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

In summary: And in QM, position is not an observable, and even if it were, the uncertainty principle guarantees a nonzero spread. And if you try to remedy that by picking a state with arbitrarily small uncertainty, it will still be nonzero.In summary, the conversation discusses the divergence of intensity at a point light source due to its finite number of photons and the concept of point particles in classical and quantum mechanics. While classical mechanics allows for the existence of point particles, quantum mechanics and relativistic dynamics show that they are idealizations and do not accurately describe physical particles.
  • #1
not my name
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...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!
 
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  • #2
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.
 
  • #3
(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?
 
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  • #4
not my name said:
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.
 
  • #5
not my name said:
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.
 
  • #6
not my name said:
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?
 
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  • #7
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.
 
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  • #8
@vanhees71 Do the problems you described go away when you make the particles non-zero size?
 
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  • #9
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).
 
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  • #10
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#)
 
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  • #11
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).
 
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  • #12
Plus it explicitly invokes quantum mechanics, not classical physics.
 

1. Why is the inverse square law important in Newtonian gravity?

The inverse square law in Newtonian gravity is important because it explains the relationship between the force of gravity and the distance between two objects. It states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This law helps us understand the motion of celestial bodies and is the basis for many calculations and theories in astrophysics.

2. What is the mathematical equation for the inverse square law?

The mathematical equation for the inverse square law is F = G(m1m2)/r^2, where F is the force of gravity, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them. This equation shows that as the distance between two objects increases, the force of gravity decreases exponentially.

3. Why do we often see the inverse square law in other areas of science besides gravity?

The inverse square law is a fundamental law of physics that applies to any force that follows a radial pattern, meaning it acts in a straight line from a central point. This includes not only gravity, but also electrostatic forces, radiation, and sound waves. In all of these cases, the intensity of the force decreases as the distance from the source increases, following the inverse square relationship.

4. How did Newton discover the inverse square law?

Isaac Newton discovered the inverse square law by observing the motion of planets and their moons. He noticed that the force of gravity between these celestial bodies decreased as the distance between them increased. He then conducted experiments and used mathematical calculations to confirm this relationship, which became known as the inverse square law.

5. Are there any exceptions to the inverse square law?

There are some cases where the inverse square law does not apply, such as when objects are very close together or when there are other forces at play. For example, when two objects are in contact, the force of gravity between them is not inversely proportional to the square of the distance. Additionally, the inverse square law does not apply to the force of gravity between objects that are very small or very far apart, as the effects of relativity and other forces become more significant.

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