Refraction and Dispersion by a Prism notes: ISC Class 12 Physics

1. Refraction through a prism: r₁ + r₂ = A and δ = i + e − A

A prism has two plane refracting faces meeting at the refracting edge; the angle between them is the refracting angle A; the face opposite the edge is the base. Rays are drawn in a principal section (a cut at right angles to the edge). A prism denser than its surroundings bends rays towards the base at both faces.
Snell’s law at the two faces: $\sin i = n\sin r_1$ and $n\sin r_2 = \sin e$ (angles from the normals).
Geometry: the normals at the entry point Q and exit point R meet at N. In the quadrilateral formed by the refracting edge, Q, N and R, the angles at Q and R are 90°, so ∠QNR = 180° − A; in triangle QNR, $r_1 + r_2 = A$ .
Deviation: the ray turns by $(i - r_1)$ at Q and $(e - r_2)$ at R; these are the inside angles of triangle QDR (D: where the incident ray produced meets the emergent ray produced backwards), so the exterior angle $\delta = i + e - A$ .
Special cases: normal incidence on the first face ( $i = 0$ , $r_1 = 0$ , $r_2 = A$ ); a ray that retraces its path from a silvered second face meets it normally ( $r_2 = 0$ , $r_1 = A$ ).

Formulas

Angles inside the prism
$r_1 + r_2 = A$
Angle of deviation
$\delta = i + e - A$

Key terms

Refracting angle: The angle A between the two refracting faces of a prism.
Angle of deviation: The angle between the incident ray produced forward and the emergent ray; δ = i + e − A.

2. The i–δ graph and the limits of emergence

As i increases, δ falls to a minimum δm and then rises: a U-shaped (not quite symmetric) curve.
Reversibility: a ray entering at $i$ and leaving at $e$ has the same δ as one entering at $e$ ; so each δ > δm occurs for two angles of incidence ( $i$ and $e$ ), which coincide only at the minimum.
Emergence: at the second face the light goes from glass to air, so it emerges only if $r_2 \le C$ ( $\sin C = 1/n$ ). The least angle of incidence for emergence (grazing emergence, $e = 90^\circ$ ) satisfies $\sin i = n\sin(A - C)$ ; for smaller i the ray is totally reflected at the second face.
Since $r_1 \le C$ always, $r_2 = A - r_1 \ge A - C$ : if $A > 2C$ , no ray emerges from the second face for any i.
Example: A = 60°, n = 1.52: C = 41.1°, least i = 29.4°, δm = 38.9° at i = 49.5°.

A = 60°, n = 1.52 (traced with Snell’s law)

Apart from the minimum, each deviation occurs for two angles of incidence, i and e.

Formulas

Least angle of incidence for emergence
$\sin i_{\min} = n\sin(A - C)$
Condition for any emergence
$A \le 2C$

Key terms

Grazing emergence: Emergence along the second face (e = 90°), when the ray meets that face at the critical angle.
Limiting angle of a prism: Twice the critical angle; a prism with a larger refracting angle lets no ray out of its second face.

3. Minimum deviation and the prism formula

At minimum deviation the path is symmetric: $i = e$ , $r_1 = r_2 = A/2$ ; in an isosceles prism the ray inside runs parallel to the base.
$\delta_m = 2i - A$ , so $i = (A + \delta_m)/2$ . Snell’s law $n = \sin i/\sin r$ gives $n = \dfrac{\sin\frac{A + \delta_m}{2}}{\sin\frac{A}{2}}$ .
For a given A, a larger n gives a larger δm; for the same glass, a larger A gives a larger δm. In a liquid, use the relative index $n_{glass}/n_{liquid}$ : δm falls.
Spectrometer: measure A; rotate the table until the image of the slit stops and turns back; that is δm. The reading is precise because δ changes very slowly near the minimum. A hollow prism with thin walls, filled with a liquid, measures the liquid’s index.
If $\delta_m = A$ , then $n = 2\cos(A/2)$ (e.g. A = 60°: n = √3).

Formulas

Prism formula
$n = \dfrac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
At minimum deviation
$r = \dfrac{A}{2},\; i = \dfrac{A + \delta_m}{2}$

Key terms

Angle of minimum deviation: The least deviation a prism produces, reached when the ray passes symmetrically (i = e).
Spectrometer: Instrument with a collimator, turntable and telescope used to measure A and δm accurately.

4. The thin prism

For a thin prism (small A) and small angles of incidence, $\sin\theta \approx \theta$ : $i = n r_1$ , $e = n r_2$ , so $\delta = n(r_1 + r_2) - A = (n - 1)A$ .
δ does not depend on i (for small i), and is proportional to A and to (n − 1).
In a liquid of index $n_l$ : $\delta = (n/n_l - 1)A$ ; zero when $n_l = n$ , negative (bent towards the edge) when $n_l > n$ .
Two thin prisms in contact add their deviations, with opposite signs if one is inverted: $\delta = (n_1 - 1)A_1 - (n_2 - 1)A_2$ .
Uses: weak prisms in spectacles to correct a squint; beam steering.

thin prism, A = 8°, n = 1.523 (exact)

For small angles of incidence δ ≈ (n − 1)A.

Formulas

Thin prism
$\delta = (n - 1)A$
Thin prism in a liquid
$\delta = \left(\dfrac{n}{n_l} - 1\right)A$

Key terms

Thin prism: A prism of small refracting angle (a few degrees), for which δ = (n − 1)A.

5. Dispersion, angular dispersion and dispersive power

Dispersion: splitting of white light into its colours, because the refractive index depends on wavelength. In glass $n_v > n_r$ (n falls as λ rises), so violet is deviated most and red least (VIBGYOR). In a vacuum there is no dispersion.
A parallel-sided slab gives no spectrum: every colour leaves parallel to the incident ray. An inverted second prism recombines the colours (Newton).
Thin prism: angular dispersion $\theta = \delta_v - \delta_r = (n_v - n_r)A$ ; mean deviation $\delta = (n - 1)A$ , n for yellow (often the mean of $n_v$ and $n_r$ ).
Dispersive power $\omega = \dfrac{\delta_v - \delta_r}{\delta} = \dfrac{n_v - n_r}{n - 1}$ : no unit, depends only on the material (crown about 0.034, flint about 0.040).
For a thick prism, trace each colour with Snell’s law; (nᵥ − nᵣ)A applies only to thin prisms.

a crown glass

Shorter wavelength, larger n, larger deviation.

Formulas

Angular dispersion
$\theta = (n_v - n_r)A$
Dispersive power
$\omega = \dfrac{n_v - n_r}{n - 1}$

Key terms

Angular dispersion: The angle between the violet and red rays emerging from a prism; (n_v − n_r)A for a thin prism.
Dispersive power: Ratio of angular dispersion to mean deviation, (n_v − n_r)/(n − 1); a property of the material.

6. The rainbow

A primary rainbow forms when sunlight entering raindrops is refracted (and dispersed), reflected once inside, and refracted out. Rays bunch at about 42° for red and 40° for violet from the antisolar direction (the direction pointing away from the sun), so light from drops at those angles is bright.
The sun must be behind the observer and the rain in front. Drops on a cone of half-angle 42° round the antisolar direction send red to the eye; those at 40° send violet: red outside, violet inside; the bow is an arc of a circle.
The top of the bow is (42° − sun’s elevation) above the horizon: no primary bow is seen when the sun is higher than about 42° (around noon in summer). Evening rainbows appear in the east, morning ones in the west.
A fainter secondary bow (two internal reflections) lies outside, at about 50°–54°, with the colours reversed.
Only the ray diagram and simple explanation are required; the derivation of the angles is not in the syllabus.

Key terms

Primary rainbow: Bow formed by one internal reflection in raindrops; red outside at about 42°, violet inside at about 40°.
Antisolar direction: The direction from the observer pointing directly away from the sun; the centre of the rainbow’s circle.

Exam tips

Derivations: r₁ + r₂ = A from the quadrilateral and triangle; δ = i + e − A from the exterior angle; at minimum deviation i = e and r = A/2.
Emergence needs r₂ ≤ C; no emergence for any i if A > 2C.
Thin prism: δ = (n − 1)A, independent of small i; in a liquid use n/nₗ.
Dispersive power (nᵥ − nᵣ)/(n − 1) has no unit and does not depend on A.
Rainbow: refraction, one reflection, refraction; sun behind; red outside at about 42°.

1. Refraction through a prism: r₁ + r₂ = A and δ = i + e − A

Formulas

Key terms

2. The i–δ graph and the limits of emergence

Formulas

Key terms

3. Minimum deviation and the prism formula

Formulas

Key terms

4. The thin prism

Formulas

Key terms

5. Dispersion, angular dispersion and dispersive power

Formulas

Key terms

6. The rainbow

Key terms

Exam tips

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