Edit 1:
I was assuming the correctness of maxwells equations, how flux remains same in the absence of additional charge. So is it to say that that is 100% correct or just simple it can be a really really good approximation and we dont know?
Edit 2:
The best explanation I got till now: We simply do not know if it is really 1/r^2, but experimentally we have found that this model is very accurate to several decimal points(2.0000…). The 1/r^2 and maxwells equations are same, they are correct or wrong together. And in my opinion, the divergence equation for electric field seems very simple, and we found its high accuracy through experiments, so I have a feeling that it must all be exact 1/r^2. But still, I don’t think we have a way to find out because the equations are themselves the starting point which describes the experiments very well, we don’t have derivation for them and I don’t think we will ever have, and on the other hand, our only way to check is by experiments which won’t give 100% confidence accuracy.
Edit 3:
Another perception is about unit-dimentions. Like ( c^2 ) * e0*u0 = 1, which binds the units of e0 and u0, so electric and magnetic formulas must together be right or wrong. I can’t think or anything more right now, but there can be some other unit-dimension constraints that I am missing which will complete the picture.
In: 40
We don’t. We basically made a guess that has held up for a few hundred years. But that guess was based on many other guesses that have held up for even longer (e.g. the concept of the inverse square law) so it’s not like we were just shooting in the dark.
As for what if we were off by just a bit? It’s possible, but in practical terms, it hasn’t mattered. I vaguely recall an anecdote that we only need pi to around 15 decimal places in order to send a probe to the other side of the galaxy usefully. And if you want even more egregious intentional fudging, check out amplifier linearity.
Probably the best answer is because the formula isn’t the reason for the effect, the effect is *caused* by something about spatial relationships and geometry.
Simple geometric formulas like 1/r² are the mathematical descriptions of those spatial relationships.
This one specifically means that the strength of the effect is evenly distributed in all directions in 3 dimensions around a point.
We know “surely” because two independent lines of hard logic provide identical results to any degree of precision. The *inverse square law* is very simple geometric reasoning; the recieved quantity of anything from a point source decreases by the square of the distance, as the available volume to propogate that limited quantity of anything increases the amount it increases at any given distance. Independently of that, in this and many other cases, the quantifiable results of empirical experiments and demonstrations produce exactly those results.
Because of units and dimensions. `r` is a unit of distance, and `r^2` is a unit of area. `r^2.00000001` is a unit that doesn’t really make any sense.
The uncertainty of the preciseness of the formula is represented by the uncertainty of the preciseness of the leading constant `k`.
Your question holds true for any physical phenomena that we have mathematical relations for. The trouble with asking “is it 100% accurate” versus “or a really good approximation and don’t know any better” is that there is no experimental difference. Experiments reveal physical reality up to a measurement uncertainty, and the best that we can claim is that a certain model is consistent with measurements and other results that are derived from that assumption. This is the closest we can get to “100% accurate”. Any deviations from that are buried within uncertainty, assuming the experimental measurements are free of artifacts. Asking “what if it’s slightly different” is then the objective of further research.
Now if I were to reinterpret your question less philosophically and more practically, where we assert that 1/r^2 scaling exactly true, we could also talk about how unlikely it is to fail. Electromagnetism is so fundamental to all subfields of modern physics that no field is completely free of it. Maxwell’s equations are not only directly tested, but also indirectly tested when we use those in experiments and models. For example, the Coulomb interaction is incredibly important in atomic and solid state physics. Its functional dependence is incredibly sensitive due to small distances (i.e. big forces bc small r) and the many-body nature of these systems (i.e. summing over ensembles). Using this 1/r^2 form, we derive electronic orbitals, use those to build our quantum theories, and are used for analysis in pretty much all of condensed matter physics.
First, all measurements are limited by our scope of the ability to measure. If I take a scrap-bin of lids and measure it on the scale then within a 2-minute span it will measure between 15.05 and 15.10 pounds before settling on 15.06 pounds. If I take it off the scale for a minute, or jostle it while getting my pen out of my pocket, then it might settle on 15.04 pounds or even 15.13 pounds.
So, for like 80% of cases the difference between an exponent of 2 and an exponent of 2.00000000000001 will probably not be significant enough to make a difference.
Second, when the substantially-miniscule decimal would actually make a significant meaningful difference, we can test for those values. Which the results of which would require a decimal value much much much smaller than even my proposal of your example.
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And then there’s the fact that we can apply mathematical transformations, such as Integration and Derivation from Calculus, to calculate different variables with different experiments. For example, if you know how far something traveled and how long it was traveling for… you can determine its speed. If you know how long it took to get to that speed, you can determine its acceleration. If you know its mass, you know how much force it had while it was accelerating and thus what force you needed to get it to move at that speed as quickly as it took to get up-to-speed.
And the same two points from above apply to those calculations as well… so if for some reason the value is actually 2.00000000000000000000000000000001 instead of 2, then it would be 2.00000000000000000000000000000001 for all the associated equations. But, nope, there are some where it’s no larger than 2.0000000000000000000000000000000000000000001. Which means for the equation you ask about, it must be smaller than 2.0000000000000000000000000000000000000000001
How the hell is this an ELI5 question?!?
It correlates beautifully to the diminishing percentage of the surface area of a sphere that a hypothetical object intersects as r increases.
I guess that we technically don’t know, but exactly 2 has a theoretical basis. The force between two objects, be it electrical/ gravitational / etc, seems to be the product of the electrical force / mass / etc of object A times that of object B, divided by the distance from A to B, divided again by the distance from B to A. Since those two distances are the same, the equation is effectively r^2. That’s how I think of it.
Reminds me a lot of veritasium’s video on the 2-way speed of light. Highly recommend checking it out when you want to further break your brain on how all of physics is dependent on indirect knowledge at best.
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