Reflection and Refraction — Question 9

Question 9

Weak dispersion changes the refracted angle with a calculable sensitivity

A beam enters a transparent material from air at fixed incidence angle i. The index changes from n to n plus a small delta n between two colors. The refracted angle is r, measured from the normal.

1. Differentiate Snells law to obtain the first-order change in r.

2. For n=1.50, r=30 degrees, and delta n=0.010, estimate the angular change in radians and degrees.

Question 9 - Solution

Weak dispersion changes the refracted angle with a calculable sensitivity

(a) Method and calculation

$$\sin i=n\sin r\Rightarrow0=\sin r\,dn+n\cos r\,dr.$$$$\boxed{dr=-\frac{\tan r}{n}\,dn}.$$

(b) Interpretation and further calculation

$$\Delta r\simeq-\frac{\tan30^\circ}{1.50}(0.010)=\boxed{-3.84900\times10^{-3}\,\mathrm{rad}}.$$$$\boxed{\Delta r\simeq-0.220532^\circ}.$$

Physical check

The higher-index color bends closer to the normal; this derivative holds at fixed incident angle.