Rainbows from square drops. Suppose that, on some surreal world, raindrops had a square cross section and always fell with one face horizontal. Figure shows such a falling drop with a white beam of sunlight incident at $\text{θ}=65.0°$ at point $P.$ The part of the light that enters the drop then travels to point $A,$ where some of it refracts out into the air and the rest reflects. That reflected light then travels to point $B,$ where again some of the light refracts out into the air and the rest reflects. What is the difference in the angle of the red light$(n=1.331)$ and the blue light $(n=1.343)$ that emerge at (a) point $A$ and (b) point  $B?$ (This angular difference in the light emerging at point $A$ would be the rainbow's angular width.)

In the figure, a $2.50 \mathrm{m}$ long vertical pole extends from the bottom of a swimming pool to a point $50.0 \mathrm{cm}$ above the water. Sunlight is incident at angle $\text{θ}=55.0°$. What is the length of the shadow of the pole on the level bottom of the pool?

In the figure (a), a beam of light in material $1$ is incident on a boundary at an angle ${\text{θ}}_1=40°.$ Some of the light travels through material $2$ and then some of it emerges into material $3$. The two boundaries between the three materials are parallel. The final direction of the beam depends, in part, on the index of refraction $n_3$ of the third material. Figure (b) gives the angle of refraction ${\text{θ}}_3$ in that material versus $n_3$ for a range of possible $n_3$ values. The vertical axis scale is set by ${\text{θ}}_{3\mathrm{a}}=30.0°$ and ${\text{θ}}_{3\mathrm{b}}=50.0°.$ (a) What is the index of refraction of a material $1,$ or is the index impossible to calculate without more information? (b) What is the index of refraction of material  $2,$ or is the index impossible to calculate without more information? (c) If ${\text{θ}}_1$ is changed to $75°$ and the index of refraction of material $3$ is $2.4,$ what is ${\text{θ}}_3?$

(a)                                                                 (b)

In the figure, the light is incident at an angle ${\text{θ}}_1=40.1°$ on a boundary between two transparent materials. Some of the light travels down through the next three layers of transparent materials, while some of it reflects upward and then escapes into the air. If $n_1=1.30, n_2=1.40$, $n_3=1.32$ and $n_4=1.75,$ what is the value of (a) ${\text{θ}}_5$ in the air and (b) ${\text{θ}}_4$ in the bottom material?

An isotropic point source emits light at wavelength $500 \mathrm{nm}$ at the rate of $300 \mathrm{W}.$ A light detector is positioned $400 \mathrm{m}$ from the source. What is the maximum rate $\frac{\partial B}{\partial t}$ at which the magnetic component of the light changes with time at the detector's location?

Dispersion in a window pane. In the figure (a), a beam of white light is incident at angle $\mathrm{\theta }=40°$ on a common window pane (shown in cross section). For the pane's type of glass, the index of refraction for visible light ranges from $1.524$ at the blue end of the spectrum to $1.509$ at the red end. The two sides of the pane are parallel. What is the angular spread of the colors in the beam (a) when the light enters the pane and (b) when it emerges from the opposite side? (Hint: When you look at an object through a window pane are the colors in the light from the object dispersed as shown in figure (b, c)?)

(a)

(b) 

(c) 

In the figure, a ray of light is perpendicular to the face $ab$ $(n=1.81).$ Find the largest value for the angle $\mathrm{\varphi }$, so that the ray is totally reflected at face $ac$ if the prism is immersed (a) in air and (b) in water.