Question

Figure $33-67$ shows a light ray entering and then leaving a falling, spherical raindrop after one internal reflection (see Fig. $33-21 a$ ). The final direction of travel is deviated (turned) from the initial direction of travel by angular deviation $\theta_{\mathrm{dov}}$ (a) Show that $\theta_{\mathrm{dev}}$ is $$ \theta_{\text {dev }}=180^{\circ}+2 \theta_{i}-4 \theta_{r} $$ where $\theta_{i}$ is the angle of incidence of the ray on the drop and $\theta_{r}$ is the angle of refraction of the ray within the drop. (b) Using Snell's law, substitute for $\theta_{r}$ in terms of $\theta_{i}$ and the index of refraction $n$ of the water. Then, on a graphing calculator or with a computer graphing package, graph $\theta_{\text {dev }}$ versus $\theta_{i}$ for the range of possible $\theta_{i}$ values and for $n=1.331$ for red light (at one end of the visible spectrum) and $n=1.343$ for blue light (at the other end). The red-light curve and the blue-light curve have different minima, which means that there is a different angle of minimum deviation for each color. The light of any given color that leaves the drop at that color's angle of minimum deviation is especially bright because rays bunch up at that angle. Thus, the bright red light leaves the drop at one angle and the bright blue light leaves it at another angle. Determine the angle of minimum deviation from the $\theta_{\text {dv}}$ curve for (c) red light and (d) blue light. (e) If these colors form the inner and outer edges of a rainbow (Fig. $33-21 a$ ), what is the angular width of the rainbow?

Name: Problem 77, Chapter 33: Fundamentals of Physics
Uploaded: 2020-02-09T18:27:32-08:00
Duration: 6 min 56 s
Channel: Carlos Henrique De Lima

   Figure $33-67$ shows a light ray entering and then leaving a falling, spherical raindrop after one internal reflection (see Fig. $33-21 a$ ). The final direction of travel is deviated (turned) from the initial direction of travel by angular deviation $\theta_{\mathrm{dov}}$ (a) Show that $\theta_{\mathrm{dev}}$ is
$$
\theta_{\text {dev }}=180^{\circ}+2 \theta_{i}-4 \theta_{r}
$$
where $\theta_{i}$ is the angle of incidence of the ray on the drop and $\theta_{r}$ is the angle of refraction of the ray within the drop. (b) Using Snell's law, substitute for $\theta_{r}$ in terms of $\theta_{i}$ and the index of refraction $n$ of the water. Then, on a graphing calculator or with a computer graphing package, graph $\theta_{\text {dev }}$ versus $\theta_{i}$ for the range of possible $\theta_{i}$ values and for $n=1.331$ for red light (at one end of the visible spectrum) and $n=1.343$ for blue light (at the other end). The red-light curve and the blue-light curve have different minima, which means that there is a different angle of minimum deviation for each color. The light of any given color that leaves the drop at that color's angle of minimum deviation is especially bright because rays bunch up at that angle. Thus, the bright red light leaves the drop at one angle and the bright blue light leaves it at another angle. Determine the angle of minimum deviation from the $\theta_{\text {dv}}$ curve for (c) red light and (d) blue light. (e) If these colors form the inner and outer edges of a rainbow (Fig. $33-21 a$ ), what is the angular width of the rainbow?

Fundamentals of Physics

David Halliday,… 11th Edition

Chapter 33, Problem 77

Instant Answer

Solved by Expert Carlos Henrique De Lima

02/09/2020

Step 1

The light ray enters the raindrop at an angle of incidence $\theta_i$ and refracts inside the raindrop at an angle $\theta_r$. Show more…

Show all steps

Thanks for your feedback!

Figure $33-67$ shows a light ray entering and then leaving a falling, spherical raindrop after one internal reflection (see Fig. $33-21 a$ ). The final direction of travel is deviated (turned) from the initial direction of travel by angular deviation $\theta_{\mathrm{dov}}$ (a) Show that $\theta_{\mathrm{dev}}$ is $$ \theta_{\text {dev }}=180^{\circ}+2 \theta_{i}-4 \theta_{r} $$ where $\theta_{i}$ is the angle of incidence of the ray on the drop and $\theta_{r}$ is the angle of refraction of the ray within the drop. (b) Using Snell's law, substitute for $\theta_{r}$ in terms of $\theta_{i}$ and the index of refraction $n$ of the water. Then, on a graphing calculator or with a computer graphing package, graph $\theta_{\text {dev }}$ versus $\theta_{i}$ for the range of possible $\theta_{i}$ values and for $n=1.331$ for red light (at one end of the visible spectrum) and $n=1.343$ for blue light (at the other end). The red-light curve and the blue-light curve have different minima, which means that there is a different angle of minimum deviation for each color. The light of any given color that leaves the drop at that color's angle of minimum deviation is especially bright because rays bunch up at that angle. Thus, the bright red light leaves the drop at one angle and the bright blue light leaves it at another angle. Determine the angle of minimum deviation from the $\theta_{\text {dv}}$ curve for (c) red light and (d) blue light. (e) If these colors form the inner and outer edges of a rainbow (Fig. $33-21 a$ ), what is the angular width of the rainbow?

Play audio

Feedback

Carlos Henrique De Lima and 71 other subject Physics 102 Electricity and Magnetism educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Snell's Law

Snell's Law is a fundamental principle in optics that relates the angle of incidence to the angle of refraction when a wave passes from one medium to another with a different refractive index. It is expressed as n1*sin(?1) = n2*sin(?2), where n1 and n2 are the refractive indices of the initial and secondary media, and ?1 and ?2 are the respective angles of incidence and refraction. This law is essential for predicting how light bends as it enters a different medium, such as when light enters a water droplet from air.

Angular Deviation

Angular deviation refers to the difference in direction between an incoming light ray and its final direction after undergoing processes like refraction and reflection. In the context of spherical droplets, the overall deviation angle is determined by the angles at which light enters, internally reflects, and exits the droplet. Analyzing this deviation is crucial in understanding phenomena such as rainbows, where the concentration of light at certain angles results in brighter colors.

Internal Reflection

Internal reflection occurs when a light ray, after being refracted into a medium such as a water droplet, reflects off the internal surface of the droplet rather than passing straight through. This reflection is governed by the laws of reflection, and it plays a significant role in the formation of the rainbow by affecting the path and exit angle of the light ray. The interplay between refraction and internal reflection determines the overall angular deviation of the ray.

Index of Refraction

The index of refraction (n) is a measure of how much a medium slows down light relative to its speed in a vacuum. It is a key factor in determining the bending of light rays through refraction. In droplets, variations in the index of refraction with wavelength (dispersion) lead to different bending for different colors, which is central to the formation of colorful rainbows.

Dispersion

Dispersion is the phenomenon where the index of refraction depends on the wavelength of light, resulting in different colors of light refracting at slightly different angles. This wavelength-dependent bending causes colors to spread out when light exits a droplet, leading to the separation of white light into its constituent spectral colors, a process that is fundamental in the creation of rainbows.

Minimum Deviation

The minimum deviation concept refers to the specific angle of incidence at which the deviation of a light ray, after being refracted and internally reflected within a medium, reaches its minimum value. At this angle, the density of outgoing light rays increases, making the corresponding color appear particularly bright. This is an important concept in the study of rainbows, as it explains why certain angles exhibit enhanced brightness for specific colors.

Rainbow Formation

Rainbow formation is a natural phenomenon that occurs when sunlight interacts with water droplets in the atmosphere. The combination of refraction, internal reflection, and dispersion within the droplets causes white light to split into its component colors, creating a spectrum of red, orange, yellow, green, blue, indigo, and violet. The angular differences in the paths of these rays due to the minimum deviation angles for each color result in the colorful arc seen in the sky.

Recommended Videos

Transcript

00:01 So in this problem, we have the light ray entering a spherical raindrop and undergoing internal reflection.

00:13 And the part a, they want to know how to express the angular deviation in terms of the the incidence and refracted angles.

00:27 So the first contribution that we get for the deviation comes from the from the first refraction.

00:35 So, delta, theta, 1 is a minus theta r.

00:45 The next contribution to the overall deviation is the reflection that occurs here, that we call point b.

00:52 And here we call point a, and here is c.

00:56 And noting that the angle between the right before the reflection and the axis normal to the back surface of this sphere is equal to teta -r, teta r, here, teta r.

01:10 And recalling the lower fraction, we conclude that the angle by which the rate turns is teta 2, 180 degrees minus 2 teta r.

01:27 And the final contribution for the deviation comes from the refraction coming at point c.

01:35 So we get at the final deviation is teta i minus teta r so we just need to sum the three deviations here we get that delta teta is delta theta 1 plus delta theta 2 plus delta theta 3 and this is 180 degrees minus 2 tita i minus 2 teta r so this is the equation related the incidence and refracted angles and the deviation of the angle of the light.

02:24 For part b, they want you to use nel's law to substitute the refractor angle in terms of the incidence angle and the refraction index of water that will use as 1 .333.

02:43 And use a graph calculator to graph the deviation angle versus deta.

02:50 We can draw a scratch of it.

02:53 So let's do this.

02:55 So the substitution using linear law is teta r is equal minus 1 of 1 over n sine of tita r.

03:10 And we substitute this in the deviation equation.

03:13 So this means delta is 180 degrees plus minus plus 2 theta i minus 2 times sine minus 1 of sine theta i over m.

03:39 And we use the values that they gave us for for n.

03:44 So we can do a quick sketch of what this graph should look like...

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace