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
So obviously we might discover tomorrow that there’s something fundamentally wrong in physics, but assuming that we don’t, there’s an easy way to understand why it’s exactly 2.
Imagine you had a point charge sitting in an empty universe. It’s going to radiate electric field (per Maxwell and really per all our discoveries that field exists), and there’s no field being created anywhere else. That field is going to go out equally in all directions, because there are no preferred directions in this picture to make it skew one way or the other. That means that if you imagine a sphere surrounding the dot, the total amount of field hitting the sphere is the same, no matter how big the sphere is.
So now imagine there’s some charged object on the sphere. It has a fixed size, so the surface area of it that’s pointed at the center – that is, the surface area that could intersect those radially expanding lines of field – is fixed too. But as the sphere gets bigger, that fixed supply of field lines gets spread over more and more sphere. The fraction of those lines that hit the object – and those are the lines that pull the objects together! – is the fraction of the sphere’s area that’s covered by the object.
The object’s area is fixed, but the sphere’s area grows as the square of the radius, because of simple geometry. So the fraction of the sphere it covers goes down as 1/r^2, and so does the force.
Yes, I did fudge a bit in this description: why do we think about a finite supply of field lines, why is force proportional to their number? There are great answers to this but they’re really not ELI5. The Maxwell equations definitely show them mathematically, and there are even deeper (and more advanced) answers – the ELI25andagradstudent answer is that all forces are mediated by the exchange of force particles; that the electromagnetic force, in particular, is an unbroken U(1) gauge force which means that the force particles in question (photons) are massless and don’t interact with each other, they just radiate out from and get absorbed by charges; that means their Lagrangian (their equation of motion) has a very simple form, and their propagator can be exactly calculated as 1/q^2, where q is their relativistic momentum vector; and one Fourier transform later that shows that in a k-dimensional spacetime, the force scales as r^(2-k), or log r if k=2, and the last bit is a very mathematical version of exactly that story with spheres that we started with.
Which I like, because the simple “intuition” answer turns out to be very deeply true even in the most advanced version.
Of course, if you’re doing the full grad school version, you’ll note that the interaction isn’t just from single photons, but you have to think about multiple virtual photons as well – and the math here gets ferociously complicated, but the result is that the actual force _isn’t_ 1/r^2, there are very small correction terms. Those are actually important in some kinds of atomic physics, for example, because they cause changes in the orbits of electrons in atoms and therefore to all sorts of physical and chemical properties of matter.
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