The above illustration
shows the refraction and reflection of a ray in a drop that
causes the primary rainbow. Using the laws of reflection and
refraction we can find the angle between the ray from the sun and
the ray that we observe (the rainbow). This angle is called *a*.

At point A, the ray is refracted, and the law of refraction gives

,

where *i* is the
angle of incidence, *r* the angle of refraction and *n*
the index of refraction for the drop of water. The ray's
deflection is the angle *i - r*.

At point B, the ray is
reflected within the drop (angle of reflection = angle of
incidence = *r*); the ray's deflection is the angle 180° - 2*r*.

At point C the ray is
refracted out of the drop, here the angle of incidence is *r*
and the angle of refraction is *i*. The ray's deflection is
the angle *i - r*.

In total, the ray's deflection becomes:

u= (i-r) + (180° - 2r) + (i-r) = 180° + 2i- 4r.

From the above figure,
it is evident that the angle *a* equals 180° - *u*, i.e.

a= 180°- (180° +2i- 4r) = 4r- 2i.

Thus, the angle of the
primary rainbow is __a____ = 4____r____ ____-____ 2__* i*.

The angle *a* is
shown on the upper right graph as a function of the wavelength l for both the primary and the secondary
rainbow. Furthermore, the angle is displayed in the little figure
in the upper left.

In the secondary
rainbow, the ray is reflected twice within the drop, so the ray
is deflected with an angle *u*, see the blue arcs on the
figure.

u= (i-r) + 2(180°-2r) + (i-r) = 360° + 2i- 6r

u= 360° - (6r- 2i).

The deflection of the sunray is thus

u= 360° - (6r-2i).

The angle of the secondary rainbow becomes:

a=180°-(6r-2i).

For both rainbows, the
angle of incidence *i* can vary between 0°, as the ray
falls on the center of the drop (*p*=0), and 90°, as the
ray brushes the drop (*p*=1). We can now use the program to
examine the variation of the angle of refraction and thus the
angle of deflection (see also Alexander's band and Colors of the
rainbow).

Exercise 7

Between which values does the angle,

a, vary for the primary rainbow?Between which values does the angle,

a, vary for the secondary rainbow?Select »

marks« and »axes« and »normals«and print a graph for both the primary and the secondary rainbow. Add to the print angles of incidence and refraction and find the »blue« angles as a function ofiandr.Hereby show the equations:

a= 4r- 2ifor the primary rainbowand

a= 180° - (6r-2i) for the secondary rainbow.

Exercise 8

Use the program to fill out a form like this:

color |
wave- length |
index
of refraction |
maximum angle of deflection for the primary rainbow | minimum angle of deflection for the secondary rainbow |

The explanation of the rainbow is that different colors have different indices of refraction and are thus refracted differently. But there are certain angles that no light is refracted to, and this interval of angles is also dependent of the wavelength.

Exercise 9

Use your results from the table in exercise 8 to draw a graph that corresponds to the little graph in the upper right corner of the program's illustration of the drop.

Notice that the track bar in the upper left switches between the primary and the secondary
rainbow. Also notice that pressing **F6** will draw a pencil
of parallel rays falling on the drop. This clearly shows where
the scattered rays go. Also notice that the density of the rays
is highest near the maximal (minimal) angle.

On this and other
displays, the ray with an angle of deflection equal to the
rainbow angle is drawn in red. This ray can be turned on and off
with **F5**. The axes of incidence (brown lines) can be turned
on and off with **F3**, and the axes of the coordinate system
can be selected with **F4**.