All answers should be in degrees.
(a) Using the result of Lecture 31 Question 3, and Snell's law, derive and evaluate for the secondary rainbow for the following values of the incidence angle and index of refraction .
1 , 1.4
incorrect
45 , 1.3
incorrect
87 , 1.55
incorrect
(b) For a given index of refraction , determine the value of the incidence angle that minimized , and the correspondent value of .
1.4
incorrect
incorrect
1.3
incorrect
incorrect
(c) What is the angle of incidence of light on a spherical raindrop that will lead to the red ( ) in the secondary rainbow?
incorrect
What is the associated value of ?
anyone
To derive and evaluate the angle of the secondary rainbow, we can start with Snell's law:
\(\frac{{\sin(\theta_2)}}{{\sin(\theta_1)}} = \frac{{n_1}}{{n_2}}\)
where:
- \(\theta_1\) is the incidence angle
- \(\theta_2\) is the refraction angle
- \(n_1\) is the index of refraction of the medium before refraction
- \(n_2\) is the index of refraction of the medium after refraction
For the secondary rainbow, we have the following values:
- \(n_1 = 1\) (air)
- \(n_2 = 1.4\) (water droplet)
(a) To obtain the angle of the secondary rainbow for various incidence angles, we need to solve Snell's law. Let's use the result from Lecture 31 Question 3:
\(\theta_2 = \sin^{-1}\left(\frac{{n_1}}{{n_2}} \sin(\theta_1)\right)\)
Now we can evaluate \(\theta_2\) for the given incidence angles and \(n_2 = 1.4\):
1. For an incidence angle of 1 degree:
\(\theta_2 = \sin^{-1}\left(\frac{{1}}{{1.4}} \sin(1)\right)\)
2. For an incidence angle of 45 degrees:
\(\theta_2 = \sin^{-1}\left(\frac{{1}}{{1.4}} \sin(45)\right)\)
3. For an incidence angle of 87 degrees:
\(\theta_2 = \sin^{-1}\left(\frac{{1}}{{1.4}} \sin(87)\right)\)
(b) To determine the value of the incidence angle that minimizes \(\theta_2\), we can set the derivative of \(\theta_2\) with respect to \(\theta_1\) equal to zero. However, we need to know the specific index of refraction (\(n_2\)) to calculate this. Could you please provide the value of \(n_2\) for this part?
(c) To find the angle of incidence for the red light in the secondary rainbow, we need to know the associated value of \(\theta_2\) or the index of refraction \(n_2\). Could you please provide either of these values?