Refraction through a prism

How a prism deviates light

A prism is a transparent block with two plane refracting faces that meet at an angle A, the angle of the prism. A ray entering the first face bends towards the normal. At the second face it bends away from the normal. Both bends turn the ray towards the base. The angle between the incident ray and the emergent ray is the angle of deviation, δ. As the angle of incidence i increases from a small value, δ first decreases, reaches a minimum Dm, and then increases. At minimum deviation the ray passes symmetrically through the prism.

Derivation: the prism formula

A ray PQ falls on face AB at angle i. It refracts at angle r₁, meets face AC at angle r₂ from inside, and leaves along RS at angle e. The normals at Q and R meet at N.

1. 1
   In the quadrilateral AQNR, the angles at Q and R are 90°. So ∠A + ∠QNR = 180°.
2. 2
   In triangle QNR: r₁ + r₂ + ∠QNR = 180°.
3. 3
   Comparing the two: r₁ + r₂ = A.
4. 4
   The deviation at the first face is i − r₁. The deviation at the second face is e − r₂.
5. 5
   Total deviation δ = (i − r₁) + (e − r₂) = i + e − A. So A + δ = i + e.
6. 6
   At minimum deviation, δ = Dm, i = e and r₁ = r₂ = r. So 2r = A, giving r = A/2.
7. 7
   Also A + Dm = 2i, giving i = (A + Dm)/2.
8. 8★
   Snell's law at the first face: n₂₁ = sin i / sin r, so n₂₁ = sin[(A + Dm)/2] / sin(A/2).
9. 9
   For a thin prism, A and Dm are small, so sin θ ≈ θ: n₂₁ = (A + Dm)/A, which gives Dm = (n₂₁ − 1)A.

★ key step

How to draw the prism ray diagram and the i-δ graph

• Draw a triangle with the apex A at the top and the base at the bottom. Mark the angle A at the apex.
• Draw the incident ray PQ striking the left face at Q. Draw the normal at Q as a dotted line and mark i.
• Draw the refracted ray QR inside the prism, bent towards the normal and towards the base. Mark r₁.
• Draw the normal at R on the right face. Mark r₂ inside, then draw the emergent ray RS bent away from the normal. Mark e.
• Extend PQ forwards and SR backwards as dotted lines until they meet. Mark the angle between them as δ.
• Mark N where the two normals meet inside the prism.
• Graph: put i on the x-axis and δ on the y-axis. Draw a U-shaped curve. Mark its lowest point as Dm, with the matching angle of incidence below it.

Worked example: the prism formula

Activity: angle of minimum deviation from the i-δ graph

A practical experiment with a glass prism, pins, a protractor and a drawing board.

1. 1
   Fix white paper on a drawing board. Place the prism on it, trace its outline ABC, and remove the prism.
2. 2
   Draw a normal at a point Q on face AB. Draw an incident line at i = 30° to the normal, and fix two pins P₁ and P₂ on it.
3. 3
   Replace the prism. Look through face AC and fix pins P₃ and P₄ in line with the images of P₁ and P₂.
4. 4
   Remove the prism. Join P₃ and P₄ to get the emergent ray. Extend the incident and emergent rays until they meet, and measure δ with a protractor.
5. 5
   Repeat for i = 35°, 40°, 45°, 50°, 55° and 60°.
6. 6
   Plot δ against i. The curve is U-shaped, and its lowest point gives Dm.
7. 7
   Result: for a 60° glass prism with n ≈ 1.5, Dm is about 37°. Then n = sin[(A + Dm)/2] / sin(A/2).