Why does ##F## often appear as inverse square laws such as Newtonian gravity?

[not my name]

...y and Coulomb's law diverge as $r\rightarrow$0? I mean, if a point light source emits light omnidirectionally, the intensity converges at the source, right?

THIS is how I should've worded my previous post!

 

[Dale]

It is just straightforward math:
$$ \lim_{x \to 0+} \frac{k}{x^2} =\infty$$

A point light source has no surface area so the intensity must diverge at the source for it to be a finite intensity elsewhere.

 

[not my name]

(Oh, nevermind, it does diverge)

Edit: WAIT nevermind I still don't get it. Suppose that a point is rapidly emitting photons at random directions every set interval. Of course there are finite photons in that point, right?

 

[jbriggs444]

Suppose that a point is rapidly emitting photons at random directions every set interval. Of course there are finite photons in that point, right?

Keep going. There is nothing wrong with this reasoning yet. A finitely massive point source can only ever emit finitely many photons of a particular energy.

How is this incompatible with an infinite intensity?

An infinite intensity multiplied by an infinitesimal surface area is compatible with a finite result.

 

[Dale]

Suppose that a point is rapidly emitting photons at random directions every set interval. Of course there are finite photons in that point, right?

Intensity is power per area. As area goes to zero the intensity becomes infinite, even with a single photon and thus small finite power.

 

[Drakkith]

I mean, if a point light source emits light omnidirectionally, the intensity converges at the source, right?

You do realize there are no real point sources, correct?

 

[vanhees71]

There are even not only no classical point particles. Their assumption is even incompatible with relativistic dynamics. This manifests itself in two mathematical facts: (a) if you try to build a relativistic dynamics for a system of interacting point particles, it turns out that a fully consistent model must be for non-interacting point particles (pretty boring in a sense) and (b) even the motion of a single particle in an external field, which is an approximation which works pretty well under the right circumstances (e.g., you can watch an electron moving in a tube with a gas making nice trajectories in electric and magnetic fields, and this is well described by relativistic point-particle mechanics), strictly speaking it's not self-consistent, i.e., as soon as you try to include the reaction of the accelerating particle to its own electromagnetic field, i.e., to include radiation damping, you get equations of motion that are inconsistent with the phenomena (the notorious Lorentz-Abraham-Dirac (LAD) equation), being "acausal". The best approximation from a quantum-(field)-theoretical point of view in fact is the socalled Landau-Lifshitz approximation of the LAD equation.

Relativistic Quantum Field Theory is a bit better than that, because at least you can describe the motion of particles and the electromagnetic field order by order in perturbation theory, which for QED is an amazingly successful description. Nevertheless from a more puristic point of view there's no rigorous mathematical formulation for interacing QFTs in (1+3) spacetime dimensions.

 

[Drakkith]

@vanhees71 Do the problems you described go away when you make the particles non-zero size?

 

[vanhees71]

In principle yes, but then it gets also pretty complicated. AFAIK it gets consistent using continuum-mechanical descriptions for the matter (like (magneto-)hydrodynamics or (semi-)classical transport theory).

 

[not my name]

Point particles exists.

Even if an elementary particle has a delocalized wavepacket, the wavepacket can be represented as a quantum superposition of quantum states wherein the particle is exactly localized. Moreover, the interactions of the particle can be represented as a superposition of interactions of individual states which are localized. [...] It is in this sense that physicists can discuss the intrinsic "size" of a particle: The size of its internal structure, not the size of its wavepacket. The "size" of an elementary particle, in this sense, is exactly zero.

For example, for the electron, experimental evidence shows that the size of an electron is less than 10−18 m.[7] This is consistent with the expected value of exactly zero.

(Quote source: https://en.wikipedia.org/wiki/Point_particle#)

 

[Vanadium 50]

Read your own reference. The very first sentence is "A point particle (ideal particle[1] or point-like particle, often spelled pointlike particle) is an idealizatiom" (emphasis mine).

 

[weirdoguy]

Plus it explicitly invokes quantum mechanics, not classical physics.