VISUALIZE Figure 34.19 showed the geometry. A ray of any wavelength is incident on the hypotenuse of the prism at ${\theta }_1=30\text{°}$.

SOLVE

1. If $n_1=1.54$ for deep red light, the refraction angle is

   $${\theta }_2={\text{sin}}^{-1}\left({\displaystyle \frac{n_1\text{sin}{\theta }_1}{n_2}}\right)={\text{sin}}^{-1}\left({\displaystyle \frac{1.54\text{sin}30\text{°}}{1.00}}\right)=50.4\text{°}$$

   Example 34.4 showed that the deflection angle is $\varphi ={\theta }_2-{\theta }_1$, so deep red light is deflected by ${\varphi }_{\text{red}}=20.4\text{°}$. This angle is slightly smaller than the previously observed ${\varphi }_{\text{violet}}=22.6\text{°}$.

2. The entire spectrum is spread between ${\varphi }_{\text{red}}=20.4\text{°}$ and ${\varphi }_{\text{violet}}=22.6\text{°}$. The angular spread is

   $$\delta ={\varphi }_{\text{violet}}-{\varphi }_{\text{red}}=2.2\text{°}=0.038\text{rad}$$

   At distance $r$, the spectrum spans an arc length

   $$s=r\delta =(2.0\text{m})\left(0.038\text{rad}\right)=0.076\text{m}=7.6\text{cm}$$

REVIEW The angle is so small that there’s no appreciable difference between arc length and a straight line. The spectrum will be 7.6 cm wide at a distance of 2.0 m.