Rainbows Rainbows are created when sunlight of different...

Rainbows Rainbows are created when sunlight of different wavelengths (colors) is refracted and reflected in raindrops. The angle of elevation $\theta$ of a rainbow is always the same. It can be shown that $\theta=4 \beta-2 \alpha,$ where $$ \sin \alpha=k \sin \beta $$ and $\alpha=59.4^{\circ}$ and $k=1.33$ is the index of refraction of water. Use the given information to find the angle of elevation $\theta$ of a rainbow. (For a mathematical explanation of rainbows see Calculus Early Transcendentals, 7 th Edition, by James Stewart, page 282 .)

Rainbows Rainbows are created when sunlight of different wavelengths (colors) is refracted and reflected in raindrops. The angle of elevation $\theta$ of a rainbow is always the same. It can be shown that $\theta=4 \beta-2 \alpha,$ where
$$
\sin \alpha=k \sin \beta
$$
and $\alpha=59.4^{\circ}$ and $k=1.33$ is the index of refraction of water. Use the given information to find the angle of elevation $\theta$ of a rainbow. (For a mathematical explanation of rainbows see Calculus Early Transcendentals, 7 th Edition, by James Stewart, page 282 .)

Key Concepts

Trigonometric Relationships

Trigonometric relationships are used to connect the various angles involved in the path of light through a raindrop. These relationships allow for the derivation of formulas, such as the one relating the angle of elevation of the rainbow to the angles of incidence and refraction, enabling precise calculations of phenomena in geometric optics.

Geometric Optics

Geometric optics deals with the propagation of light in terms of rays and explains phenomena like reflection and refraction. It is fundamental in understanding how light interacts with objects such as raindrops to produce effects like rainbows by following precise paths determined by the laws of reflection and refraction.

Refraction

Refraction is the bending of light as it transitions from one medium to another with different optical densities, quantified by their refractive indices. This concept is crucial for explaining how sunlight disperses into different colors when it enters and exits raindrops, thereby forming the spectrum seen in a rainbow.

Snell's Law

Snell's Law mathematically relates the angles of incidence and refraction to the refractive indices of two media. This law is used to determine the change in direction of light as it passes from air into water, which is essential for calculating the angles involved in rainbow formation.

Reflection

Reflection is the process by which light bounces off surfaces. In the context of a rainbow, internal reflection within raindrops redirects the refracted light, contributing to the specific pattern and angle at which the light finally emerges to create the observed arc.

Transcript

00:01 From the work problem given, make sure that you get the main information that you need.

00:04 So the first thing that you need to know is that you can find data by doing 4 beta minus to alpha.

00:12 And then you also have another equation in there that says the sign of data equals k sign of beta.

00:19 Now they're giving you the alpha and they're giving you the k.

00:22 So we're going to use that to find beta because we're going to need that in order to plug it in into my first equation.

00:30 To find beta.

00:32 So again, my goal is to find beta.

00:37 So again, the first step is i am going to do the second equation because i can do anything here because i don't know beta, but with my second equation, i could find beta.

00:48 So the first thing that i am going to do is substitute what i know.

00:53 So i know that sign of 59 .4 is equal to 1 .33.

01:00 Times sign of beta.

01:04 My next step then is to find what sign of 59 .4.

01:09 Again, make sure that you are using degrees in your calculator.

01:15 So if you put in sign of 59 .4, that gives you 0 .86.

01:21 So this is 0 .86 about.

01:26 I'm rounding.

01:27 And this gives you 1 .33...