Hypothetical light in a sphere

This is a question that’s popped into my mind a few times in the past, and now, it finally pops in while I have the opportunity to ask the great minds of the Dope!!

Okay, here goes;

Assuming a hollow sphere, say, about a meter in diameter, with the inside mirrored, and flawless.
Assuming a light source that’s also a perfect sphere, floating perfectly in the middle of the mirror-sphere. It’s about 2 inches in diameter (actual sizes don’t matter, this is just so people have the same relative dimensions in mind.)
Also, this light source is impervious to heat, so it won’t self-destruct during our experiment. Also, it’s made of a basically non-reflective material.

Now, create your theories on light intensity;
If we move the light source off-centre of the mirror-sphere, what happens to the light intensitywithin the sphere?

I already have my theories for this, which I’ll type up in Notepad (before reading any of the replies in here), and copy/paste in here later. Also, you may add anything you believe pertinent to the discussion.

S^G

By your desrciption, it seems that the first instance of the sphere will have all light rays exiting a source, reflecting off the mirror back towards the center, and be absorbed by the source. In the second instance, the light source being off center, the light rays presumably will not be reflected towards the center and absorbed, but will reflect multiple times, and thus more light will be present.

I think that would be true if this was a point-source of light, but a point-source is an idealization (David Falk, Seeing the Light), and we can never have a true point-source. At best, their would never be a condition in which every ray is reflected to the center of the sphere (even if the sphere’s surface is perfect and infinitely thin).

So, I theorize that the difference in light intensity would be zero, because no matter where the light source is, it should absorb the same amount of light.

I’ve thought about this too! In my version: you have a sphere with a mirrored inner surface containing nothing in the center, and shine a laser beam inside, at an angle through a very small external “hole” with a one-way reflective surface. Could you “trap” most of the light inside? What would happen if you suddenly shattered the sphere?

There isn’t any way to construct or arrange such a device so that the light wouldn’t quite quickly find its way back out of the same small hole as it entered (I think).

True, but if you have a shutter that can close in a nanosecond or so, you will be fast enough - light only travels about a foot in that time. Note that I said “if”. :smiley:

Why this fascination with light rattling around inside a sphere? The light bounces around. In the real world, some invariably gwets lost with each bounce. If your reflectivity is rho, then you typically have on the order of (rho/1-rho) bounces inside before the light gets absorbed. If your walls are made of magic material that doesn’t absorb, then it rattles around forever, or until whatever’s inside your sphere manages to absorb it all.
satellite Guy’s premise in the OP actually showed up as a problem on one of my courses in radiometry, only with both the outside sphere and inner sphere made of blackbody absorbers, which makes it both more physical and more interesting. Photons going from the inner sphere see a hemisphere of the outer sphere and are absorbed by the outer sphere. Photons from the outer sphere re=radiated inwards have a chance of running into (and being absorbed by) the inner sphere that you can work out by geometry. Call it x%. They then have a (100-x)% chance of hitting (and being absorbed by) the outer sphere itself.
You can handle all these cases for an off-center sphere as well, but the math is a hell of a lot more complex, and it makes a difference where you are on each sphere. That added complexity in calculation doesn’t add a bit to your understanding, which is why they don’t usually assign problems like this.
CoG888 is quite right about there not being true point sources in the real world (which is why, even with a perfect parabolic reflector, searchlights invariably spread out. See Miles V. Klein’s book Optics, or Boyd’s stealing of the same problem in his Radiation and Detectors. But the OP did specify an inner sphere, not a point source.
And you don’t have to make your spheres out of indestructible unobtainium. Just don’t use a buttload of light in the first place.

This one is a bit different to the usual light-in-a-sphere threads - they normally pose the question with the sphere as some kind of energy storage device (often with magic properties such as perfect one-way transmission of light) - but this one is a bit different (and interesting) because it’s really about the way light distributes itself inside a closed spherical reflector.

Okay, here goes my take on the subject;

I believe that with the light source perfectly centered, the light exiting from the source will bounce off the mirrored surface of the sphere, and come back to hit the light source, and be absorbed as heat. Even the light that exits from the edge of the light source will be blocked by the opposite edge, since the source is perfectly spherical.
Once you move the light source off-center, the light will be free to bounce around within the sphere many times, to add itself to the new light always coming from the light source. as a result, the light intensity will increase drastically, while the light source won’t have to absorb as much of it as heat. Mind you, the sphere itself wil probably heat up quite a bit.

Carry on

S^G

How would the wavelength of the light factor into this? Would the construction of the sphere (i.e. its radius) have any effect on the interference waves that would be created?

If we’re talking human-dimensioned sphere, then we’re so far from the size of the wavelength of light that it’s pretty much irrelevant. It’s when you get to dimensions comparable to the size of the sphere that interesting things happen. This is why interference becomes very important in the formation of rainbows when the drop dimensions become such that the path length differences are only a couple of wavelengths, something first pointed out by Thomas Young in 1804 when he was promoting the wave theory of light. (Even though that’s a refractive situation).
I have to admit that’s it’s not perfectly clear to me what happens when you try to shove light into a sphere that is itself only a couple of wavelengths across (or, even more interestingly, less than that), yet perfectly reflecting. There’s probably an idal resonance at a wavelength or so. I’;ll bet someone’s done a paper on this at some point or other.

There’s been some interesting work on light in nanostructures. This isn’t a sphere, but here’s news from Cornell on light trapped in volumes orders of magnitude smaller than a cubic wavelength:

http://nanophotonics.ece.cornell.edu/research.html

I think the light would pulse. If the source is non reflective it should absorb the light its emitting, but light takes time to travel so it would be visible for a short time. I’m uncertain how light travels through itself but that may affect the outcome

There is nothing special about light being reflected multiple times between two mirrors. Your sphere can be considered as 2 concave mirrors having the same radius and a common centre. In your case, the intensity of light will keep increasing until the mirror starts melting, unless it is perfectly reflective, in which case the intensity will increase forever.

 In fact, that's how a laser works: a gain medium (that is, a medium in which stimulated emission can take place) is placed in between two mirrors. One of the mirrors has a very high reflectivity (as close to 100% as possible); the other one also reflects most of the light, but some of it goes through. A photon can bounce between the two mirrors thousands of times (and ideally stimulating the emission of other photons at each trip) before getting out through the second mirror. 

Now, to put a spin on the OP, lets assume that the smaller sphere has the outside coated with a reflective layer and that the two spheres are concentric. In this configuration you’ll get a convex mirror in the centre and a concave mirror surrounding it. In fact, that’s one of the configuration used for lasers (have a look at the second figure on this page ) , so in principle you can get a spherical laser, provided that you fill the space between the spheres with a gain medium (e.g., a mixture of gases).
If you make the outer sphere semi-transparent, then your laser will look like a ball of light. If you make the inner sphere semi-transparent, then all the light will be focused in the common centre of the two spheres and you’d get a very intensity (which could be used for trivial purposes, like cold fusion :slight_smile: ).
Also, from the wiki link above, it looks like someone managed to make a laser out of a drop of solution.

As for what's happening when light meets sub-wavelength structures... well, many things can happen :). As examples, you can have a look at [micro-structured optical fibres](http://en.wikipedia.org/wiki/Photonic-crystal_fiber) or even optical fibres having a [diameter inferior to the wavelength of light](http://en.wikipedia.org/wiki/Subwavelength-diameter_optical_fiber)propagating trough them. You get some interesting effects by illuminating semiconductor spheres several nanometers in diameter, aka [quantum dots.](http://en.wikipedia.org/wiki/Quantum_dot)

I admire your enthusiasm and interest, but there is much that is incorrect in this lengthy statement (and much that isn’t quite so wrong, but still misleading). Unfortunately, this is a bad and busy day, and I haven’t the time to comment on it right now.
But, to begin with, this statement is completely incorrect, unless you’re making it tongue-in-cheek, in which case smilies are your friend:

Quoth CalMeacham:

Wouldn’t you just get spherical harmonic standing waves? It seems like a textbook problem to me.

Back to the OP, what are the properties of the light source? Does it get a constant amount of power from some energy source (a battery or something), or is it calibrated to emit a constant amount of light? If the former, then it’ll get brighter with time, since the light that it absorbs after it’s bounced back to it will heat it up. If the latter, then the drain on the battery will decrease to nothing, once it gets to the point where its incandescence is enough to provide the specified amount of light and it can’t cool off from that temperature.

Unfortunately, I wasn’t making a joke; you’re right, I’ve got it wrong. Should have read more carefully the OP too. My apologies.

So, may I have a second try? :slight_smile:

As described by the OP, it seems that all the light emitted by the light source stays inside the sphere. I don’t see any way for it to get out, as the mirror is said to be perfect (by which I understand that it reflects everything). So some of the light* will get absorbed by the light source, which will become hotter and it will start radiating IR light. If the mirror reflects perfectly all the wavelengths, then the IR thermally emitted from the light source will also be reflected, absorbed again by the light source which will become even hotter and so on, as Chronos said.
If the mirror does not reflect IR, then there is a way for the light source to cool down. The question is if it would reach an equilibrium temperature low enough** so that it does not start emitting significant amounts of visible light (which would be reflected back by the mirror). This would also depend on what kind of light source this is (thermal, LED, fluorescent…).

  • I wonder if it is possible to place the light source in such a way that some of the light emitted will never be reflected back on it… This wouldn’t change the overall situation, but the temperature increase may be slower.
    ** A higher temperature may also allow cooling, if the mirror does not reflect UV…

I’ve also said that, if the mirror is not perfect and absorbs some of the light, it would start melting at some point. That may not be the case; it will became hotter, but it will also start emitting blackbody radiation. If it emits towards the outside the same power as the light source emits, then it would also reach an equilibrium.

This being said and with the hope that this post is less wrong than the previous, I’ll go take a nap. Apparently I really need it. :slight_smile:

The more a laser bounces the slower it goes. Check it out.
http://www.sixtysymbols.com/videos/lasercooling.htm
So its possible that if your sphere was somehow made of polished robidium, or whatever’s practical, that you could cool it to absolute zero. Idk if the same is true for the original light question but either way it would be cool

What kind of light it emits will depend on its temperature-- There’s nothing special about infrared in that regard. If we’re assuming, for instance, that the light source in the center is an incandescent light bulb, then it’s already more than hot enough to emit infrared-- It is, in fact, hot enough to emit visible light, too. If we increase the temperature further, then it’ll emit even more visible light (the amount of infrared emitted will also increase, but not by as much as the visible increases). Heat it up enough, and you’ll start getting significant amounts of ultraviolet or beyond.

Well, only the center of the inner sphere is at the center of the outer sphere, so the surface will emit light that reflects in directions as random as the emission characteristics. The center sphere is non-reflective (absolutely?) so it heats up, and emits more, and higher frequency light. Since it has some magical light production of its own things get hot pretty fast. If the outer sphere is not perfectly evacuated you get a magic heat source in a ball of plasma, inside a perfect reflector, which better be magically strong too.

If the center sphere is held in place magically, then it stays in place, magically. (of course, if the outer sphere is not held in place magically, then neither is the inner sphere.) If not, it is only stable when the entire system is stable, and any off center position will remain imbalanced, and it will eventually begin random walking around the interior plasma. I suppose all this perfectly strong material will allow it to ricochet around endlessly.

Tris