The inverse square law for point light sources inside an opaque medium

In summary, the inverse square of the distance to a point light source is still a constant, but the power passing through a sphere of radius r around the source is reduced as exp(-λr)/r².
  • #1
mgamito
8
0
Hello all,

I already know that the radiant intensity of a point light source falls off with the inverse square of the distance to the source. This, however, only happens in a vacuum. My question is, what is the more general law for a point source inside an opaque medium with a known absorption coefficient σ(x) that may vary across space. From symmetry considerations alone, I would expect that the result will still be a function only of the distance to the light source, as before, just not an inverse square power anymore. The actual function will, of course, depend on σ(x) and should be the outcome of some 1D differential equation whose control variable is the distance to the source - it is the form of this 1D equation that I am looking for.

To clarify a bit further, the above absorption coefficient occurs in the light transport equation when stating that the derivative of radiance L(t) along a light ray parameterised by t is:

dL(t)/dt = -σ(t) L(t)

or stating the same in 3D space:

(ω.∇) L(x, ω) = -σ(x) L(x, ω)

where L(x, ω) is the radiance at point x in the direction ω.

Thank you,
manuel
 
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  • #2
Both factors apply. The inverse square plus (or, rather, multiplied by) the absorption per metre travelled.
 
  • #3
Hi Sophie, you mean that there is an effective absorption coefficient given by σ(t)/t^2 ? I don't see how that would work because in an optically transparent medium (σ = 0), the inverse square term would vanish together with the σ, which we know is not true.

manuel
 
  • #4
Absorption PER METRE. It's exponential with distance (in addition to the spreading loss).
 
  • #5
Better way to look at it, perhaps, is that in vacuum, the total power of the radiation passing through a spherical shell around a point source is a constant regardless of shell radius. This gives you inverse square per unit area. In opaque medium, the total power drops of as exp(-λr). Therefore, power per unit area drops as exp(-λr)/r²
 
  • #6
Thank you both - I understand it now. I didn't imagine the solution was so simple as to just multiply the inverse square with the absorption (after application of the exponential to the latter).
 

What is the inverse square law for point light sources inside an opaque medium?

The inverse square law for point light sources inside an opaque medium states that the intensity of light at a given point is inversely proportional to the square of the distance from the point source. This means that as the distance from the source increases, the intensity of light decreases rapidly.

How does the inverse square law affect the brightness of a light source inside an opaque medium?

The inverse square law directly affects the brightness of a light source inside an opaque medium. As the distance from the source increases, the brightness decreases significantly. This is because the same amount of light is spread out over a larger area, resulting in a lower intensity of light.

Does the inverse square law apply to all sources of light inside an opaque medium?

Yes, the inverse square law applies to all point light sources inside an opaque medium. This includes sources such as light bulbs, candles, and stars. However, it does not apply to extended sources of light, such as a fluorescent tube or a LED strip.

How is the inverse square law for point light sources inside an opaque medium different from the inverse square law for point sources in a vacuum?

The inverse square law for point light sources inside an opaque medium is different from the inverse square law for point sources in a vacuum because in an opaque medium, the light is absorbed and scattered by the particles in the medium. This means that the intensity of light decreases even more rapidly with distance compared to a vacuum, where there is no interference from a medium.

What are some real-world applications of the inverse square law for point light sources inside an opaque medium?

The inverse square law for point light sources inside an opaque medium is crucial in understanding the behavior of light in various situations. Some real-world applications include photography, where the distance between the subject and the light source affects the exposure and brightness of the photo. It is also important in understanding the intensity of light in medical imaging techniques such as X-rays and CT scans.

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