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Dispersion and Wavelength-Dependent Refraction

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Refractive Index as a Material PropertyDispersion and PrismsDispersion: Wavelength and Refractive Index
dispersion wavelength-dependent prism

Core Idea

Dispersion is the wavelength-dependent variation of refractive index in a material: shorter wavelengths (blue light) have higher refractive indices than longer wavelengths (red light) in normal dispersion. This causes white light to separate into its component colors when passing through a prism. Dispersion is the origin of rainbows and explains why different colors refract at different angles.

Explainer

Dispersion builds directly on what you know about refractive index. You learned that n = c/v, where c is the speed of light in vacuum and v is its speed in the medium. When light enters glass, it slows down, and n captures how much. Dispersion extends this by revealing that n is not a single fixed number for a given material — it depends on the frequency (and therefore wavelength) of the light. The refractive index of glass for blue light is measurably higher than for red light.

The physical reason is that light interacts with the electrons in the material, and this interaction is strongly frequency-dependent. Higher-frequency light (shorter wavelengths, bluer) drives electron oscillations closer to their natural resonance frequency, producing a stronger interaction and more slowing. This is called normal dispersion and is the behavior of glass, water, and most transparent solids at visible wavelengths. The relationship between n and wavelength is not linear — it curves steeply toward the ultraviolet end of the spectrum. Empirical formulas like the Cauchy equation (n ≈ A + B/λ²) capture this behavior well for visible light.

The consequence is that Snell's law — n₁ sin θ₁ = n₂ sin θ₂ — produces a different refraction angle for each color. When white light enters a prism, each wavelength bends by a different amount at both the entry and exit surfaces. Blue light, with the highest n, bends the most; red light, with the lowest n, bends the least. The cumulative effect of two refractions (entry and exit) spreads the colors into a continuous spectrum. The angular spread between red and violet across the visible spectrum is the dispersion of the material, and it varies widely between glass types — which is why lens designers combine different glass types to cancel dispersion while preserving focusing power.

Rainbows arise from the same physics in spherical water droplets. Sunlight enters a droplet, reflects off the back interior surface, and refracts again on exit. Because each wavelength exits at a slightly different angle (red at ~42°, violet at ~40° from the antisolar point), different colors reach your eye from droplets at different positions in the sky. The result is the colored arc. The key insight in both prisms and rainbows is the same: dispersion is not an imperfection or side effect — it is a fundamental consequence of how electromagnetic waves interact with bound electrons, and it is the mechanism behind spectroscopy, optical fiber chromatic dispersion, and the chromatic aberration that lens designers work to correct.

Practice Questions 5 questions

Prerequisite Chain

Understanding ZeroThe Number ZeroCounting to FiveCounting to 10Counting to 20Counting a Set of Objects Up to 20Cardinality: The Last Number CountedMatching Numerals to QuantitiesSubitizing Small QuantitiesAddition Within 10Number Bonds to 10Addition Within 20Doubles and Near DoublesDoubles Facts Within 10Near Doubles Facts Within 20Mental Math Strategies for AdditionMental Math: Adding and Subtracting TensAddition Within 100Repeated Addition as MultiplicationMultiplication as Equal GroupsMultiplication: ArraysBasic Multiplication Facts (0s, 1s, 2s, 5s, 10s)Multiplication Facts Within 100Division as Equal SharingDivision as Grouping (Measurement Division)Division: Grouping (Repeated Subtraction) ModelDivision: Fair Sharing ModelDivision as Equal SharingDivision as GroupingBasic Division FactsDivision Facts Within 100Multiplication and Division Fact FamiliesRelationship Between Multiplication and DivisionDivision Facts as Inverse of MultiplicationRemainders and Quotients in DivisionDivision Word ProblemsMulti-Step Word ProblemsSolving Multi-Step Word ProblemsMultiplication Word ProblemsDivision Word ProblemsIntroduction to Long DivisionFactors and MultiplesPrime and Composite NumbersEquivalent FractionsRelating Fractions and DecimalsDecimal Place ValueIntegers and the Number LineComparing and Ordering IntegersAbsolute ValueAdding IntegersSubtracting IntegersMultiplying IntegersDividing IntegersUnit RatesProportionsPercent ConceptConverting Between Fractions, Decimals, and PercentsOperations with Rational NumbersTwo-Step EquationsSolving Multi-Step EquationsEquations with Variables on Both SidesAngle Pairs: Complementary, Supplementary, and VerticalParallel Lines and TransversalsCorresponding AnglesAlternate Interior AnglesTriangle Angle Sum TheoremExterior Angle TheoremTriangle Inequality TheoremSimilar Triangles: AA SimilaritySimilar Triangles: SSS and SAS SimilarityProportions in Similar TrianglesRight Triangle Trigonometry IntroductionSine, Cosine, and Tangent RatiosTrigonometric Ratios ReviewRadian MeasureConverting Between Degrees and RadiansThe Unit CircleGraphing Sine and CosineGraphing Tangent and Reciprocal Trigonometric FunctionsDerivatives of Trigonometric FunctionsAntiderivativesIndefinite IntegralsBasic Integration RulesRiemann SumsDefinite Integral DefinitionDouble Integrals: Definition and SetupIterated Integrals and Fubini's TheoremDouble Integrals over Rectangular RegionsDouble Integrals over General RegionsApplications of Double Integrals: Area, Mass, and MomentsCenter of MassConservation of Linear MomentumElastic CollisionsInelastic CollisionsCoefficient of RestitutionCollision Analysis and Real-World ApplicationsTwo-Body Collisions in the Center-of-Mass FrameReduced Mass and Two-Body ProblemsKinematics in Two DimensionsProjectile MotionCircular Motion: KinematicsSimple Harmonic MotionWave Motion: Definition and ClassificationTransverse Wave Characteristics and PropertiesWavelength, Frequency, and Wave SpeedWave Speed in Elastic MediaAcoustic Impedance and Mechanical ImpedanceImpedance Matching and Wave Reflection at BoundariesReflection and the Law of ReflectionGeometric Optics and the Ray ApproximationWavefronts and Ray Description of Wave PropagationHuygens's Principle and WavefrontsRefraction of WavesSnell's LawTotal Internal ReflectionDispersion and PrismsDispersion and Wavelength-Dependent Refraction

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