Dispersive Power and Achromatism
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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.