Prisms, Dispersion and the Rainbow: Quick Revision

A brief note on Prisms, Dispersion and the Rainbow. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• A prism bends a ray toward its base at both faces. $r_1 + r_2 = A$ and $\delta = i + e - A$.
• The deviation is least when the ray passes symmetrically: $r_1 = r_2 = \dfrac{A}{2}$ and $i = e$.
• Dispersion: glass has a larger refractive index for violet than for red, so white light fans out into a spectrum, red bent least, violet most.
• A second prism upside down joins the colours into white; a pure colour cannot be split further.
• A rainbow: refraction, one reflection, refraction in each drop. Red at about $42^\circ$ from the point opposite the Sun, violet at $40^\circ$; the Sun must be behind you.

The facts at a glance

A worked example

A ray passes symmetrically through a $60^\circ$ prism of glass with $n = 1.58$. Find the angle of incidence and the deviation.

1. Inside. $r_1 = r_2 = 30^\circ$.
2. Incidence. $\sin i = 1.58 \times \sin 30^\circ = 0.79$, so $i = 52.2^\circ$.
3. Deviation. $\delta = 2i - A = 104.4^\circ - 60^\circ = 44.4^\circ$, the least this prism can give.

Practice with new numbers

A program to try

The program follows a ray through a prism for several angles of incidence and finds the least deviation.

Mistakes to avoid

• 'The prism makes the colours.' It only separates the colours already in white light.
• Forgetting to take away $A$ in $\delta = i + e - A$.
• Putting violet on the outside of the main bow. Red is outside.
• Looking for a rainbow toward the Sun. The Sun must be behind you.

Check yourself

1. In a $58^\circ$ prism, $r_1 = 26^\circ$. Find $r_2$.

2. A ray enters a $60^\circ$ prism at $51^\circ$ and leaves at $47^\circ$. Find the deviation.

3. Why does a prism split white light, but a single colour passes through as one beam?

4. The Sun is $27^\circ$ above the horizon. How high is the top of the red bow?