Waves/Geometrical Problems

Problems
Figure 3.14: Refraction through multiple parallel layers with different refractive indices. Figure 3.15: Refraction through a $$45^{\circ }$$-$$45^{\circ }$$-$$90^{\circ }$$ prism. Figure 3.16: Focusing of parallel rays by a parabolic mirror. Figure 3.17: Refraction through a wedge-shaped prism.


 * 1) The index of refraction varies as shown in figure 3.14:
 * 2) Given $$\theta_1$$, use Snell's law to find $$\theta_2$$.
 * 3) Given $$\theta_2$$, use Snell's law to find $$\theta_3$$.
 * 4) From the above results, find $$\theta_3$$, given $$\theta_1$$. Do $$n_2$$ or $$\theta_2$$ matter?
 * 5) A $$45^{\circ }$$-$$45^{\circ }$$-$$90^{\circ }$$ prism is used to totally reflect light through $$90^{\circ }$$ as shown in figure 3.15. What is the minimum index of refraction of the prism needed for this to work?
 * 6) Show graphically which way the wave vector must point inside the calcite crystal of figure 3.3 for a light ray to be horizontally oriented.
 * 7) The human eye is a lens which focuses images on a screen called the retina. Suppose that the normal focal length of this lens is $$4 \mbox{ cm}$$ and that this focuses images from far away objects on the retina. Let us assume that the eye is able to focus on nearby objects by changing the shape of the lens, and thus its focal length. If an object is $$20 \mbox{ cm}$$ from the eye, what must the altered focal length of the eye be in order for the image of this object to be in focus on the retina?
 * 8) Show that a concave parabolic mirror focuses incoming rays which are parallel to the optical axis of the mirror precisely at a focal point on the optical axis, as illustrated in figure 3.16. Hint: Since rays following different paths all move from the distant source to the focal point of the mirror, Fermat's principle implies that all of these rays take the same time to do so (why is this?), and therefore all traverse the same distance.
 * 9) Use Fermat's principle to explain qualitatively why a ray of light follows the solid rather than the dashed line through the wedge of glass shown in figure 3.17.
 * 10) Test your knowledge of Fermat's principle by finding the value of $$y$$ for which $$t$$ is a minimum in equation (3.14). Use this to derive Snell's law.