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I'm trying to design a script for a Unity game that can accurately (give or take) simulate heat based on distance from a source. Using the inverse square law seems to be a way I can do this, but setting distance to zero always seems to set intensity to infinity. Am I misunderstanding something about the inverse square law?

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    $\begingroup$ It suggests that the law breaks down close to the source. $\endgroup$ – Mozibur Ullah Aug 20 '16 at 10:24
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    $\begingroup$ The inverse square law is based on the area of a sphere with the radius being the distance from the source. It's meaningless to say you have a sphere with a radius of 0, that's a point. You sphere has to be at least large enough to contain the source for it to work correctly. $\endgroup$ – Jason Goemaat Aug 21 '16 at 6:33
  • $\begingroup$ Indeed, what's the surface area of a sphere with r=0... Does that even make sense geometrically? $\endgroup$ – Mark K Cowan Aug 22 '16 at 10:54
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This is just an artefact from assuming your heat source is infinitesimal in size. Also, more generally, physics tends to break down in the limit $r\to 0$.

So, in many cases, you have the luxury of doing 'something else' in the vicinity of $r=0$. For example, it is common in various simulation settings (n-body, free energy perturbation, etc) to use so-called soft-core potentials. These are potentials that are modified so that the singularities disappear.

For a $1/r^n$ potential, it's common to replace it with something of the form $$ \frac{1}{(1+r^{ns})^{1/s}} $$ Here is a plot for $n=2,s=3$.

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    $\begingroup$ At close proximity you need to consider Quantum Field Theory. In essence all of our particle physics devices (colliders such as Fermilab, LHC, and others) try to bring these things closer and closer together. $\endgroup$ – Viktor Aug 20 '16 at 14:18
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    $\begingroup$ @Viktor There's quite a range of distances between the point where the inverse law breaks down and quantum effects become important. This is the "near field". No quantum effects needed here. To calculate the effect in the near field, divide the source into many many small volumes, treat each of the small volumes as a point source, and add the effect from each. The size of the small volumes must be much smaller than the distance from the source to the observation point. $\endgroup$ – garyp Aug 20 '16 at 15:32
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    $\begingroup$ I get the first part, but I don't understand what the variables in the second part of your answer mean. I think r represents distance from the source, but I have no idea what s and n represent. $\endgroup$ – Andrew Pampuch Aug 20 '16 at 18:27
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    $\begingroup$ I think @lemon is suggesting that you effectively add a maximum temperature starting at some small distance from the heat source through to the center of the heat source. Even the center of the sun or the middle of a torch flame has a maximum effective temperature as the non-zero volume of these sources distributes the heat sources as you enter inside them. $\endgroup$ – Brad Cooper - Purpose Nation Aug 20 '16 at 23:14
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The inverse square law applies to point sources. In nature there are no point sources, so strictly speaking it never applies.

In nature we have extended sources. If you are far away from the source, then it "looks like" a point source, so the inverse square law is a good approximation. Close to the source, the inverse square law doesn't apply unless the source happens to be perfectly spherical. Even in that case the law breaks down once you cross the surface and are inside the source.

Update

@AnonymousCoward (below) reminds me that electrons do appear to be point sources. I point this out for completeness. This fact doesn't change things for your question, though.

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Yes. An inverse square law is a law stating that some physical quantity is proportional to $1/r^2$, i.e. $X = Y/r^2$ is the general formula.

And so, the limit as distance [$r$] approaches 0 is "equal to infinity".

That is to say, the strength of a force that obeys the inverse-square law (like the electromagnetic force) approaches infinity as the distance to the source of the force tends to 0.

NOTE: The distance term in the inverse square law, usually denoted $\mathbf{r}$, represents the distance from the source, which is why right AT the source, $r=0$, so the magnitude of the force would be something divided by 0, which would be 'infinity'.

So two electrically charged particles with the same charge (i.e. both positive or both negative), for example, cannot 'touch' as it were (unless other forces were brought into play), because the electromagnetic force is an inverse-square law force: $$E = \frac{kQq}{r^2}$$

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  • $\begingroup$ actually nucleons can touch. That is the concept behind fusion. using the short range strong nuclear forces to overcome the electrostatic repulsion. $\endgroup$ – Lelouch Aug 20 '16 at 14:03
  • $\begingroup$ Ah, yes of course. Edited. $\endgroup$ – K. T. Aug 20 '16 at 16:45
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    $\begingroup$ @Lelouch nucleons can touch but their origins are still separated by the distance of the sum of their radii, so the electromagnetic force being applied from each to the other is still less than infinity. $\endgroup$ – Patrick Roberts Aug 20 '16 at 22:25
  • $\begingroup$ At those distances , quantum pressure like the degenracy pressure acts as well. $\endgroup$ – Lelouch Aug 21 '16 at 3:37
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There actually are point sources in nature (electrons, for example, are effectively point sources), but when you're talking about really small distances, you have to think in terms of probabilities for a particle to be at a particular location. And when you talk about probabilities, you're really talking about probability densities. In other words, you're talking about the probability for a particle to be between x and x+dx, where dx is arbitrarily small but not zero. The probability for a particle to be at an exact location, with zero uncertainty, is zero.

So even though the inverse square law predicts an infinite force at zero distance between two point particles, the probability of exactly zero distance is zero.

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