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Curve of Rainbow
Name: Lee
Status: educator
Age: 50s
Location: N/A
Country: N/A
Date: 10/31/2004
Question:
A friend of mine was visiting his young granddaughter
and they viewed a rainbow together. Her question to him was not about
the colors but was fascinating to me for such a young observer. She
asked "Why is a rainbow curved and why is it in a continuous curve and
not in small sections?" The grandfather seeks help and I would love to
see this myself. The idea that light is broken down in passing
through raindrops which are basically spherical may be part of this
but somehow it seems very incomplete.
Replies:
The reason a rainbow is a smooth curve is because it is the *angle*
between the light incident on a raindrop, and the light exiting the
raindrop, that determines what color is seen. And, as you note, it
also depends on the fact that raindrops are, on average, roughly
spherical. The path from sun to raindrop to viewer's eye includes
the angle, and all the raindrops located so that the angle has a
particular value, will contribute to a particular color of the
rainbow. This might be difficult for a child to visualize, but it is
a simple geometrical notion that you can demonstrate with a coat
hanger.
Here is a maybe more detail than you really want about how it works:
First, some background: a ray of light that hits the middle of the
raindrop is partially reflected and partially transmitted. The
reflected part goes back directly toward the source; the transmitted
part goes through the raindrop and is partially reflected at the back
surface. Again the reflected part goes back directly toward the
source; the transmitted part leaves the raindrop and we don't care
about it anymore. That ray that reflected from the back surface is
now heading back toward the front surface, where the partial-reflection
thing will happen again -- many times, in fact. On each round trip,
some light will be transmitted through the front surface back to the
source.
Second bit of background: What if the light ray hits off center?
Then things get very interesting. First, the surface of the raindrop
acts as a prism, since the light is incident at an angle, and the
light will be refracted (its path will be bent) as it passes through
the surface. Second, the first reflection from the back surface
will not be in the same place as the second reflection from the back
surface; the light will sort of bounce around within the drop,
reflecting here and there in a very predictable manner and sequence.
Third bit of background: each reflection from the back surface will
make its own rainbow. Normally, you can only see the first-reflection
rainbow, and maybe a faint suggestion of the second-reflection rainbow
(at a different angle).
Fourth bit of background: The angle through which a light ray is bent
as it passes through the air-water surface depends on the index of
refraction of the water (and the air). The index of refraction of
water is different for different colors of light. This is because
electrons in the water molecules are tuned, like guitar strings, and
a light ray tries to vibrate them at the frequency (color) of the
light ray. Light rays that are nearer in frequency to the electron's
"tune" are affected differently -- i.e., they see a different index of
refraction -- i.e., they are slowed down by a different amount, and
therefore refracted by a different angle -- than light rays farther
away in frequency.
So a light ray hitting the raindrop off center will be refracted,
and part of it will exit the raindrop at an angle that depends on
how off-center the incident ray was. In fact, as the ray hits further
and further off center, the exiting ray will make a greater and greater
angle with the incident ray -- up to a point. There is a maximum
exit angle, which corresponds to a certain off-center distance, and
the exiting light is concentrated near this maximum angle, because
the variation of exit angle with off-center distance is smallest near
this angle. You can demonstrate this with a laser and a cylinder of
something clear. The maximum exit angle depends on the color of
light because the refraction angle generally depends on the color.
Tim Mooney
I think Ask-A-Scientist has an answer to this one:
http://www.newton.dep.anl.gov/askasci/phy00/phy00555.htm
Todd Clark, Office of Science
U.S. Department of Energy
This was answer I wrote on Aug 22, 2002. I had to search the Argonne
archives to find it; (I just did not want to type it all again). Anyway, I
hope this helps answer your friend's granddaughter.
Martha Croll
Yes your instincts are correct to suspect that there is "something more
going on". The "explanation" you asked for sent me on a search, and the
answer is a bit complicated and long to give here. However, the good news
is, there is a magnificent web site that walks through the "curvature
question" and several other properties of rainbows that you will (read
that: I did) find very interesting and enlightening.
Check out: http://my.unidata.ucar.edu/content/staff/blynds/rnbw.html?printer
Vince Calder
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Update: June 2012
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