# Dispersive Power and Achromatism

> ω = (μv − μr)/(μy − 1), angular dispersion, achromatic prism and lens pairs with flint and crown glass — JEE Physics dispersion optics.

- Canonical URL: https://prepelephant.com/topics/jee/physics/dispersive-power
- Exam / course: JEE · Subject: Physics
- Publisher: PrepElephant (https://prepelephant.com) — Prepared and reviewed by the PrepElephant Academic Review Team
- First published: 2026-10-02
- Last updated: 2026-10-02
- How to cite: "Dispersive Power and Achromatism", PrepElephant, https://prepelephant.com/topics/jee/physics/dispersive-power

## Direct answer

Dispersive power measures how strongly a material splits white light relative to how much it deviates it: ω = (μv − μr)/(μy − 1), the difference of refractive indices for violet and red divided by the mean deviation factor. A thin prism of angle A deviates yellow light by δy = (μy − 1)A and spreads the spectrum by the angular dispersion δv − δr = (μv − μr)A = ω δy. Flint glass, with higher dispersive power than crown, splits more per unit deviation — the exploitable asymmetry behind the achromatic combination, where a flint component cancels a crown component's dispersion while leaving some net deviation. Achromatic lenses obey ω1/f1 + ω2/f2 = 0: a converging crown lens cemented to a weaker diverging flint lens brings all colours to one focus.

## What you must remember

- **Definition:** ω = (μv − μr)/(μy − 1); it is dimensionless, and the μy − 1 in the denominator (not μv − μr alone) is what students mis-quote under time pressure.
- **Thin-prism deviation:** δ = (μ − 1)A for small angles; angular dispersion = (μv − μr)A = ω × δy, so dispersion grows with prism angle and with dispersive power.
- **Achromatic prism pair:** two prisms of different glasses with (μv − μr)1A1 = (μv − μr)2A2 cancel dispersion; since flint disperses more, its prism angle is smaller, and the pair still deviates — the direct-vision prism.
- **Deviation without dispersion:** same condition, net deviation δ = (δ1 + δ2) nonzero, dispersion zero — white light emerges white but bent.
- **Achromatic doublet (lenses):** ω1/f1 + ω2/f2 = 0; a crown converging lens with a flint diverging lens of shorter magnitude focal length gives a net converging, colour-corrected system, F = f1f2/(f1 + f2).
- **Typical numbers:** crown glass ω ≈ 0.02, flint ω ≈ 0.04 with similar mean indices near 1.5-1.6 — flint disperses about twice as strongly per unit deviation.
- **Spectrum order:** violet bends most (μv largest), red least; blue-ward refraction increases with frequency in normal dispersion, the pattern Cauchy's formula μ = A + B/λ² encodes.

## Building an achromat

Design the classic: a crown converging lens, f1 = +20 cm, ω1 = 0.02, cemented to a flint diverging lens with ω2 = 0.04. Achromatism demands ω1/f1 + ω2/f2 = 0, so f2 = −(ω2/ω1) f1 = −0.04 × 20/0.02 = −40 cm. The combination's power is 1/20 − 1/40 = 1/40, an equivalent focal length of +40 cm: still converging, but every colour now focuses at that same 40 cm to first order. The flint lens contributes only half the crown's power yet exactly its dispersion — precisely because its dispersive power is double, which is the entire reason flint glass exists in optical catalogues.

The prism version runs the same asymmetry in angle. A crown prism of 6 degrees (μv − μr = 0.008) needs a flint prism satisfying 0.008 × 6 = (μv − μr)_flint × A2; with flint's dispersion 0.016 per unit, A2 = 3 degrees, oppositely oriented. The pair sends white light out deviated but undispersed — a direct-vision spectroscope arrangement. Flip the emphasis — choose the two angles so deviation cancels instead — and dispersion survives: the arrangement of two similar-prism dispersion amplification, occasionally asked as a numerical.

## Where students slip

The definition's denominator is the recurring error: writing ω = (μv − μr)/μy or dividing by (μv + μr) mis-scales every subsequent number, and JEE options are tuned to catch it. Second, achromatism is about equal dispersion, not equal deviation — the flint element must have the opposite sign but a different magnitude of power, and students who set f1 = f2 with opposite signs produce a zero-power system, not an achromat. In prism pairs, remember the prisms oppose: their deviations subtract for achromatism-oriented designs (net deviation survives) or add for the deviation-cancelled design (net dispersion survives) — the question stem always states which, so read the target before choosing the sign. Finally, dispersive power depends on the material and the wavelength pair chosen; quoting ω without the v-r-y convention is meaningless, and assertion–reason questions probe exactly whether students know ω is a material property, not a lens property — a lens of any shape ground from the same glass carries the same ω.

## Frequently asked questions

### How is dispersive power defined?

ω = (μv − μr)/(μy − 1), the ratio of the violet-red index spread to the mean deviation factor, a dimensionless material property.

### What is angular dispersion of a thin prism?

(μv − μr)A, the difference of the violet and red deviations; it equals ω times the mean yellow deviation, linking the two measurable spreads.

### What condition makes a lens pair achromatic?

ω1/f1 + ω2/f2 = 0 — the dispersions cancel while the powers only partially cancel, so a crown converging lens with a weaker flint diverging lens remains net converging.

### Why does the achromatic doublet use flint and crown specifically?

Flint glass has roughly double the dispersive power of crown at comparable mean index, so a small-power flint element can cancel a large-power crown element's colour spread and leave useful net convergence.

### Can a prism combination deviate light without dispersing it?

Yes — the direct-vision combination: oppositely oriented crown and flint prisms with (μv − μr)A equal for both send white light out deviated but with zero net angular dispersion.