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Why is it that when an object has a small hole, the hole appears dark during the day? Why doesn't sunlight fill the enclosed space and make the hole appear to be the color of the interior walls?

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Long story short: because points in the hole can't see very many bright things that could light them up.

Objects in the room are lit during the day by the sun if they have a direct line of sight to the sun. These sunlit objects receive a large number of photons from the sun. They absorb some of them and emit the rest of them in all directions. These "secondary" light rays hit other objects, lighting them up, but less than the sun lit up the sunlit objects because of that absorption. These objects also absorb light and emit the rest, so on and so forth.

The "brightness" of an object is based on the total number of photons that hit it. If we add up all of the primary, secondary, tertiary, etc. rays we can see how bright any one point will be.

If you have a hole, it's hard to get light in there. If you think about the perspective of a point in the hole, it's going to be lit up by all of the rays from the room which can hit this point. Very few of those come from outside the hole. If you look most directions inside the hole, you see one of the other walls of the hole. That means the illumination from most directions inside the hole has undergone many many bounces to get there, each one diminishing the light.

If you have a hole constructed of a material which absorbs very little light, such as white styrofoam, you'll find that the darkening of the hole is very minimal. This is because less light is absorbed with every bounce, so there's more opportunity for the light to take paths which lead to any nook or cranny within the hole.

You may also notice an interesting effect with color here as well. If, in your very white hole, you paint one green wall, the entire hole will look green. This is because many of the paths bounce off of the green wall, absorbing non-green light. The light that succeeds at bouncing many times (thus lighting the hole effectively) will mostly have hit the green wall at least once, causing the entire hole to become green.

You can see a similar effect if you hold a colored object up to a white wall. You'll see that the wall takes on the color of the object. This is called "diffuse reflections."

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Shallow holes ARE well illuminated in daylight.
One reason 'a hole' is dark, is that a deep hole has much more surface area than the aperture area, and only light that shines on the aperture (proportional to the aperture area and the illumination) illuminates that area (hole-bottom area plus sidewall area). So, on average, brightness of the hole is that of the material times the ratio of aperture area to the aperture-plus-sidewall.

For a 1/4 inch cylinder hole, 1.5 inches deep, that ratio is 1:7

A second, trivial, reason that holes are dark, is that one often examines such a hole by directing light at it, and looking straight down. Alas, your head blocks any light directed straight down. So, the light is shined from the side and directly illuminates only the sidewall of the hole; you don't see that because your viewing angle is edge-on to that side.

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