Describe the dispersion of light as illustrated by the refraction of white light by a glass prism

3.2 Dispersion of Light – Refraction of White Light by a Glass Prism

Learning objective

Describe the dispersion of light as illustrated by the refraction of white light by a glass prism, and use this knowledge to solve quantitative problems, plan a simple prism experiment and recognise related applications.

1. Definition of dispersion

2. The seven visible colours (order of decreasing wavelength)

1. Red (≈ 700 nm)
2. Orange (≈ 620 nm)
3. Yellow (≈ 580 nm)
4. Green (≈ 530 nm)
5. Blue (≈ 470 nm)
6. Indigo (≈ 445 nm)
7. Violet (≈ 400 nm)

3. Refractive index versus wavelength

For most glasses the index decreases smoothly as the wavelength increases. A convenient empirical form is Cauchy’s equation:

\[ n(\lambda)=A+\frac{B}{\lambda^{2}}+\frac{C}{\lambda^{4}}\quad(\lambda\text{ in µm}) \]

Typical values for crown glass (to three decimal places) are shown below. The uncertainties are ±0.001, the usual precision for IGCSE tables.

Colour (approx.)	Wavelength λ (nm)	Refractive index n (±0.001)
Violet	400	1.545
Blue	470	1.540
Green	530	1.535
Yellow	580	1.532
Orange	610	1.531
Red	650	1.528

4. Fundamental relationships

• Snell’s law (refraction at a single surface): \[ n_{1}\sin i = n_{2}\sin r \] where i is the angle of incidence, r the angle of refraction, and n₁, n₂ the refractive indices of the two media.
• Deviation of a prism (for a ray of wavelength λ): \[ \delta(\lambda)=i+e(\lambda)-A \]
  • i = angle of incidence on the first face
  • e(λ) = angle of emergence from the second face (depends on λ)
  • A = apex angle of the prism (geometrical angle between the two refracting faces)
• Minimum deviation (symmetric path, r₁ = r₂): \[ \delta_{\min}=2i_{\min}-A \qquad\text{and}\qquad n(\lambda)=\frac{\sin\!\bigl(\frac{A+\delta_{\min}}{2}\bigr)}{\sin\!\bigl(\frac{A}{2}\bigr)}. \] This formula is the basis of laboratory determinations of n(λ).
• For a thin prism (small A) the deviation is approximately \(\delta\approx (n-1)A\).
• Angular separation of two colours (λ₁, λ₂) to first order: \[ \Delta\theta \approx \bigl[n(\lambda_{1})-n(\lambda_{2})\bigr]\sin A . \]

5. How a prism produces a spectrum

1. White light (a mixture of all visible λ) strikes the first face at angle i.
2. Each component λ has its own index n(λ), giving a colour‑dependent refraction angle r₁(λ) via Snell’s law.
3. The ray traverses the glass and reaches the second face, where it is refracted again to an emergence angle e(λ).
4. The total deviation δ(λ) = i + e(λ) − A. Because n(blue) > n(red), δ(blue) > δ(red). The colours fan out, producing a continuous spectrum (red least deviated, violet most deviated).

6. Practical activity – Prism dispersion experiment

Objective

Determine the refractive index of crown glass for two colours (e.g., red and blue) using the minimum‑deviation method.

Apparatus

• Equilateral glass prism (apex angle A ≈ 60°)
• Spectral lamp or white LED with a narrow slit
• Goniometer or protractor with a rotating table (±0.1° accuracy)
• Screen or white paper to observe the emerging spectrum
• Ruler or measuring tape (for distance measurements, if required)
• Safety goggles and lab coat

Procedure

1. Place the prism on the rotating table and align the incident white beam so that it strikes the first face near the centre.
2. Rotate the prism slowly and observe the emerging spectrum on the screen. Adjust the prism until the red and blue components are each at a minimum angular deviation (the spectrum is most compact). Record the corresponding angle of incidence i_min for each colour.
3. Measure the apex angle A of the prism (provided by the manufacturer or measured with a protractor).
4. Calculate the minimum deviation for each colour: \(\delta_{\min}=2i_{\min}-A\).
5. Use the minimum‑deviation formula to compute n(λ) for red and blue.
6. Estimate the experimental uncertainty by repeating the measurement three times and calculating the mean ± standard deviation.

Safety & practical considerations

• Wear safety goggles to protect eyes from stray beams.
• Handle the prism with both hands; glass edges are sharp.
• Ensure the light source is securely mounted to avoid accidental movement.
• When using a spectral lamp, allow it to cool before handling.
• Record angles to the nearest 0.1° and keep significant figures consistent with the instrument precision.

Data‑analysis suggestions

• Plot n versus 1/λ² to check the linearity predicted by Cauchy’s equation.
• Compare your experimental n values with the table in section 3; discuss possible sources of error (e.g., prism imperfections, mis‑alignment, finite slit width).

7. Applications of dispersion

• Optical fibres – Material dispersion causes different wavelengths to travel at slightly different speeds, spreading a short pulse. Low‑dispersion (e.g., fluorine‑doped) glass and graded‑index profiles are used to minimise this effect.
• Chromatic aberration in lenses – Because n(blue) > n(red), a simple converging lens brings blue light to a focus nearer the lens than red light (longitudinal chromatic aberration). A secondary effect is transverse chromatic aberration, where colour fringes appear at the image edges. Achromatic doublets, apochromatic lenses and special low‑dispersion glasses are employed to correct these problems.
• Rainbows and atmospheric phenomena – Water droplets act as tiny prisms, separating sunlight into a spectrum.

8. Common misconceptions

• “All colours travel at the same speed in glass.” – False. The speed is \(v=c/n(\lambda)\) and varies with wavelength.
• “A prism creates new colours.” – False. It only separates the colours already present in white light.
• “Dispersion occurs only in prisms.” – False. Any medium where n depends on λ (lenses, fibres, rain droplets, diffraction gratings) exhibits dispersion.
• “The deviation formula works for any prism shape.” – It is derived for a triangular prism with a single apex angle A; other shapes require modified geometry.

9. Link to thin lenses – chromatic aberration revisited

Because the focal length of a thin lens is \(f(\lambda)=\dfrac{R}{2\,(n(\lambda)-1)}\) (R = radius of curvature), the focal length is shorter for blue light than for red light. This produces:

• Longitudinal chromatic aberration – different focal planes for different colours, seen as coloured halos when the eye cannot accommodate.
• Transverse chromatic aberrure – colour fringes at the periphery of the image because magnification varies with λ.

Understanding prism dispersion therefore underpins the design of achromatic doublets (two glasses with opposite dispersion) and modern photographic/telescope optics.

10. Assessment‑style questions

Question 1 – Multiple Choice (AO1)

When white light passes through a glass prism, which statement is correct?

1. The violet component is deviated less than the red component because violet has a longer wavelength.
2. All colours emerge at the same angle because the prism material is homogeneous.
3. The violet component is deviated more than the red component because the refractive index for violet is larger.
4. Dispersion only occurs if the prism is made of a metal.

Answer: C

Question 2 – Calculation (AO2)

A triangular prism of apex angle A = 60.0° is used in a laboratory. For blue light (λ ≈ 470 nm) the measured minimum deviation is \(\delta_{\min}=58.2°\). Calculate the refractive index of the glass for this wavelength. Give your answer to three significant figures.

Solution:

\[ n = \frac{\sin\!\bigl(\frac{A+\delta_{\min}}{2}\bigr)}{\sin\!\bigl(\frac{A}{2}\bigr)} = \frac{\sin\!\bigl(\frac{60.0°+58.2°}{2}\bigr)}{\sin\!\bigl(30.0°\bigr)} = \frac{\sin(59.1°)}{0.500} = \frac{0.857}{0.500} = 1.71\;( \text{to three s.f.} ) \]

Thus, \(n_{\text{blue}}\approx1.71\).

11. Summary checklist

1. State the definition of dispersion (variation of n with λ).
2. List the seven visible colours in order of decreasing wavelength.
3. Explain why shorter wavelengths have a larger refractive index and are bent more sharply.
4. Write Snell’s law and the prism deviation formula, defining every symbol.
5. Describe the minimum‑deviation method and show how it yields n(λ).
6. Perform a simple prism experiment, noting safety, measurement, and uncertainty considerations.
7. Identify two real‑world applications (optical fibres, chromatic aberration) and explain the role of dispersion.
8. Correct the listed misconceptions.
9. Connect prism dispersion to longitudinal and transverse chromatic aberration in thin lenses.
10. Answer exam‑style questions confidently, using the appropriate formulas and significant‑figure rules.