PhysicsISC Class 12 · Ray Optics and Optical Instruments

Refraction Through a Prism: Minimum Deviation, Thin Prism and Dispersion

Updated 15 October 2026 · Fact-checked

For a prism of angle A, the refraction angles inside obey $r_1 + r_2 = A$ and the deviation is $\delta = i + e - A$. At minimum deviation the ray passes symmetrically ($i = e$, $r_1 = r_2 = \frac{A}{2}$), giving $n = \frac{\sin\frac{A + \delta_m}{2}}{\sin\frac{A}{2}}$. A thin prism deviates light by $\delta = (n - 1)A$.

Understand Refraction Through a Prism and Dispersion

Geometry of a prism. A ray strikes face AB at angle of incidence i, refracts at $r_{1}$ , meets face AC at $r_{2}$ and emerges at angle e. The two normals and the faces form a quadrilateral whose angles give
$r_{1} + r_{2} = A .$
The ray turns by $(i - r_{1})$ at the first face and $(e - r_{2})$ at the second, so the total deviation is
$δ = (i - r_{1}) + (e - r_{2}) = i + e - A .$

The i-δ graph. As i increases from a small value, δ first decreases, reaches a minimum $δ_{m}$ , then increases. The curve is not symmetric. For every δ other than $δ_{m}$ there are two values of i (the ray can be reversed: i and e swap). At the minimum the two coincide, so $i = e$ .

Minimum deviation and the prism formula (derivation, asked in ISC 2025 Q15). At $δ_{m}$ : $i = e$ , so by Snell's law $r_{1} = r_{2} = r$ . Then $2 r = A$ , so $r = \frac{A}{2}$ , and $δ_{m} = 2 i - A$ , so $i = \frac{A + δ _{m}}{2}$ . Snell's law at the first face, $n = \frac{s i n i}{s i n r}$ , gives
$n = \frac{s i n ( \frac{A + δ _{m}}{2} )}{s i n ( \frac{A}{2} )} .$
In an equilateral prism at minimum deviation the ray inside is parallel to the base.

Grazing emergence (asked in ISC 2024 Q14). If the emergent ray just grazes the second face, e = 90° and $r_{2}$ equals the critical angle c, where $sin c = \frac{1}{n}$ . Then $r_{1} = A - c$ and $sin i = n sin (A - c)$ . This i is the least angle of incidence for which light emerges; for smaller i the ray is totally internally reflected at the second face.

Thin prism. For small A (a few degrees) and near-normal incidence, $sin θ \approx θ$ , so $i = n r_{1}$ , $e = n r_{2}$ and
$δ = (n - 1) A,$
independent of the angle of incidence.

Dispersion. The refractive index of glass is larger for violet than for red, so violet is deviated most. White light spreads into a spectrum (VIBGYOR, violet nearest the base). For a thin prism:

angular dispersion $= δ_{v} - δ_{r} = (n_{v} - n_{r}) A$ ;
mean deviation $δ_{y} = (n_{y} - 1) A$ ;
dispersive power $ω = \frac{δ _{v} - δ _{r}}{δ _{y}} = \frac{n _{v} - n _{r}}{n _{y} - 1}$ , a property of the material only.

Rainbow (ray diagram, no derivation). Sunlight enters a raindrop, refracts, is internally reflected and refracts out, dispersing the colours.

Primary rainbow: two refractions and one internal reflection; red on the outside (about 42° from the anti-solar direction) and violet inside (about 40°); brighter.
Secondary rainbow: two refractions and two internal reflections; colours reversed (red inside); at about 51° to 54°; fainter.

A rainbow is seen with the Sun behind the observer.

Refraction Through a Prism and Dispersion: formulas and key facts

Prism angle: $r_{1} + r_{2} = A$; Geometry inside the prism.
Deviation: $δ = i + e - A$; Any incidence.
Prism formula: $n = \frac{s i n \frac{A + δ _{m}}{2}}{s i n \frac{A}{2}}$; At minimum deviation.
Grazing emergence: $sin i = n sin (A - c), sin c = \frac{1}{n}$; Least i for emergence.
Thin prism: $δ = (n - 1) A$; Small A only.
Angular dispersion: $δ_{v} - δ_{r} = (n_{v} - n_{r}) A$; Thin prism.
Dispersive power: $ω = \frac{n _{v} - n _{r}}{n _{y} - 1}$; No unit.

How to solve Refraction Through a Prism and Dispersion questions

Identify the special case first: minimum deviation, grazing emergence, thin prism or a general ray.

1Minimum deviation: $r = \frac{A}{2}$ , $i = \frac{A + δ _{m}}{2}$ ; use the prism formula either way round.
2Grazing emergence: find c from $sin c = \frac{1}{n}$ ; then $r_{1} = A - c$ and $sin i = n sin r_{1}$ . Calculate c; do not quote 42° without working.
3General ray: Snell at face 1 for $r_{1}$ ; $r_{2} = A - r_{1}$ ; if $r_{2} > c$ the ray is totally reflected; else Snell for e; $δ = i + e - A$ .
4Thin prism: $δ = (n - 1) A$ ; for dispersion use $n_{v}$ and $n_{r}$ separately.
5Prism in a liquid: replace n by the relative index $\frac{n _{g l a ss}}{n _{l i q u i d}}$ .
6Keep three or four significant figures in intermediate angles; round only at the end.

Refraction Through a Prism and Dispersion in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Where to look for a rainbow

Suppose it has just rained in the late afternoon and the Sun is low in the west behind you.

Where in the sky should you look, and which colour will be on the outside of the bright bow?

Try it first, then show the solutionSolution

A rainbow forms opposite the Sun, round the anti-solar point (the shadow of your head).
The primary bow lies at about 40° to 42° from that point, with red on the outside.
A fainter secondary bow may appear higher, at about 51° to 54°, with red on the inside.

Answer: Look east, about 42° from the shadow of your head; red outside in the primary bow.

Memory hook: Sun at your back, rainbow in front; primary: red on top.

Prism spectacles

Imagine an optometrist who prescribes a thin glass prism of angle 3° (n = 1.5) in a spectacle lens to help a patient whose eyes do not point quite together.

By how much does the prism shift the direction of light?

Try it first, then show the solutionSolution

Thin prism: $δ = (n - 1) A = 0.5 \times 3° = 1.5°$ .
The image appears shifted towards the apex by this angle, whatever the angle of incidence (for small angles).

Answer: 1.5°.

Memory hook: Thin prism: deviation is half the angle for ordinary glass.

Tricks to remember Refraction Through a Prism and Dispersion

Short memory aids. Say each one aloud once, then use it on a question the same day.

Add the r's, add i and e
$r_{1} + r_{2} = A$ ; $i + e = A + δ$ .
Use it when: Every prism problem.
Minimum means mirror image
At $δ_{m}$ the path is symmetric: i = e, $r_{1} = r_{2}$ .
Use it when: Deriving the prism formula.
Grazing means critical
e = 90° ⇔ $r_{2} = c$ .
Use it when: 2024-type numericals.
Violet bends most
Higher n, more deviation; violet ends nearest the base.
Use it when: Dispersion order.

Refraction Through a Prism and Dispersion: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Condition at minimum deviation · 1 mark · MCQBoard level

At minimum deviation, the angles of refraction inside a prism of angle A are:
(a) $r_{1} = A$ , $r_{2} = 0$
(b) $r_{1} = r_{2} = \frac{A}{2}$
(c) $r_{1} = r_{2} = A$
(d) $r_{1} = 2 r_{2}$

Try it first, then show the solutionSolution

The path is symmetric, so $r_{1} = r_{2}$ , and $r_{1} + r_{2} = A$ .

Answer: (b).

Example 2: Minimum deviation from n · 2 marks · Short answerBoard level

A prism of angle 60° is made of glass of refractive index 1.6. Find the angle of minimum deviation.

Try it first, then show the solutionSolution

$sin \frac{A + δ _{m}}{2} = n sin \frac{A}{2} = 1.6 \times 0.5 = 0.8$ , so $\frac{60° + δ _{m}}{2} \approx 53.13°$ . (1)
$δ_{m} \approx 106.26° - 60° = 46.26°$ . (1)

Answer: ≈ 46.3°.

Example 3: Derivation of the prism formula · 3 marks · Long answerHarder board level

With a labelled ray diagram, show that $n = \frac{s i n \frac{A + δ _{m}}{2}}{s i n \frac{A}{2}}$ for a prism.

Try it first, then show the solutionSolution

Diagram: incident ray, refracted ray inside, emergent ray with arrows; mark i, $r_{1}$ , $r_{2}$ , e, A and δ. From the geometry, $r_{1} + r_{2} = A$ and $δ = i + e - A$ . (1)
At minimum deviation i = e, so $r_{1} = r_{2} = r$ : $r = \frac{A}{2}$ and $i = \frac{A + δ _{m}}{2}$ . (1)
Snell's law: $n = \frac{s i n i}{s i n r} = \frac{s i n \frac{A + δ _{m}}{2}}{s i n \frac{A}{2}}$ . (1)

Answer: As shown.

Example 4: Least angle of incidence for emergence · 3 marks · Short answerHarder board level

A prism of angle 50° has refractive index 1.6. Find the smallest angle of incidence for which a ray can emerge from the second face.

Try it first, then show the solutionSolution

At the limit the ray grazes the second face: $r_{2} = c$ , $sin c = \frac{1}{1.6} = 0.625$ , so c ≈ 38.68°. (1)
$r_{1} = A - c \approx 50° - 38.68° = 11.32°$ . (1)
$sin i = 1.6 sin 11.32° \approx 0.314$ , so i ≈ 18.3°. (1)

Answer: ≈ 18.3°.

Example 5: Dispersion in a thin prism · 3 marks · Short answerHOTS / competency

A thin prism of angle 4° is made of flint glass with $n_{v}$ = 1.66, $n_{r}$ = 1.64 and $n_{y}$ = 1.65. Find the deviation of violet, red and yellow light, the angular dispersion and the dispersive power.

Try it first, then show the solutionSolution

$δ_{v} = 0.66 \times 4° = 2.64°$ ; $δ_{r} = 0.64 \times 4° = 2.56°$ ; $δ_{y} = 0.65 \times 4° = 2.60°$ . (1)
Angular dispersion $= 2.64° - 2.56° = 0.08°$ . (1)
$ω = \frac{0.08}{2.60} = \frac{0.02}{0.65} \approx 0.031$ . (1)

Answer: 2.64°, 2.56°, 2.60°; 0.08°; ≈ 0.031.

Example 6: The same prism in water · 4 marks · Case-basedHOTS / competency

Suppose a student finds that a 60° glass prism gives a minimum deviation of 40° in air.
(i) Find the refractive index of the glass. (1 mark)
(ii) The prism is placed in water (n = 1.33). Find the refractive index of glass relative to water. (1 mark)
(iii) Find the new angle of minimum deviation. (1 mark)
(iv) Explain why the deviation is so much smaller. (1 mark)

Try it first, then show the solutionSolution

(i) $n = \frac{s i n 50°}{s i n 30°} \approx 1.532$ .
(ii) $n_{r e l} = \frac{1.532}{1.33} \approx 1.152$ .
(iii) $sin \frac{60° + δ _{m}}{2} = 1.152 \times 0.5 = 0.576$ , so $\frac{60° + δ _{m}}{2} \approx 35.2°$ and $δ_{m} \approx 10.3°$ .
(iv) Bending depends on the ratio of refractive indices; glass and water differ much less than glass and air.

Answer: (i) ≈ 1.532; (ii) ≈ 1.152; (iii) ≈ 10.3°; (iv) smaller relative index.

Where students lose marks in Refraction Through a Prism and Dispersion

The slips examiners see most often, and what to write instead.

Mistake: Quoting the critical angle without calculating it
The ISC 2024 examiners (Q14) saw "c = 42°" written with no working, or the prism formula used for a grazing ray.
Wrong working
Use $n = \frac{s i n \frac{A + δ _{m}}{2}}{s i n \frac{A}{2}}$ for grazing emergence.
Right working
For n = 1.6, A = 60°: $sin c = \frac{1}{1.6}$ , c ≈ 38.7°; $r_{1} = 60° - 38.7° = 21.3°$ ; $sin i = 1.6 sin 21.3°$ , i ≈ 35.6°.
Do this: Grazing emergence is not minimum deviation; show $sin c = \frac{1}{n}$ .
Usually costs: 1 to 2 marks.
Mistake: Rounding too early
Also 2024: rounding intermediate angles gave a deviated final answer.
Do this: Carry two decimal places in angles until the last step.
Usually costs: 1 mark.
Mistake: Incomplete derivation diagram
In 2025 Q15 diagrams lacked arrows, and i, $r_{1}$ , $r_{2}$ , e and $δ_{m}$ were not marked; the minimum-deviation condition was not stated.
Do this: Label every angle and write "at $δ_{m}$ , i = e and $r_{1} = r_{2}$ ".
Usually costs: 1 mark.
Mistake: Rainbow colour order reversed
The 2025 examiners (Q1(B)(iii)) noted wrong colour orders, and "diffraction" written for refraction.
Do this: Primary: red outside; secondary: red inside; refraction plus internal reflection.
Usually costs: 1 mark.

Exam tips

In derivations, write $r_{1} + r_{2} = A$ and $δ = i + e - A$ before imposing minimum deviation.
Draw the i-δ graph with a clear minimum and unequal arms if asked.
Give angles in degrees with one or two decimal places.
For dispersion, state that dispersive power depends only on the material.

Practice questions on Refraction Through a Prism and Dispersion

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The deviation produced by a thin prism of angle A and refractive index n is:
(a) nA
(b) (n − 1)A
(c) (n + 1)A
(d) $\frac{A}{n}$
Hint
Small-angle approximation.
Answer
(b).
Q2 · 1 mark · Short answerBoard level
Find the deviation produced by a thin glass prism of angle 8° and refractive index 1.5.
Hint
(n − 1)A.
Answer
4°.
Q3 · 2 marks · Short answerHarder board level
A 60° prism gives a minimum deviation of 30°. Find its refractive index.
Hint
$\frac{s i n 45°}{s i n 30°}$ .
Answer
$2 \approx 1.414$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): In a glass prism, violet light is deviated more than red light.
Reason (R): The refractive index of glass is greater for violet than for red.
(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.
Hint
$δ = (n - 1) A$ .
Answer
(a).
Q5 · 2 marks · Short answerHarder board level
For an equilateral prism of refractive index 1.5 set at minimum deviation, find the angle of incidence and $δ_{m}$ .
Hint
r = 30°.
Answer
$sin i = 0.75$ , i ≈ 48.6°; $δ_{m} = 2 i - A \approx 37.2°$ .
Q6 · 2 marks · Short answerHOTS / competency
A ray falls normally on one face of a 60° prism of refractive index 1.5. Show that it cannot emerge from the second face.
Hint
Compare $r_{2}$ with c.
Answer
$r_{1} = 0$ , so $r_{2} = 60°$ ; c ≈ 41.8°. Since $r_{2} > c$ , the ray is totally internally reflected at the second face.
Q7 · 2 marks · Short answerHOTS / competency
State two differences between a primary and a secondary rainbow.
Hint
Number of reflections; colour order.
Answer
Primary: one internal reflection, red outside, about 42°, brighter. Secondary: two internal reflections, colours reversed (red inside), about 51° to 54°, fainter.
Q8 · 4 marks · Case-basedHOTS / competency
Case study: Suppose a spectrometer is used with a 60° prism. Minimum deviation is 38.0° for red light and 40.2° for violet light.
(i) Find $n_{r}$ . (1 mark)
(ii) Find $n_{v}$ . (1 mark)
(iii) Find the angular dispersion a thin 5° prism of this glass would produce. (1 mark)
(iv) Find the dispersive power, taking the mean n as the average of $n_{r}$ and $n_{v}$ . (1 mark)
Hint
$n = \frac{s i n \frac{A + δ _{m}}{2}}{s i n 30°}$ .
Answer
(i) $2 sin 49° \approx 1.509$ . (ii) $2 sin 50.1° \approx 1.534$ . (iii) $(1.534 - 1.509) \times 5° \approx 0.125°$ . (iv) $\frac{0.0249}{0.522} \approx 0.048$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Relation between $r_{1}$ , $r_{2}$ and A?Show answer
$r_{1} + r_{2} = A$ .
2.Deviation of a thin prism?Show answer
$δ = (n - 1) A$ .
3.Ray inside an equilateral prism at $δ_{m}$ ?Show answer
Parallel to the base.
4.Dispersive power formula?Show answer
$ω = \frac{n _{v} - n _{r}}{n _{y} - 1}$ .
5.Internal reflections in a primary rainbow?Show answer
One.

Refraction Through a Prism and Dispersion: frequently asked questions

Why is there a minimum deviation?

As i increases, the deviation at the first face rises but that at the second falls; their sum is least when the ray passes symmetrically through the prism.

Why is the i-δ graph not symmetric?

Deviation changes at different rates on either side of the minimum because sine is not linear; only the point i = e is special.

Does a thin prism's deviation depend on the angle of incidence?

Not for small angles near normal incidence; then δ = (n − 1)A.

What does the ISC 2027 syllabus need here?

Refraction through a prism, minimum deviation and the derivation of the n-A-\delta_m relation, the i-δ graph, the thin prism, dispersion, angular dispersion, dispersive power and the rainbow ray diagram (no derivation).

How was the prism asked recently?

ISC 2024 Q14 (3 marks) on grazing emergence from an equilateral prism; ISC 2025 Q15 (3 marks) on deriving the prism formula; and 1-mark items on dispersion (2024) and rainbows (2025).