MHT-CET Physics · Optics (Ray)

The Prism: Deviation and Dispersion

A prism of angle A refracts a ray twice, with r₁ + r₂ = A and deviation δ = i + e − A; the deviation is least when the ray passes symmetrically, μ = sin((A + δₘ)/2)/sin(A/2), and a thin prism deviates every ray by (μ − 1)A, different for each colour.

All Optics (Ray) notes MHT-CET Physics strategy

Why this matters

17 PYQs, 8 of them HARD. Thirteen are deviation: minimum deviation, a thin prism in air and in water, a ray grazing the second face, a silvered face that sends the ray back, two rays through a double prism. Four are dispersion — dispersive powers of two prisms, an achromatic doublet, the rainbow. Two cards.

Concept 1 of 2: Deviation and Minimum Deviation

Inside a prism the two refraction angles add to the prism angle, r₁ + r₂ = A, and the total deviation is δ = i + e − A. At minimum deviation the path is symmetric: i = e, r = A/2, and the ray inside runs parallel to the base, which gives μ = sin((A + δₘ)/2)/sin(A/2); for an equilateral prism μ = 2 sin((60° + δₘ)/2). A thin prism deviates by (μ − 1)A at small angles; in water, use the relative index μ_g/μ_w, which cuts the deviation of a 3/2 prism to a quarter. If the emergent ray just grazes the second face, r₂ is the critical angle. If the second face is silvered and the ray returns along its path, it must strike that face normally, so r₁ = A and μ = sin i / sin A.

Definition

$r_1 + r_2 = A$ ; $\delta = i + e - A$ .
Minimum deviation: $i = e$ , $\mu = \dfrac{\sin\frac{A + \delta_m}{2}}{\sin\frac{A}{2}}$ (equilateral, i = 50° ⇒ δₘ = 40°).
Thin prism: $\delta = (\mu - 1)A$ ; in water $\mu' = \dfrac{3/2}{4/3} = \dfrac{9}{8}$ ⇒ δ/4.
Grazing emergence: $r_2 = C$ (μ = √2, A = 60° ⇒ $r_1 = 15^\circ$ , $i = \sin^{-1}(\sqrt{2}\sin 15^\circ)$ ).
Silvered second face, ray retraces: $r_1 = A$ ; at incidence 2A, $\mu = 2\cos A$ .

Prism

\mu = \frac{\sin\frac{A + \delta_m}{2}}{\sin\frac{A}{2}}, \qquad \delta_{\text{thin}} = (\mu - 1)A

Worked example

An equilateral prism gives a minimum deviation of 30°. Its refractive index?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 4th May Shift 1 · Q11Easy

Example 1 · Optics (Ray) · Prism — Deviation, Dispersion, and Refractive Index

In an equilateral prism the ray undergoes minimum deviation when it is incident at an angle of

50^\circ

. The angle of minimum deviation is

Using the glass's own index in water

In water the prism bends light by the RELATIVE index, μ_glass/μ_water = 9/8. The deviation drops from 0.5A to A/8.

Forgetting that a retracing ray meets the silvered face normally

To come straight back, the ray inside must hit the silvered face at 90°, so r₁ equals the prism angle A.

Drill 12 more on deviation and minimum deviation

Concept 2 of 2: Dispersion, Dispersive Power and Achromatism

Violet bends more than red because its refractive index is higher. The spread between them relative to the mean deviation is the dispersive power: ω = (δ_V − δ_R)/δ_Y, with δ_Y the average of the two. Two lenses in contact are achromatic — the same focus for all colours — when ω₁P₁ + ω₂P₂ = 0, so the dispersive powers are in the inverse ratio of the powers. A rainbow is refraction, dispersion and reflection inside drops; the angular deviation of the ray is part of it too.

Definition

$\omega = \dfrac{\delta_V - \delta_R}{\delta_Y}$ , $\delta_Y = \dfrac{\delta_V + \delta_R}{2}$ (9° and 11° against 11° and 13° ⇒ 5 : 6).
Achromatic pair in contact: $\omega_1P_1 + \omega_2P_2 = 0$ (+2 D from +5 D and −3 D ⇒ ω ratio 3 : 5).

Dispersive power

\omega = \frac{\delta_V - \delta_R}{\delta_Y}, \qquad \omega_1P_1 + \omega_2P_2 = 0

Worked example

A prism deviates red by 8° and violet by 12°. Its dispersive power?

Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 15th May Shift 2 · Q34Moderate

Example 2 · Optics (Ray) · Prism — Deviation, Dispersion, and Refractive Index

A glass prism 'A' deviates the red and blue rays through

10^\circ

and

12^\circ

respectively. A second prism 'B' deviates them through

8^\circ

and

10^\circ

respectively. The ratio of their dispersive powers is (A to B)

Dividing by the violet or red deviation

Dispersive power divides the spread by the MEAN (yellow) deviation. Using δ_V or δ_R changes the ratio of two prisms.

Drill 3 more on dispersion, dispersive power and achromatism

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Deviation and Minimum Deviation
Prism
$\mu = \frac{\sin\frac{A + \delta_m}{2}}{\sin\frac{A}{2}}, \qquad \delta_{\text{thin}} = (\mu - 1)A$
Dispersion, Dispersive Power and Achromatism
Dispersive power
$\omega = \frac{\delta_V - \delta_R}{\delta_Y}, \qquad \omega_1P_1 + \omega_2P_2 = 0$

Watch out for (3)

Using the glass's own index in water
→ Deviation and Minimum Deviation
Forgetting that a retracing ray meets the silvered face normally
→ Deviation and Minimum Deviation
Dividing by the violet or red deviation
→ Dispersion, Dispersive Power and Achromatism

Test yourself on Optics (Ray)

20 past MHT-CET questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Optics (Ray) chapter test Drill all 17 questions