Refraction through a Prism & Dispersion

PHYPrism, Dispersion & Optical Instruments

When light passes through a prism it bends towards the base, and the total turn from the original path is the angle of deviation $\delta$ . As the angle of incidence is varied, $\delta$ falls to a single minimum value $\delta_m$ , at which the ray travels symmetrically through the prism and $r_1 = r_2$ ; this special case gives the prism formula used to measure refractive index.

Because $\mu$ depends on wavelength (it is larger for violet than for red), white light splits into its colours — dispersion — and the spread is summarised by the dispersive power $\omega$ . ISC examines this through deviation numericals, minimum-deviation $\mu$ calculations, the thin-prism approximation $\delta = (\mu-1)A$ , and grazing-emergence (critical-angle) problems.

Prism relations (geometry of refraction)

A = r_1 + r_2 \qquad \delta = i + e - A

A

= refracting (apex) angle,

r_1, r_2

= angles of refraction at the two faces,

i

= angle of incidence,

e

= angle of emergence,

\delta

= angle of deviation. Valid for any single ray passing through the prism.

Refractive index at minimum deviation

\mu = \dfrac{\sin\!\left(\dfrac{A + \delta_m}{2}\right)}{\sin\!\left(\dfrac{A}{2}\right)}

\delta_m

= angle of minimum deviation; at minimum deviation the ray is symmetric, so

r_1 = r_2 = \dfrac{A}{2}

and

i = e

. Standard formula for measuring

\mu

of the prism material.

Thin-prism deviation

\delta = (\mu - 1)\,A

Holds only for a small refracting angle

A

(so all angles are small and

\sin\theta \approx \theta

);

\delta

is then independent of the angle of incidence.

A

and

\delta

are expressed in the same units.

Dispersive power

\omega = \dfrac{\mu_V - \mu_R}{\mu_Y - 1} = \dfrac{\delta_V - \delta_R}{\delta_Y}

\mu_V, \mu_R, \mu_Y

= refractive indices for violet, red and mean (yellow) light;

\delta_V, \delta_R, \delta_Y

= corresponding deviations. The second form holds for a thin prism.

\omega

is dimensionless and depends only on the material, not on

A

Angular dispersion (thin prism)

\Delta\delta = \delta_V - \delta_R = (\mu_V - \mu_R)\,A

\Delta\delta

= angular spread between the violet and red emergent rays for a thin prism;

A

= refracting angle. Note that

\Delta\delta = \omega\,\delta_Y

, where

\delta_Y = (\mu_Y - 1)A

is the mean deviation.

Critical angle (for grazing emergence / TIR check)

\sin C = \dfrac{1}{\mu}

C

= critical angle for the glass-air interface (

\mu

= refractive index of glass relative to air). When the emergent ray just grazes a face, the angle of incidence inside the glass at that face equals

C

; used in grazing-emergence and total-internal-reflection prism problems.

Two master equations carry most numericals: $A = r_1 + r_2$ (refraction geometry) and $\delta = i + e - A$ (deviation). For an equilateral prism $A = 60^{\circ}$ , so the deviation follows directly from $i$ and $e$ without finding $\mu$ .
At minimum deviation the path is symmetric: $i = e$ and $r_1 = r_2 = \dfrac{A}{2}$ . This is the ONLY condition under which $\mu = \dfrac{\sin\frac{A+\delta_m}{2}}{\sin\frac{A}{2}}$ applies — do not use it for a general (non-minimum) ray.
The $\delta$ versus $i$ curve is U-shaped: as $i$ increases, $\delta$ first decreases to a single minimum $\delta_m$ and then rises. A given deviation (other than $\delta_m$ ) is produced by two different angles of incidence, with the values of $i$ and $e$ interchanged.
The thin-prism rule $\delta = (\mu - 1)A$ is the small-angle limit of the minimum-deviation formula; here deviation does not depend on the angle of incidence. Apply it only for small $A$ (e.g. $A = 5^{\circ}$ ), never for a $60^{\circ}$ or $45^{\circ}$ prism.
At normal incidence $i = 0$ and $r_1 = 0$ , so $r_2 = A$ . If the ray then just grazes the second face, the internal angle of incidence there equals the critical angle: $r_2 = C = A$ , giving $\mu = \dfrac{1}{\sin A}$ .
Dispersive power $\omega = \dfrac{\mu_V - \mu_R}{\mu_Y - 1}$ is a pure material property: it is dimensionless, independent of the refracting angle $A$ , and uses the MEAN index $\mu_Y$ (yellow) minus 1 in the denominator — not $\mu_V$ or $\mu_R$ .
Refractive index rises from red to violet ( $\mu_V > \mu_Y > \mu_R$ ), so violet deviates most and red least; the violet ray emerges nearer the base. By Cauchy's relation, $\mu$ increases as the wavelength $\lambda$ decreases.
Distinguish the two outputs of dispersion: angular dispersion $\Delta\delta = (\mu_V - \mu_R)A$ has the dimensions of an angle and depends on $A$ , whereas dispersive power $\omega = \dfrac{\Delta\delta}{\delta_Y}$ is dimensionless and depends only on the glass.

Worked example · 3 marks

A ray of monochromatic light is incident on the first face of an equilateral glass prism (

\mu = 1.5

) and the emergent ray just grazes the second face. Calculate the angle of incidence.

Grazing emergence means the ray strikes the second face exactly at the critical angle, so $r_2 = C$ .
$\sin C = \dfrac{1}{\mu} = \dfrac{1}{1.5} = 0.667$ , giving $C = 41.8^{\circ}$ .
For a prism $r_1 + r_2 = A$ , and an equilateral prism has $A = 60^{\circ}$ , so $r_1 = 60^{\circ} - 41.8^{\circ} = 18.2^{\circ}$ .
Snell's law at the first face: $\sin i = \mu \sin r_1 = 1.5 \times \sin 18.2^{\circ} = 1.5 \times 0.3123 = 0.4685$ .

Answer

i = \sin^{-1}(0.4685) \approx 27.9^{\circ}

, about

28^{\circ}

	Primary rainbow	Secondary rainbow
Refractions inside the drop	Two	Two
Total internal reflections	One	Two
Colour order (from the top)	Red outside, violet inside	Reversed — violet outside, red inside
Brightness	Brighter	Fainter, because one more reflection loses light

The rainbow is the one place the board reliably tests refraction, dispersion and total internal reflection together.

Where the marks go

Using $\mu = \dfrac{\sin\frac{A+\delta}{2}}{\sin\frac{A}{2}}$ for a ray that is NOT at minimum deviation. That formula needs the symmetric case; for a general ray use $A = r_1 + r_2$ together with Snell's law $\mu = \dfrac{\sin i}{\sin r_1} = \dfrac{\sin e}{\sin r_2}$ at each face.
Putting the wrong term in the dispersive-power denominator — writing $\mu_V - 1$ or $\mu_R - 1$ instead of the mean $\mu_Y - 1$ , or forgetting the $-1$ entirely and dividing by $\mu_Y$ .
Applying the thin-prism formula $\delta = (\mu-1)A$ to a thick prism, or confusing $(\mu-1)$ with $\mu$ — the deviation is proportional to $(\mu-1)$ , so glass of $\mu = 1.5$ gives $\delta = 0.5A$ , not $1.5A$ .
In grazing-emergence problems, forgetting that 'just emerges along the face' fixes the angle of incidence inside the glass equal to the critical angle $C$ (with $\sin C = \dfrac{1}{\mu}$ ), and mishandling $\delta = i + e - A$ when $e = 90^{\circ}$ .

How the board asks it

Numerical $A = r_1 + r_2$ and $\delta = i + e - A$
A ray of light is incident at $40^{\circ}$ on one face of an equilateral glass prism and emerges from the opposite face at $52^{\circ}$ . Calculate the angle of deviation produced by the prism.
Numericalrefractive index at minimum deviation3 mkAsked 2023 · 2025
An equilateral prism of refracting angle $A = 60^{\circ}$ produces a minimum deviation of $40^{\circ}$ . Calculate the refractive index of the material of the prism.
Derive / proveprism formula at minimum deviation3 mkAsked 2023 · 2025
Derive the prism formula $\mu = \dfrac{\sin\frac{A + \delta_m}{2}}{\sin\frac{A}{2}}$ for a ray passing symmetrically through a prism of refracting angle $A$ at minimum deviation $\delta_m$ .
Diagram / graphthe $\delta$ versus $i$ curve
Draw a graph showing the variation of the angle of deviation $\delta$ with the angle of incidence $i$ for a glass prism, and use it to explain why a given deviation (other than $\delta_m$ ) can be obtained for two different angles of incidence.
Numericalgrazing emergence at the critical angle3 mkAsked 2024
A ray falls normally on the first face of a prism of refracting angle $A$ and just grazes the second face on emergence. If the refractive index of the glass is $1.5$ , calculate the refracting angle $A$ of the prism.
Give reasonsdispersion and dispersive power1 mkAsked 2024
White light is passed through a prism. Give reasons why violet light is deviated more than red light, and state, with justification, whether the dispersive power $\omega$ depends on the refracting angle $A$ of the prism.
Distinguishthe rainbow — refraction, dispersion and TIR in one phenomenon2 mkAsked 2023 · 2025
Name the phenomena involved in the formation of a rainbow, and state one difference between a primary and a secondary rainbow.

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.