Problems

1. Derive

$$\frac{\eta_2}{\eta_1}=\frac{v_1}{v_2}$$

where $\eta_i$ is the refractive index of material $i$ and $v_i$ is the speed of propagation in material $i$.

2. Suppose light of frequency $F$ in a vacuum enters a medium with refractive index $a$. What is the light’s frequency in this new medium?

3. Joe shined a laser through a medium with a refractive index of $1.82$. In the same conditions, Bob shined a different laser through the same medium, with the same angle of incidence. Yet, he observed the beam of the different laser refracted at a different angle. Explain why.

4. Suppose light of frequency $F$ enters a medium with refractive index $a$. What is the wavelength of the light in this new medium, in terms of $F$, $a$, and $c$ (speed of light in vacuum)?

5. Suppose a light beam hits a mirror. The incident beam forms an angle of $X$ degrees with the reflected beam. What is the angle between the incident beam and the normal of the mirror?

6. Light goes from air $(\eta_1=1.00)$ into glass $(\eta_2=1.50)$ at $30^\circ$ to the normal. Find the refracted angle $\theta_2$.

7. A beam in glass $(\eta_1=1.50)$ heads toward air $(\eta_2=1.00)$. Find the critical angle for total internal reflection.

8. A coin is $12\text{ cm}$ below the surface of still water $(\eta\approx 1.33)$. Viewed from above at near-normal incidence, what apparent depth does it have?

9. Green light with frequency $f=5.5\times10^{14}\,\text{Hz}$ enters water $(\eta=1.33)$. Find its speed $v$ and wavelength $\lambda$ in the water.

Solutions

1) Use the definition $\eta_i=\dfrac{c}{v_i}$.

$$\frac{\eta_2}{\eta_1}=\frac{c/v_2}{c/v_1}=\frac{v_1}{v_2}.$$

2) $F$. Frequency does not change at a boundary.

3) Refractive index depends on wavelength (dispersion). Different laser wavelength $\Rightarrow$ different $\eta(\lambda)$ $\Rightarrow$ different refraction angle. Prism rainbow is the classic example.

4) $v_2=\dfrac{c}{a}$, so

$$\lambda_2=\frac{v_2}{F}=\frac{c}{aF}.$$

5) Law of reflection: angles (to the normal) match. The two-ray angle is $2\theta_i=X$, so $\theta_i=\dfrac{X}{2}$.

6) Snell: $\eta_1\sin\theta_1=\eta_2\sin\theta_2$.

$$\sin\theta_2=\frac{1.00}{1.50}\sin30^\circ=\frac{1}{3}\ \Rightarrow\ \theta_2\approx 19.47^\circ.$$

7) $\sin\theta_c=\dfrac{\eta_2}{\eta_1}=\dfrac{1.00}{1.50}=\dfrac{2}{3}$, so $\theta_c\approx 41.8^\circ$.

8) Near-normal viewing: $d_{\text{app}}\approx \dfrac{d_{\text{real}}}{\eta}=\dfrac{12\,\text{cm}}{1.33}\approx 9.0\,\text{cm}.$

9) $v=\dfrac{c}{\eta}\approx \dfrac{3.00\times10^8}{1.33}\approx 2.26\times10^8\,\text{m/s}$.
$\lambda=\dfrac{v}{f}\approx \dfrac{2.26\times10^8}{5.5\times10^{14}}\approx 4.10\times10^{-7}\,\text{m}\approx 410\,\text{nm}.$