Physics Photo of the Week

Physics Photo of the Week

November 17, 2023

Brilliant Rainbow - Photos and diagrams by Donald F. Collins

This brilliant rainbow was seen several weeks ago (October 20, 2023).  The rain that day (0.10 inch) was the final rain before the onset of a severe drought - not relieved until a light rainfall over three weeks later (0.21 inch) on November 11.

Rainbows appear only when both rain and sunlight are present at the same time.  The rain drops must be illuminated directly by the sunlight for the rainbow to be visible.  In this image the rainbow appears to be between Warren Wilson College (near the foot of the rainbow) and the mountains behind the base of the rainbow.  Both the foreground, where a building of WWC is visible and the immediate background mountains as well as the rain are all lit up by direct sunshine.

The picture at right shows a zoomed-in photo of the base of the rainbow.  The colors are extremely brilliant!  All the colors from red to blue are shown in this photo.  Notice that the rainbow appears  in front of the mountains beyond.  If there were rain and sunlight in the foreground dark landscape, the rainbow would appear in front of that foreground.  Many of us have played with making rainbows with the spray from a garden hose.  It's fun!

Since rain falls from clouds, usually directly overhead, the sunlight can only reach the rain if the Sun is low enough in the sky, to appear from beyond the back of the clouds - behind the observer - in order for the sunlight to reach the rain shower.  As a result rainbows are usually seen only in the late afternoon or early morning when the Sun is low in the sky near the horizon.

A rainbow only forms in the direction opposite the direction of the Sun.  In the photo above, the camera is looking east, and the Sun is in the west behind the camera.  The rainbow arc is essentially produced by the internal reflection of the sunlight from the far side of the raindrops.  The color is produced by refraction of light by the rainbow similar to the dispersion of white light by refraction from transversing a triangular prism.

The physics of a rainbow is "simple" physics, but geometrically complicated.  The drawing at left shows the optical path of sunlight as it enters the raindrop; the light ray is bent by refraction as it enters the spherical raindrop.  The refracted sunlight travels internally to the far side of the raindrop where it is partially internally reflected.  The internal light continues after the internal reflection to the bottom interior surface of the raindrop where the light that leaves the raindrop is bent again.  At each point where light strikes the surface of the raindrop, it is split into a reflected ray and a transmitted ray.  Only the pertinent rays that contribute to the rainbow are shown in the diagram.  The angle between the emitted light and the incoming light is approximately 42 deg.  At each of the two internal reflections in the drawing, some of the light is transmitted and not shown because that light does not contribute to the rainbow.  At the point where light enters the raindrop, some of the light is reflected and is also not shown, because it reflects away from the observer and doesn't contribute to the rainbow.

The important property of the optics is that at each refraction (where the light enters the drop, and where the light leaves the drop, the light is dispersed into the different component colors that compose white light.  The amount that the light is bent at each refraction point (where light either enters the water or exits the water) depends on the refractive index of the water.  The refractive index if the ratio of the speed of light in air (or vacuum) divided by the slower speed of light in the water.  This index is approximately 1.3 for water.  The important property of the refractive index is that for most transparent materials, the index depends on the color or the wavelength of the light.  Blue light (short wavelength) exhibits a larger index of refraction than red light (long wavelength).  Thus blue light is bent more than red light when it traverses the raindrop.  Thus we see the refracted light at about 42 degrees from the the Sunlight's initial direction, but the blue light and red light are refracted at slightly different angles and the rainbow arc appears as a spectrum.  A triangular prism is often used to disperse light into its component colors and is simpler geometry than the spherical raindrop.

A large model of a raindrop to demonstrate the optics was made by filling a 1 liter spherical flask borrowed from the WWC chemistry lab.   The flask, filled with water, is held by a clamp on a laboratory stand in a darkened room.  The only light is that from a projector shining directly on the flask.  The projector is behind the camera directed toward the flask diagonally from the right.  On the left side of the flask near the edge we see a spot of light that is colored red. This is the light that exits the drop that is drawn in the drawing above.  if the camera were moved slightly to the right the color of the emitted light would be blue.  Also in the picture above we also see a small white dot in the right center of the spherical flask.  This is a partial reflection from the point where the projector light entered the "drop".  This directly reflected ray is fainter than the refracted light that emerged from near the left edge and doesn't affect the appearance of the rainbow much.  The broad whitish areas around the perimeter of the flask are the direct reflections of the walls and ceiling of the room distorted by the spherical mirror of the external surface of the spherical glass flask.  The room was not completely dark.

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Physics Photo of the Week is published weekly during the academic year on Fridays by the Warren Wilson College Physics Department. These photos feature interesting phenomena in the world around us.  Students, faculty, and others are invited to submit digital (or film) photographs for publication and explanation. Atmospheric phenomena are especially welcome. Please send any photos to dcollins@warren-wilson.edu.

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