9 ms·

The Inverse Square Law (2000)

48 points by tucaz 7y ago

puzzledobserver 7y ago
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 7y ago
  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.
  - fhars 7y ago
    „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 7y ago
      Gravity might have extra dimensions on a much larger scale than that, potentially up to the millimeter scale or so.
OldGuyInTheClub 7y ago
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.
rkachowski 7y ago
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.
amelius 7y ago
Does the power of a laser beam follow the inverse square law?
- frank2 7y ago
  Yes: https://www.quora.com/Is-the-light-from-lasers-reduced-by-the-inverse-square-law-as-distance-grows-similar-to-other-light-sources https://www.quora.com/Is-the-light-from-lasers-reduced-by-th... ADDED. Yes at distances from the source greater than 3.3 meters
  - amelius 7y ago
    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-8VmDlY45g https://groups.google.com/forum/#!topic/sci.optics/P-8VmDlY4...
    - Noughmad 7y ago
      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 7y ago
        Yes, makes sense.
- doublement 7y ago
  The first sentence of the piece says that "Any point source which spreads its influence equally in all directions without a limit to its range will obey the inverse square law." A laser beam has a direction, as such doesn't spread its influence equally in all directions.
madengr 7y ago
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).
greens 7y ago
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.
- philzook 7y ago
  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
  - nabla9 7y ago
    But does the inverse square law hold in non-euclidean space?
    - FreeFull 7y ago
      It doesn't. As far as we can tell though, our universe doesn't have much curvature, so the problem doesn't come up.
      - nabla9 7y ago
        It comes up in our solar system. The anomalous motion of Mercury can explained only if the gravity does not follow inverse square law.
        evanb 7y ago
        That's a short-distance effect, though, not a long-distance one.
  - evanb 7y ago
    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)
donutdan4114 7y ago
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 7y ago
  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.
- evanb 7y ago
  The photons (let's say) that cause the inverse square law of the E+M force aren't on-shell photons [on-shell]. So, they're not photons that are "traveling" or are "emitted" in the same way that radiation is---they don't carry any energy away. Similar with or whatever. The 'force law' isn't susceptible to low-intensity counting issues. In the last case---radiation---the photons are on-shell (so you can, for example, count them) and inverse square does degrade when you hit low counts. However, time-averaging restores it on average. [on-shell] https://en.wikipedia.org/wiki/On_shell_and_off_shell https://en.wikipedia.org/wiki/On_shell_and_off_shell
dahart 7y ago
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?
- samatman 7y ago
  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.
  - dahart 7y ago
    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.
- magicalhippo 7y ago
  Often overlooked in such representations is just how much larger and more massive the sun is relative to the planets. Uranus is "only" about 20 times further from the sun than the earth, which according to the inverse square law means the suns gravity (and light) is 1/400th of what the earth experiences. But the sun is about 330000 times more massive than the earth, while Uranus is "only" 14 times more massive.
lurquer 7y ago
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.

48 points by tucaz 7y ago

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 7y ago
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.
- fhars 7y ago
  „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 7y ago
    Gravity might have extra dimensions on a much larger scale than that, potentially up to the millimeter scale or so.

OldGuyInTheClub 7y ago

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.

rkachowski 7y ago

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.

amelius 7y ago

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

frank2 7y ago
Yes: https://www.quora.com/Is-the-light-from-lasers-reduced-by-the-inverse-square-law-as-distance-grows-similar-to-other-light-sources https://www.quora.com/Is-the-light-from-lasers-reduced-by-th... ADDED. Yes at distances from the source greater than 3.3 meters
- amelius 7y ago
  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-8VmDlY45g https://groups.google.com/forum/#!topic/sci.optics/P-8VmDlY4...
  - Noughmad 7y ago
    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 7y ago
      Yes, makes sense.
doublement 7y ago
The first sentence of the piece says that "Any point source which spreads its influence equally in all directions without a limit to its range will obey the inverse square law." A laser beam has a direction, as such doesn't spread its influence equally in all directions.

madengr 7y ago

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).

greens 7y ago

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.

philzook 7y ago
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
- nabla9 7y ago
  But does the inverse square law hold in non-euclidean space?
  - FreeFull 7y ago
    It doesn't. As far as we can tell though, our universe doesn't have much curvature, so the problem doesn't come up.
    - nabla9 7y ago
      It comes up in our solar system. The anomalous motion of Mercury can explained only if the gravity does not follow inverse square law.
      - evanb 7y ago
        That's a short-distance effect, though, not a long-distance one.
- evanb 7y ago
  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)

donutdan4114 7y ago

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 7y ago
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.
evanb 7y ago
The photons (let's say) that cause the inverse square law of the E+M force aren't on-shell photons [on-shell]. So, they're not photons that are "traveling" or are "emitted" in the same way that radiation is---they don't carry any energy away. Similar with or whatever. The 'force law' isn't susceptible to low-intensity counting issues. In the last case---radiation---the photons are on-shell (so you can, for example, count them) and inverse square does degrade when you hit low counts. However, time-averaging restores it on average. [on-shell] https://en.wikipedia.org/wiki/On_shell_and_off_shell https://en.wikipedia.org/wiki/On_shell_and_off_shell

dahart 7y ago

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?

samatman 7y ago
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.
- dahart 7y ago
  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.
magicalhippo 7y ago
Often overlooked in such representations is just how much larger and more massive the sun is relative to the planets. Uranus is "only" about 20 times further from the sun than the earth, which according to the inverse square law means the suns gravity (and light) is 1/400th of what the earth experiences. But the sun is about 330000 times more massive than the earth, while Uranus is "only" 14 times more massive.

lurquer 7y ago

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.