
The Inverse Square Law (2000) - tucaz
http://hyperphysics.phy-astr.gsu.edu/hbase/Forces/isq.html
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dahart
While it makes intuitive sense, I find the inverse square law for gravity
really hard to comprehend when I think about the solar system, for example
Uranus, or even the earth, orbiting the sun. When you look at how far apart
they are (for example
[https://news.ycombinator.com/item?id=21735528](https://news.ycombinator.com/item?id=21735528)),
the radius of the orbits relative to the sizes of the sun & planets, for me it
seems to defy the inverse square law that there is still enough force at those
distances to keep the planets orbiting.

If you built a scale model of the solar system out in the middle of space,
would you have the same orbital periods? Like with a scale model of the moon,
would a tennis ball sized rock orbit a soccer ball sized rock from a distance
of ~25 feet in about a month?

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samatman
Long story short: yes.

To expand a bit: lowering the masses by so many orders of magnitude would make
the allowable range of vectors much more precise, and the degree of
perturbation which the system could withstand would be much lower.

I wouldn't expect such a system to be robust with scale-model comets flying
around it, for example.

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dahart
That's interesting; I wouldn't mind hearing a spoiler of the long story. Is
there something in the system besides time that doesn't scale relatively?
comets at this scale would probably be dust-particle sized, right? Based on
the soccer ball / tennis ball scale, my estimate is that a 1km diameter comet
would be ~0.02 millimeters in the scale model.

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OldGuyInTheClub
Hyperphysics is a delight. I've worked in the physical sciences for years but
keep going back to refresh myself on the basics. I always learn something
again for the first time.

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puzzledobserver
That space is three dimensional is one of those propositions drilled into us
as schoolchildren, and which most of us accept on faith.

Yet, since it is a statement of the real world, it can only be validated by
experiment and not by a mathematical proof. Trying to contort one's fingers or
sticks in various mutually perpendicular orientations isn't satisfactory since
we could be limited by lack of imagination of the dexterity of our hands.

The existence of physical phenomena satisfying the inverse square laws is one
of those observations which constrains the geometry of the world.

~~~
evanb
On long distances, anyway. The evidence for inverse square laws over short
distance is much weaker, and so there is room for curled-up "extra dimensions"
we sometimes hear about, both in extensions to the standard model and in
stringy ideas.

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fhars
„Long distance“ here includes anything longer than a 1000th of the classical
radius of the hydrogen atom, which is to a very high accuracy described by an
inverse square law.

~~~
evanb
Gravity might have extra dimensions on a much larger scale than that,
potentially up to the millimeter scale or so.

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lurquer
Magnets -- not EM fields but actual magnets -- sort of follow an inverse cubed
law.

Anyway, Principia Mathematica's big conclusion was that an inverse squared
force will result in elliptical orbits.

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donutdan4114
Does the law still work perfectly at very large distances considering single
or zero photons (or gravitons or whatever) hitting a surface?

I guess is there a mathematical difference between a completely continuous and
smooth distribution of energy vs. “quantized” energy at large distances?

~~~
jakeinspace
Intuitively I'd think yes but only averaged over time. Waves in quantum fields
should pretty much be perfectly matched to the inverse square law over any
distance, but the actual observation of particles (by collapse or whichever
interpretation you prefer) will obviously have variance which will only smooth
out to an inverse square flux over time.

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madengr
So when transmit power is small enough to emit single photons, I suppose the
antenna radiation pattern is a probability density function. I suppose a wave
function for different modes on the antenna, as the photon has more energy
(higher frequency).

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rkachowski
I've never heard of the skunk on a flagpole example (does the smell drop off
according to the inverse square law), but surely this can be refuted by
pointing out that the other examples are expressions of energy transmitted
from a point source and the smell is directionally emitted from the skunks
butt.

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greens
For gravity and EM fields the inverse square law is a consequence of a
massless force carrier particle and Lorentz invariance. Quantum Field Theory
in a Nutshell by Zee talks about this early on.

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philzook
An interesting but rather overcomplicated seeming explanation. The posted
article basically is saying inverse square laws results from conversation of
flux (of any kind) in 3d. That seems like a pretty good explanation to me

~~~
evanb
For radiation, yes.

But for force laws, having a massless carrier is critical for an inverse-
square law. With massive carriers (like the carriers of the weak force, W+Z
bosons) the range of the force scales like 1/M; the force law is more like
exp(-Mr)/r ---> 1/r as M-->0\. The diminishing of the force with exp(-Mr)
means flux isn't conserved. (note I worked in units where hbar = 1 = c, so
that the W's mass ~= 80 GeV/c^2 corresponds to 1/M << 1 fm)

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philzook
But also things that aren't radiation or high energy physics. Inverse square
laws are ubiquitous in hydrodynamics, heat flow, electrostatics. Good ole
fashioned 19th century physics. And I think that's its most natural arena.

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amelius
Does the power of a laser beam follow the inverse square law?

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frank2
Yes:

[https://www.quora.com/Is-the-light-from-lasers-reduced-by-
th...](https://www.quora.com/Is-the-light-from-lasers-reduced-by-the-inverse-
square-law-as-distance-grows-similar-to-other-light-sources)

ADDED. Yes _at distances from the source greater than 3.3 meters_

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amelius
This post [1] disagrees:

> (...) if you measure the intensity of a beam with a photodiode (say) 1 inch
> from the laser and then 10 inches from the laser, you will not find 1/100th
> of the light with your fixed area detector. If you do this same experiment
> with an incandescent light bulb and make your closest measurement far enough
> away (much, much greater than the size of the filament), you will observe
> this square law decrease.

[1]
[https://groups.google.com/forum/#!topic/sci.optics/P-8VmDlY4...](https://groups.google.com/forum/#!topic/sci.optics/P-8VmDlY45g)

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Noughmad
That post is wrong though.

The real answer is, as usual, "it's complicated". But not very so.

The beam width is roughly constant with distance close to the focal point, so
there the inverse square law does not apply. Far from the focal point, it's
the 3D angle that's roughly constant, which means that the law does apply
there.

~~~
amelius
Yes, makes sense.

