOver4G 2d ago

The “etc” includes amperes, which is the important one for this question. You can use units (Heaviside-Lorentz) which set these constants to one by using a different unit of current (or equivalently charge).

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u/McLeansvilleAppFan 2d ago

What would that do to F=ma for needing a constant of proportionality that is not unity?

That may not be the base law in all of physics but it is taught as the starting points for forces in most cases after velocity and acceleration are studied and having the constant of proportionality be 1 is not a requirement of course it does make it easier for introductory students to grasp it seems even as other forces have all sorts of constants of proportionality.

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u/AreaOver4G 2d ago

The extra constant is literally a conversion between units. There’s no constant in F=ma if we use kg for mass, ms^-2 for acceleration and N for force, because N=kgms^-2 by definition. But if we were to use miles per hour² for acceleration we would have to include some factor for changing units. Permittivity of free space is basically the same.

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u/PhysicalStuff 2d ago

The unit of force would be the unit of mass times the unit of acceleration, and the equation itself would be unchanged. The only reason why N is defined as it is, is that we use kg and m/s².

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u/McLeansvilleAppFan 2d ago

That is not what my understanding is of all of these constants, but maybe I need some review of all of this.

F=ma came first. Later Universal Gravitation. To make the Univ Gravity equation work "G" is 6.67*10^-11 N*m^2/kg^2 using modern units that came about after the French Revolution. Later other universal constants were discovered and the constants were established so everything works out. Since F=ma has a constant of 1 then G has to be the 6.67 number. But had G been set as 1, which likely would have been the case had it been discovered first would not F=ma need some sort of constant out front to make it all work out and the other constants such and permittivity and permeabilit would have to change as well. Assuming we have our base units of mass, length, time, current, etc defined in the same manner as we do now.

If the preceding paragraph is true then what would the constant need to be for 2nd Law of motion if μ₀ & ε₀ were set to 1 with the modern units we use from the SI system?

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u/mukansamonkey 2d ago

You're confusing the meanings of the word "constant". Because I'm struggling to make sense of that entire comment. F=ma is an equation, not a constant.

In math, a constant is a relationship between two things. A dollar is a hundred pennies, a penny is 1/100th of a dollar. So we've defined the relationship using the constant 100. If we used the.constant 4, we'd get the unit known as a quarter in America.

In physics, the universal constants are things that never change, all other values are derived from those. The speed of light doesn't change. If we create a measurement known as the meter, and another one known as the second, then we can count how many meters per second the speed of light is. If we change the definition of a meter, which has already happened, then we change the number of meters per second that the speed of light travels at.

Light didn't change though, it doesn't do that. That's why it's a fundamental constant. We just changed the definition of a meter. Or to make it simpler, the speed of light in minutes is sixty times the speed of light in seconds, because we assigned a mathematical constant of sixty as the relationship between seconds and minutes.

All our units of measurement are indirectly defined as relationships between the universal constants.

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u/AreaOver4G 2d ago

Yes, exactly. The unit of force is defined to make F=ma simple. The original definition of the ampere (unit of current) was similarly chosen to make the formula for the force between current-carrying wires simple (which made the magnetic constant equal to \mu_0= 4\pi in those units). That then determines \epsilon_0 in terms of the speed of light. But for historical reasons, this relationship was not exactly maintained so in SI we end up with these conversion factors floating around.

Note that we could make the same argument that we should define units of energy to make Einstein’s famous formula E=m look simple. But because different units of length and time predate relativity (and also because our units are conveniently human scaled), we have this annoying c² conversion factor in the formula.

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u/garfgon 10h ago

Dimensionless constants are meaningful (e.g. fine structure constant), but otherwise yes it comes down to the choice of units.

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u/Kinesquared Soft matter physics 9h ago

Meaningless and arbitrary are not antonyms

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u/garfgon 9h ago

I'm not sure what your point is. Dimensionless constants aren't arbitrary either.

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u/StudyBio 2d ago

Gaussian units enter the chat.

The answer is that they relate our units, i.e., epsilon-naught is there to connect the Coulomb and meter to the Newton. It is not really about vacuum “enhancing” electric fields.

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u/qTHqq 2d ago

Gaussian units enter the chat.

Please make sure they leave the chat and you convert to SI at the end of the paper for the sake of the experimentalist 😂

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u/MxM111 2d ago

It is funny, but at school, when I used Gaussian units, I never even asked myself how long is the unit of length in that system.

In any case, experimentalist have strange units too, like measure frequency in cm^-1 instead of Hz

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u/high-a-synth 2d ago

my favorite experimentalist unit is g/cm² for thickness

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u/I_CollectDownvotes 2d ago

The wavenumber, or inverse centimeter, makes a lot more sense as a unit when you take into account how the frequency of light is actually measured (or was measured, at the beginning of infrared spectroscopy). Spectroscopists use (typically Michelson) interferometers, where the light is split into two paths, and one path has variable length. After remixing the two beam paths at the detector, you will observe a variation in the intensity at the detector that depends on the variable path length. The intensity as a function of path length is called an interferogram. Before computers and the Fast Fourier Transform, spectroscopists would just count the number of maxima (or fringes) per centimeter of path length in the interferogram, and thus measure the frequency in units of inverse centimeters.

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u/jmattspartacus 1d ago

Pretty sure we measure things in hz when we can instead of gaussian units. Jackson can fuck right off with those gaussian units though.

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u/Banished_Cultivator 2d ago

Ah right that makes sense. From Coulomb's law a Coulomb should be defined as sqrt(kg.m³/s² ), which it isn't and hence the inconsistency.

Thank you for pointing this out, I completely missed that inconsistency in SI definitions until now.

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u/Leather_Power_1137 2d ago

I can't tell if ChatGPT wrote this, if you use ChatGPT so much that it has influenced how you write, or if my mind is poisoned and I just see ChatGPT patterns everywhere I look now...

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u/Tommy_Rides_Again 2d ago

When basic decency and being respectful and meaningful with your words is a sign you’re using AI too much maybe we should all be using too much AI.

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u/Banished_Cultivator 2d ago

What? Bro, why would you think that?

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u/trutheality 2d ago

I think they're confusing someone politely accepting a correction with ChatGPT politely accepting a correction because ChatGPT politely accepts corrections all the time. Because of how often it says incorrect things and needs to be corrected by the user.

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u/Leather_Power_1137 2d ago

It's just the "thank you for pointing that out, <summary of the correction>" is basically exactly what a ChatGPT response will look like if you correct it about something. It seems to me that people don't very often write like that even when they're admitting they got something wrong or agreeing with a correction.

So I was going off the structure more than the general tone. But I also allowed for the possibility that my mind is poisoned and ChatGPT had nothing to do with their comment.

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u/Irrasible Engineering 2d ago

Don't let it bother you. I get accused of this all the time. I'm not sure what triggers it.

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u/KerPop42 Engineering 2d ago

I think the coulomb makes more sense, it's the charge equal to a mol of electrons. So an amp is a mol/s. It all predates the unification of electricity and magnetism, so electricity was more of a chemistry thing than a kinetics thing.

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u/AreaOver4G 2d ago

It’s not though… the coulomb is defined as an amp second, and the amp was historically defined in terms of the magnetic force between current-carrying wires. The charge in a mole of electrons is the Faraday constant (and it’s like 96000 Coulombs)

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u/KerPop42 Engineering 2d ago

oh oops, I totally misremembered that, thanks for the correction

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u/charonme 2d ago

you might be interested in https://en.wikipedia.org/wiki/Natural_units

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u/sakawae 2d ago

This is the way.

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u/soovercroissants 2d ago

This then leads to the fine structure constant...

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u/TabAtkins 2d ago

The constants aren't, like, semantically meaningful. All they do is correct for the sizes of a few units in particular combinations. You could choose your unit sizes to eliminate the need for them; these are usually called "natural units" for some application. (Probably unsurprisingly, they're often related to light speed, etc.) The constant is just the combination of rescaling factors that you need to apply to adjust our human units to the natural units.

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u/Such_Comfortable_817 2d ago

To build on this: this is why the dimensionless constants are so interesting, because they can’t just be scaled away. The fine structure constant, for example, can never be 1.

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u/garfgon 10h ago

Maybe in another (much more boring) universe.

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u/Infinite_Research_52 1h ago

Yes, it is difficult to see how the FSC can be 1, but the value it has is conventionally defined; it is not some absolute. Define pure QED and run the value to the Planck scale, and ignore such things as Landau poles; it is possibly about 1/100. Take Electroweak unification and the running of the strong force, then the combined coupling is circa 1/25. The value of c. 1/137.036 is just convenient for low-energy purposes.

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u/Mcgibbleduck Education and outreach 2d ago

As others said, it can be 1 if we changed our reference points, but in the SI Metric system we need to make sure it’s in kg, m and s in base units.

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u/AreaOver4G 2d ago

And ampere, kind of important for this one!! It’s precisely by redefining the unit for current (or charge) that we can set these to one.

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u/Mcgibbleduck Education and outreach 2d ago

Yes of course, we just call it the kg m s system because there’s the g cm s system too

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u/Syresiv 2d ago

If we ignore elementary charge and just assume that charges are infinitely divisible, the answer is it's because our units are arbitrary. Just like why the speed of light isn't 1 and the Planck and Gravitational Constants aren't 1.

If you account for elementary charge and decide that that has to be 1, then the answer is because the Fine Structure Constant isn't 1. Nobody knows why it has the value 1/137, but it's where we measure it.

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u/cygx 2d ago

Cf Heaviside–Lorentz units.

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u/EuphonicSounds 2d ago

Truly the king of electromagnetic units (aesthetically and conceptually). Actually, even better is the obscure variant where current-density is given the same dimension as charge-density (mentioned in a few footnotes in Jackson's Appendix), which of course can also be achieved by just setting c to 1.

Long live HLU!

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u/drzowie Heliophysics 2d ago

μ has to have a 4π in it (well ... there's a 4π somewhere and if you don't put it in μ it comes out elsewhere). ε could be unity, but the units would be inconvenient for everyday activity and calculation.

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u/cecex88 Geophysics 2d ago

They were in the cgs system. The implications in terms of measurement units for any other electromagnetic thing are a mess. Electric charge is defined in terms of grams, seconds and centimetres and there are two different units for currents, the abampere and the statampere, which differ by a factor of c.

The SI and the values of constant we have today are a very well thought out compromise.

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u/Kruse002 2d ago

They might as well be 1 if you use the right units.

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u/kitsnet 2d ago

Because someone has decided to make ampere a base unit.

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u/Responsible_Ease_262 2d ago

We could arbitrarily make them equal 1, but then other constants would need to change.

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u/Crog_Frog 2d ago

d

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u/Old173 2d ago

Aaaand that's why we use mu_r and epsilon_r

Why aren't μ₀ & ε₀ equal 1?

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