The Prism: Deviation, Dispersion & Dispersive Power

Isaac Newton let a sunbeam into a darkened room, put a glass prism in its path, and caught on the far wall not a brighter spot of white light but a band of pure colour — red, orange, yellow, green, blue, violet — spread out in order. That single experiment, three and a half centuries ago, is the ancestor of every prism problem you will ever solve, and of the rainbow, the spectroscope, and the colours that leak from the edge of a cheap lens. A prism does two things at once: it bends light toward its thick base, and because glass bends different colours by different amounts, it spreads white light into a spectrum. This article makes both quantitative. A ray crossing a prism is deviated by an angle of deviation $\delta$ that depends on how you tilt the prism; as you rotate it, $\delta$ passes through a minimum, and at that symmetric setting the prism gives up its refractive index through the clean relation $\mu = \sin\!\big(\tfrac{A + \delta_m}{2}\big)\big/\sin\!\big(\tfrac{A}{2}\big)$ , where $A$ is the refracting angle. For a thin prism the whole thing simplifies to $\delta = (\mu - 1)A$ , and the spreading of colours — the angular dispersion — follows immediately. The colour-spreading a glass produces per unit of deviation is its dispersive power $\omega = (\mu_v - \mu_r)/(\mu_y - 1)$ , a pure number that decides whether two prisms can be combined to bend light without colouring it, or to colour it without bending it. We build all of this from the geometry of a ray crossing two faces, using only algebra, trigonometry and Snell's law. By the end you will locate the minimum deviation, extract an index from it, compute how far a prism smears the spectrum, and design the crown–flint combinations — achromatic and direct-vision — that real instruments are built from. IGCSE candidates: your syllabus stops at the qualitative "white light splits into seven colours" picture; the minimum-deviation and dispersive-power machinery below is for CBSE, ICSE, JEE and NEET.

Key takeaways

A prism bends a ray toward its base by the angle of deviation $\delta$ ; as the prism is rotated, $\delta$ passes through a minimum at the symmetric setting where the ray crosses parallel to the base.
At minimum deviation the refractive index is $\mu = \sin\!\big(\tfrac{A + \delta_m}{2}\big)\big/\sin\!\big(\tfrac{A}{2}\big)$ , with $A$ the refracting angle; the angles are then $i = e = (A + \delta_m)/2$ and $r_1 = r_2 = A/2$ .
Crown-glass anchor ( $A = 60°$ , $\mu = 1.5$ ): $\delta_m = 37.2°$ , with $i = 48.59°$ , $r = 30°$ . And $A = 60°$ , $\delta_m = 40°$ gives $\mu = 1.532$ .
For a thin prism $\delta = (\mu - 1)A$ : a $5°$ crown prism deviates by $2.5°$ ; the angular dispersion is $(\mu_v - \mu_r)A$ , e.g. $0.054°$ for a $6°$ prism.
Dispersive power $\omega = (\mu_v - \mu_r)/(\mu_y - 1)$ is a property of the glass alone: crown ( $\mu_r = 1.514$ , $\mu_y = 1.517$ , $\mu_v = 1.523$ ) gives $\omega = 0.0174$ .
Two prisms combine for deviation without dispersion (achromatic) or dispersion without deviation (direct-vision): a $6°$ crown with flint $\mu_y' = 1.65$ needs $A' = 4.77°$ for zero net deviation.
Key idea: a prism bends light by $\delta$ and spreads colour by dispersion; at minimum deviation $\mu = \sin\!\big(\tfrac{A+\delta_m}{2}\big)/\sin\!\big(\tfrac{A}{2}\big)$ , thin prisms give $\delta = (\mu-1)A$ , and the glass's dispersive power $\omega$ decides how the two combine.

The core insight

A prism deviates a ray toward its base by an angle $\delta$ that is least at the symmetric passage, where $\mu = \sin\!\big(\tfrac{A+\delta_m}{2}\big)/\sin\!\big(\tfrac{A}{2}\big)$ ; because $\mu$ depends on colour, white light spreads into a spectrum whose angular width is $(\mu_v - \mu_r)A$ for a thin prism, and the glass's dispersive power $\omega = (\mu_v - \mu_r)/(\mu_y - 1)$ sets how prisms combine to bend without colouring or colour without bending.

Where this lives today

The prism's two jobs — bending and spreading — underpin a surprising range of instruments. A spectrometer or spectroscope uses a prism (or its cousin the diffraction grating) to fan light into a spectrum and read the chemical fingerprint of a star, a flame or a pollutant; Fraunhofer's lines, industrialised, are how we know what the Sun is made of. The direct-vision prism of Example 2 lives in pocket spectroscopes and hand-held mineral testers, splitting light without deflecting the view. The achromatic doublet — two glasses chosen to cancel dispersion by equation (6) — sits in every quality camera lens, telescope and pair of binoculars, taming the colour fringing that a single lens would show. Prism binoculars and periscopes fold light with prisms; a rainbow is dispersion by raindrops rather than glass; and the coloured edges you sometimes see through cheap optics are uncorrected dispersion at work. Even the split of a laser line in a pulse compressor or a spectrograph on a space telescope is equation (2) and its dispersion cousins, scaled to the frontier of measurement.

The physical picture

A prism is a wedge of glass with two flat refracting faces meeting at the refracting angle $A$ , the sharp edge at the top and the thick base below. A ray of light entering one face is refracted toward the normal (air into glass), crosses the glass, and is refracted again away from the normal at the second face (glass into air). The net effect of the two refractions is to swing the ray around toward the base of the prism — it emerges travelling in a noticeably different direction from the one it came in on. The angle between the original and final directions is the angle of deviation $\delta$ . Two facts make the prism interesting. First, $\delta$ is not fixed: tilt the prism and the deviation changes, sweeping down to a lowest value and back up. Second — and this is Newton's discovery — the glass bends violet light more than red, because its refractive index is slightly larger for shorter wavelengths. So a single incident ray of white light leaves the prism as a little fan of colours, red closest to the incident direction and violet furthest from it. That fan is the spectrum, and its angular width is the angular dispersion. Everything in this article is the geometry of that bending and that spreading.

Figure

White light crossing a prism at minimum deviation: the ray passes symmetrically so that the angle of incidence equals the angle of emergence and the internal ray runs parallel to the base, the refracting angle A sits at the apex, the total bend is the minimum deviation delta m, and the emerging beam fans from red to violet

Deviation and its minimum

Trace a ray across the prism and label the four angles it makes: the angle of incidence $i$ at the first face, the two internal refraction angles $r_1$ and $r_2$ , and the angle of emergence $e$ at the second face. Two geometric facts always hold. The internal angles sum to the refracting angle, $r_1 + r_2 = A$ , and the total deviation is what each face contributes:

\delta = i + e - A \qquad (1)

where: $\delta$ is the angle of deviation, $i$ and $e$ the angles of incidence and emergence, and $A$ the refracting angle. As you rotate the prism, $i$ and $e$ trade off, and $\delta$ falls to a minimum and rises again. That minimum occurs at the symmetric passage, where the ray crosses the prism parallel to its base so that $i = e$ and $r_1 = r_2 = A/2$ . At that setting equation (1) becomes $\delta_m = 2i - A$ , so the angle of incidence there is $i = (A + \delta_m)/2$ . The minimum is broad and flat — a range of incidence angles near it all give nearly the same deviation — which is exactly why the minimum-deviation setting is the reliable one to measure.

The prism formula

At the symmetric setting, Snell's law at the first face — quoted from earlier in this chapter — turns the geometry into a formula for the refractive index. With $i = (A + \delta_m)/2$ and $r_1 = A/2$ , applying $\mu = \sin i / \sin r_1$ gives the single most examined prism relation:

\mu = \frac{\sin\!\big(\tfrac{A + \delta_m}{2}\big)}{\sin\!\big(\tfrac{A}{2}\big)} \qquad (2)

where: $\mu$ is the refractive index of the prism glass, $A$ the refracting angle and $\delta_m$ the minimum deviation. This is a two-way tool: measure $A$ and $\delta_m$ on a spectrometer and it hands you $\mu$ ; or, knowing $\mu$ and $A$ , solve it for $\delta_m$ . Note the argument carefully — it is $(A + \delta_m)/2$ over $(A/2)$ , not $\delta_m/2$ over anything; misplacing the halving is the classic error. For a crown-glass prism of $A = 60°$ and $\mu = 1.5$ , equation (2) gives $\delta_m = 37.2°$ , worked in full below.

Thin prisms and the spreading of colour

When the refracting angle is small — a few degrees — the sines in equation (2) can be replaced by their angles, and the whole relation collapses to a strikingly simple one:

\delta = (\mu - 1)A \qquad (3)

where: $\delta$ is the deviation of a thin prism of refracting angle $A$ and index $\mu$ . A $5°$ crown prism ( $\mu = 1.5$ ) deviates light by $\delta = (0.5)(5°) = 2.5°$ , whatever the (small) incidence. Because $\mu$ depends on colour, each colour gets its own deviation, and the spread between the extremes — violet minus red — is the angular dispersion:

\theta = \delta_v - \delta_r = (\mu_v - \mu_r)A \qquad (4)

where: $\mu_v$ and $\mu_r$ are the indices for violet and red, and $A$ the refracting angle. For a $6°$ prism of crown glass ( $\mu_v - \mu_r = 0.009$ ), the spectrum is only $\theta = (0.009)(6°) = 0.054°$ wide — a fine fan, but enough to separate the colours. The mean deviation, taken at the yellow (middle) colour, is $\delta_y = (\mu_y - 1)A$ .

Figure

A graph of the angle of deviation against the angle of incidence for a prism: the curve falls from high values, flattens through a broad minimum where the angle of incidence equals the angle of emergence, and rises again, so that a single minimum deviation corresponds to the symmetric passage

Dispersive power: a property of the glass

The angular dispersion $(\mu_v - \mu_r)A$ depends on the prism's size $A$ ; the mean deviation $(\mu_y - 1)A$ does too. But their ratio cancels $A$ entirely, leaving a pure number that belongs to the glass alone — the dispersive power:

\omega = \frac{\mu_v - \mu_r}{\mu_y - 1} \qquad (5)

where: $\mu_v$ , $\mu_r$ and $\mu_y$ are the indices for violet, red and yellow (the mean colour). It measures how much colour-spreading a glass gives per unit of bending, and because it is independent of $A$ , a crown glass has one dispersive power whatever prism you cut from it. Crown glass ( $\mu_r = 1.514$ , $\mu_y = 1.517$ , $\mu_v = 1.523$ ) has $\omega = 0.009/0.517 = 0.0174$ ; a flint glass, richer in lead, disperses more strongly, with $\omega$ two or three times larger. That difference is exactly what lets two glasses be played against each other.

Combining prisms: bend without colour, or colour without bend

Because deviation and dispersion scale differently with the two glasses, a crown prism and a flint prism can be paired to cancel one while keeping the other. Set the flint prism base-to-apex against the crown, and choose the two refracting angles to balance:

\text{achromatic (no net dispersion):}\quad (\mu_v - \mu_r)A = (\mu_v' - \mu_r')A' \qquad (6)

\text{direct-vision (no net deviation):}\quad (\mu_y - 1)A = (\mu_y' - 1)A' \qquad (7)

where: unprimed quantities are the crown prism and primed the flint. Equation (6) cancels the colour-spreading while leaving a net bend — the principle of the achromatic prism and, in lens form, the achromatic doublet that beats chromatic aberration. Equation (7) cancels the bending while leaving the colour — the direct-vision prism in a pocket spectroscope, which spreads sunlight into its spectrum without deflecting the line of sight. For a $6°$ crown prism ( $\mu_y = 1.517$ ) paired with flint of $\mu_y' = 1.65$ , zero net deviation needs $A' = (0.517 \times 6°)/0.65 = 4.77°$ .

Derivation AThe prism formula at minimum deviation

The goal is to derive $\mu = \sin\!\big(\tfrac{A + \delta_m}{2}\big)/\sin\!\big(\tfrac{A}{2}\big)$ from the geometry of a ray crossing the two faces, and to use it to land the crown-glass anchor $\delta_m = 37.2°$ for $A = 60°$ , $\mu = 1.5$ . The only physics quoted is Snell's law; everything else is the geometry of the symmetric passage, using algebra and trigonometry alone.

Step 1. Write the two geometric facts for any ray crossing the prism: the internal angles sum to the apex, $r_1 + r_2 = A$ , and the deviation is $\delta = (i - r_1) + (e - r_2) = i + e - A$ , since each face turns the ray by the difference between its outside and inside angles.

Step 2. Impose the minimum. As the prism rotates, $\delta$ is least when the path is symmetric — $i = e$ and $r_1 = r_2$ . (One can show the turning point of $\delta$ is exactly this symmetric case.) Then $r_1 = r_2 = A/2$ from $r_1 + r_2 = A$ , and $\delta_m = 2i - A$ from $\delta = i + e - A$ , so the incidence angle is $i = (A + \delta_m)/2$ .

Step 3. Apply Snell's law at the first face, air to glass: $\mu = \dfrac{\sin i}{\sin r_1}$ . Substitute the symmetric values $i = (A + \delta_m)/2$ and $r_1 = A/2$ : $\mu = \dfrac{\sin\!\big(\tfrac{A + \delta_m}{2}\big)}{\sin\!\big(\tfrac{A}{2}\big)}$ . That is equation (2).

Step 4. Solve for the anchor. With $A = 60°$ and $\mu = 1.5$ : $\sin\!\big(\tfrac{60 + \delta_m}{2}\big) = 1.5\,\sin 30° = 1.5 \times 0.5 = 0.75$ , so $\tfrac{60 + \delta_m}{2} = \arcsin(0.75) = 48.59°$ , giving $60 + \delta_m = 97.18°$ and $\delta_m = 37.2°$ . At that setting the incidence is $i = 48.59°$ and the internal angle $r = A/2 = 30°$ . As a second use of equation (2), a measured $\delta_m = 40°$ for $A = 60°$ gives $\mu = \sin 50°/\sin 30° = 0.766/0.5 = 1.532$ . Verified: $\delta_m = 37.2°$ (with $i = 48.59°$ , $r = 30°$ ) for $A = 60°$ , $\mu = 1.5$ , and $\mu = 1.532$ for $\delta_m = 40°$ , are recomputed in verify_hs2-prism-dispersion.py.

Go deeper · Derivation B · 1 minDispersive power as dispersion over mean deviation

Derivation BDispersive power as dispersion over mean deviation

The goal is to derive the dispersive power $\omega = (\mu_v - \mu_r)/(\mu_y - 1)$ as the ratio of angular dispersion to mean deviation for a thin prism, and to land the crown-glass value $\omega = 0.0174$ . The point is that the prism angle $A$ cancels, so $\omega$ is a property of the glass, not the prism — pure algebra on the thin-prism relations.

Step 1. Write the thin-prism deviation for each colour with equation (3): red $\delta_r = (\mu_r - 1)A$ , violet $\delta_v = (\mu_v - 1)A$ , and the mean (yellow) $\delta_y = (\mu_y - 1)A$ .

Step 2. Form the angular dispersion — the spread between the extreme colours: $\theta = \delta_v - \delta_r = (\mu_v - 1)A - (\mu_r - 1)A = (\mu_v - \mu_r)A$ . The constant $-1$ and the $A$ are common to both, so only the index difference survives times $A$ .

Step 3. Divide the angular dispersion by the mean deviation and watch $A$ cancel: $\omega \equiv \dfrac{\theta}{\delta_y} = \dfrac{(\mu_v - \mu_r)A}{(\mu_y - 1)A} = \dfrac{\mu_v - \mu_r}{\mu_y - 1}$ . The refracting angle has vanished, so $\omega$ depends only on the three indices — it is the glass's own dispersive power.

Step 4. Put in the crown-glass numbers $\mu_r = 1.514$ , $\mu_y = 1.517$ , $\mu_v = 1.523$ : $\omega = \dfrac{1.523 - 1.514}{1.517 - 1} = \dfrac{0.009}{0.517} = 0.0174$ . This small number says a crown prism spreads colour only about $1.7\%$ as much as it bends — which is why a single lens shows only a faint colour fringe, and why cancelling even that faint dispersion (an achromat) is worth the trouble. Verified: $\theta = 0.054°$ for $A = 6°$ , $\omega = 0.0174$ for crown glass, and the cancellation of $A$ , are recomputed in verify_hs2-prism-dispersion.py.

Limiting cases

Case	Result	Meaning
symmetric passage $i = e$	$\delta = \delta_m$	the minimum-deviation setting used to measure $\mu$
thin prism, small $A$	$\delta = (\mu - 1)A$	deviation independent of incidence
$\mu \to 1$ (glass matches air)	$\delta \to 0$	no bending; the prism disappears optically
red vs violet	$\delta_v > \delta_r$	violet always deviates more (higher $\mu$ )
achromatic pair, eq (6)	net colour zero	deviation survives — the achromat
direct-vision pair, eq (7)	net deviation zero	colour survives — the pocket spectroscope

Common trapThe prism formula uses (A + δm)/2 over A/2 — and dispersive power belongs to the glass, not the prism

Two errors dominate this topic. The first is misreading the prism formula as $\mu = \sin(\delta_m/2)/\sin(A/2)$ or some other misplacement of the halving. It is $\mu = \sin\!\big(\tfrac{A + \delta_m}{2}\big)\big/\sin\!\big(\tfrac{A}{2}\big)$ — the whole $(A + \delta_m)$ is halved in the numerator, and $A$ alone is halved in the denominator. Write the two halvings out before substituting and the slip disappears. The second is treating $\delta_m$ as "the deviation of the prism", as if a prism had one deviation. It does not: $\delta$ depends on how the prism is tilted, and $\delta_m$ is specifically its minimum, reached only at the symmetric passage. A third trap concerns dispersive power: $\omega = (\mu_v - \mu_r)/(\mu_y - 1)$ is a property of the glass alone and is completely independent of the refracting angle $A$ — a big prism and a small prism of the same glass have the same $\omega$ , even though their dispersions differ. And remember violet bends more than red because its index is larger; students who reverse this get every spectrum upside down. Finally, the thin-prism formula $\delta = (\mu - 1)A$ holds only for small $A$ — never apply it to a $60°$ prism.

Worked examples

Board level — minimum deviation of a crown prism (CBSE and NEET pattern)

A prism of refracting angle $60°$ is made of crown glass of refractive index $1.5$ . Find its minimum deviation and the angles of incidence and internal refraction at that setting. Then find the deviation of a thin $5°$ prism of the same glass.

Step 1. Use the prism formula at minimum deviation, $\mu = \sin\!\big(\tfrac{A + \delta_m}{2}\big)/\sin\!\big(\tfrac{A}{2}\big)$ , with $A = 60°$ and $\mu = 1.5$ . Rearrange: $\sin\!\big(\tfrac{60 + \delta_m}{2}\big) = \mu\sin\!\big(\tfrac{60}{2}\big) = 1.5\sin 30° = 1.5 \times 0.5 = 0.75$ .

Step 2. Invert the sine: $\tfrac{60 + \delta_m}{2} = \arcsin(0.75) = 48.59°$ , so $60 + \delta_m = 97.18°$ and $\delta_m = \boxed{37.2°}$ .

Step 3. At that symmetric setting, the angle of incidence is $i = \tfrac{A + \delta_m}{2} = 48.59° = \boxed{48.59°}$ , and the internal refraction angle is $r = A/2 = \boxed{30°}$ at each face.

Step 4. For the thin $5°$ prism, use $\delta = (\mu - 1)A = (0.5)(5°) = \boxed{2.5°}$ — a small, incidence-independent bend, and the reason a thin prism is used in eyepieces and range-finders where a fixed, small deviation is wanted.

Competitive level — dispersive power and a direct-vision prism (JEE and NEET pattern)

For crown glass, $\mu_r = 1.514$ , $\mu_y = 1.517$ , $\mu_v = 1.523$ . Find the dispersive power. For a $6°$ crown prism, find the angular dispersion. Then find the flint refracting angle needed to make a direct-vision (zero net deviation) combination with a flint of $\mu_y' = 1.65$ .

Step 1. Dispersive power: $\omega = \dfrac{\mu_v - \mu_r}{\mu_y - 1} = \dfrac{1.523 - 1.514}{1.517 - 1} = \dfrac{0.009}{0.517} = \boxed{0.0174}$ .

Step 2. Angular dispersion of the $6°$ crown prism: $\theta = (\mu_v - \mu_r)A = (0.009)(6°) = \boxed{0.054°}$ — the full width of the spectrum it produces.

Step 3. Direct-vision condition, equation (7), demands equal mean deviations: $(\mu_y - 1)A = (\mu_y' - 1)A'$ , so $A' = \dfrac{(\mu_y - 1)A}{\mu_y' - 1} = \dfrac{(0.517)(6°)}{0.65} = \dfrac{3.102}{0.65} = \boxed{4.77°}$ .

Step 4. Interpret: a $6°$ crown prism and a $4.77°$ flint prism, set opposite each other, bend the yellow ray back to zero net deviation, so the line of sight is undeflected — yet because the two glasses have different dispersive powers, the colours do not fully cancel and a small spectrum survives. That is exactly the direct-vision prism inside a pocket spectroscope, which shows you the spectrum of a lamp while you look straight at it. Timing note: about two minutes for all three parts. The dispersive power is a plug-in; the direct-vision angle is one rearrangement of $(\mu_y - 1)A = (\mu_y' - 1)A'$ . The trap is confusing the two combination conditions — direct-vision balances the deviations $(\mu_y - 1)A$ , while an achromatic prism balances the dispersions $(\mu_v - \mu_r)A$ .

Exam traps

Misplacing the halving in the prism formula. It is $\mu = \sin\!\big(\tfrac{A + \delta_m}{2}\big)/\sin\!\big(\tfrac{A}{2}\big)$ — halve the whole $(A + \delta_m)$ on top and $A$ alone on the bottom. Writing $\sin(\delta_m/2)$ is the classic slip.

Treating $\delta_m$ as the only deviation. Deviation depends on the tilt; $\delta_m$ is its minimum, reached at the symmetric passage $i = e$ . A general incidence gives a larger $\delta$ .

Using $\delta = (\mu - 1)A$ for a large prism. The thin-prism formula holds only for small $A$ (a few degrees). Never apply it to a $60°$ prism — use the full formula (2) there.

Thinking dispersive power depends on the prism. $\omega = (\mu_v - \mu_r)/(\mu_y - 1)$ is a property of the glass alone; $A$ cancels. Two prisms of the same glass have the same $\omega$ .

Reversing red and violet. Violet has the larger index, so it deviates more than red. Getting this backward inverts every spectrum and every dispersion sign.

Confusing the two combinations. Achromatic (no colour) balances the dispersions $(\mu_v - \mu_r)A$ ; direct-vision (no bend) balances the deviations $(\mu_y - 1)A$ . Match the condition to what is being cancelled.

Board scope. The full quantitative treatment — minimum deviation, the prism formula, dispersive power and crown–flint combinations — is set by CBSE, ICSE, JEE-Main, JEE-Advanced and NEET. IGCSE 0625 covers only the qualitative half: white light dispersing through a prism into the seven-colour spectrum (topic 3.2.4), with no minimum-deviation formula and no dispersive power — so an IGCSE candidate should learn the spectrum picture here and treat the formulas as enrichment. AP Physics 2 mentions that index varies with wavelength but does not set the minimum-deviation or dispersive-power content, so this is not on the AP-2 exam; IB SL/HL and A-Level 9702 carry no prism optics of this kind at all.

Origins · 1 minJoseph von Fraunhofer and how this was discovered

Joseph von Fraunhofer

1787–1826 · Bavaria

Fraunhofer rose from an orphaned apprentice glazier to the finest optician of his age, and he made the prism a precision instrument. Examining sunlight spread by a prism through a small telescope, he mapped hundreds of fine dark lines crossing the solar spectrum — the Fraunhofer lines — and labelled the strongest with letters still used today: C in the red, D in the yellow, F in the blue. Those lines fixed standard wavelengths at which a glass's refractive index is quoted, which is precisely what the dispersive power $\omega = (\mu_v - \mu_r)/(\mu_y - 1)$ needs: definite red, yellow and blue references to measure $\mu_r$ , $\mu_y$ and $\mu_v$ against. Fraunhofer also built achromatic lenses of unmatched quality by playing crown against flint. Honest scope: the minimum-deviation formula is older, resting on Snell's and Newton's work; Fraunhofer's gift is the precise numbers — the standard wavelengths and indices — that make dispersion a measurable, designable quantity rather than a curiosity.

History

1621

— Willebrord Snellius states the law of refraction that underlies the prism formula.

1666

— Isaac Newton disperses white light with a prism, showing it is a mixture of colours.

1758

— John Dollond patents the achromatic doublet, combining crown and flint to cancel dispersion.

1814

— Joseph von Fraunhofer maps the dark solar lines, fixing the standard wavelengths for refractive index.

1859

— Gustav Kirchhoff and Robert Bunsen found spectroscopy, reading the elements from prism spectra.

Going further

Texts. NCERT Physics Class 12, Ch. 9 ("Ray Optics and Optical Instruments"), refraction through a prism, dispersion and dispersive power · H. C. Verma, Concepts of Physics Vol. 1, "Dispersion and Spectra" · D. C. Pandey / Cengage optics problem sets for the JEE and NEET prism variants · Cambridge IGCSE 0625, topic 3.2.4, dispersion of white light (qualitative).

Provenance. Standard geometric optics. Recompute: $A = 60°$ , $\mu = 1.5 \to \sin\!\big(\tfrac{60 + \delta_m}{2}\big) = 0.75$ , $\delta_m = 37.2°$ , $i = 48.59°$ , $r = 30°$ ; $A = 60°$ , $\delta_m = 40° \to \mu = \sin 50°/\sin 30° = 1.532$ ; thin $5°$ crown $\to \delta = 2.5°$ ; crown $\omega = (1.523 - 1.514)/(1.517 - 1) = 0.0174$ ; angular dispersion of a $6°$ prism $= (0.009)(6) = 0.054°$ ; direct-vision $A' = (0.517)(6)/0.65 = 4.77°$ .

Prerequisites. The Ray Model of Light and the Laws of Reflection — the ray picture the prism refracts and disperses. Leads to. Prism, Deviation and Dispersion Numericals, the archetype gym that drills these formulas; Scattering and Atmospheric Optics, where the same wavelength-dependent bending colours the sky; and the university seams below.

Go deeper (UG). The full theory of chromatic aberration and achromatic design, and the wave-optics account of dispersion (why $\mu$ depends on wavelength at all), are developed in Dispersion by Prisms and Chromatic Aberration and Achromats in the university optics track. They are enrichment only; every result here follows from Snell's law, the geometry of the two faces, and the small-angle approximation for thin prisms.

Why does the deviation of a prism pass through a minimum?

Because the two faces share the total bending, and the sharing is most even at the symmetric passage. When the prism is tilted so that the ray enters steeply, the first face does most of the deviating and the second little; tilt it the other way and the roles swap. In between there is a setting where the ray crosses symmetrically — entering and leaving at equal angles, $i = e$ , and travelling parallel to the base inside — and there the total deviation is least. Rotating past that point in either direction increases $\delta$ again, so the graph of deviation against incidence is a smooth valley. The minimum is broad and flat, which is what makes it the reliable setting to measure on a spectrometer: small errors in aligning the prism barely change the reading.

What exactly is the difference between angular dispersion and dispersive power?

Angular dispersion is the actual angle between the extreme colours a particular prism produces — violet minus red — and it is $(\mu_v - \mu_r)A$ , so it grows with the prism's refracting angle $A$ . A bigger prism of the same glass gives a wider spectrum. Dispersive power is the ratio of that dispersion to the mean deviation, $\omega = (\mu_v - \mu_r)/(\mu_y - 1)$ , and because the $A$ cancels, it is a pure number belonging to the glass alone, independent of the prism's size. So angular dispersion answers "how wide is the spectrum from this prism?" while dispersive power answers "how strongly does this glass spread colour relative to how much it bends?" You use angular dispersion to size a spectrum and dispersive power to design combinations of glasses.

Why does violet bend more than red?

Because glass has a slightly larger refractive index for shorter wavelengths — a property called normal dispersion. Violet light, with the shortest wavelength in the visible band, sees the highest index and so is bent the most at each face; red, with the longest wavelength, sees the lowest index and bends the least. The difference is small — for crown glass $\mu$ runs from about $1.514$ at the red end to $1.523$ at the violet — but it is enough to fan white light into a visible spectrum. This is also why the order of colours in a prism spectrum is always red (least deviated, nearest the original beam) through to violet (most deviated). Getting the order backward is a common slip; anchor it with "violet bends most because its index is biggest".

When can I use the thin-prism formula δ = (μ − 1)A?

Only when the refracting angle $A$ is small — a few degrees at most. The simple formula comes from replacing the sines in the full relation with the angles themselves, which is accurate only for small angles. For a thin prism the deviation then becomes independent of the angle of incidence, a great convenience, and the whole dispersion analysis — angular dispersion $(\mu_v - \mu_r)A$ and dispersive power — is built on it. But for a substantial prism, such as the standard $60°$ prism, the small-angle approximation fails badly, and you must use the full minimum-deviation formula $\mu = \sin\!\big(\tfrac{A + \delta_m}{2}\big)/\sin\!\big(\tfrac{A}{2}\big)$ instead. A good rule: if the prism angle is more than about $10°$ , do not use $\delta = (\mu - 1)A$ .

What is the difference between an achromatic and a direct-vision prism combination?

They cancel opposite things. An achromatic combination is built to remove the colour spreading while keeping a net bend: you balance the two prisms' dispersions, $(\mu_v - \mu_r)A = (\mu_v' - \mu_r')A'$ , so white light emerges deviated but not fanned into a spectrum — the principle behind the achromatic lens that fixes colour fringing. A direct-vision combination does the reverse: it removes the net deviation while keeping the colour, by balancing the mean deviations $(\mu_y - 1)A = (\mu_y' - 1)A'$ , so light emerges spread into a spectrum but travelling in the original direction — the pocket spectroscope you look straight through. Both use a crown prism against a flint prism; which quantity you balance decides which effect survives. Confusing the two conditions is the commonest error in combination problems.

How is a rainbow related to a prism?

A rainbow is dispersion by water droplets instead of a glass prism. Sunlight entering a spherical raindrop refracts (spreading the colours a little), reflects once off the back of the drop, and refracts again on the way out — a refract–reflect–refract path that acts like a tiny prism, bending violet more than red. Millions of drops, each sending its spectrum to your eye from a slightly different angle, build the familiar arc, with red on the outside and violet on the inside of the primary bow. The physics is the same wavelength-dependent index that makes a glass prism disperse; only the geometry differs, because the drop is round and includes an internal reflection. The detailed rainbow geometry — the $42°$ radius, the second bow, Alexander's dark band — is developed in the atmospheric-optics article and its university deep-dive.