Dispersion & atmospheric refraction

Dispersion & atmospheric refraction

{KEY: type=definition | title=What is Dispersion? | text=Dispersion is the phenomenon of splitting a composite light (like white light) into its constituent colours when it passes through a medium where the speed of light depends on its wavelength. The band of coloured components of a light beam is called its spectrum.}

Unpacking Dispersion: Why White Light Splits

You've seen it in rainbows and on the face of a crystal: a single beam of white sunlight entering and a brilliant splash of colours emerging. This magical separation is called dispersion. It happens because the medium (like a water droplet or a glass prism) treats each colour of light slightly differently.

The core reason for dispersion lies in the relationship between the refractive index (n) of a material and the wavelength (λ) of the light passing through it. For most transparent materials like glass, the refractive index decreases as the wavelength of light increases. This means shorter wavelength light (like violet) bends more than longer wavelength light (like red).

The Cause: Cauchy's Relation

This wavelength-dependent behaviour of the refractive index is mathematically described by Cauchy's formula. While it's an empirical formula, it provides a very good approximation for many transparent materials in the visible spectrum.

{FORMULA: expr=n(λ) = A + B/λ² + C/λ⁴ + ... | symbols=n(λ): Refractive index as a function of wavelength, A, B, C: Cauchy's constants for the material, λ: Wavelength of light}

For most practical purposes, we can simplify this to just the first two terms: n(λ) ≈ A + B/λ². This simple relation tells us everything we need to know:

• As wavelength λ increases (moving from violet towards red), the term B/λ² decreases.
• Therefore, the refractive index n decreases.
• According to Snell's Law (n₁sin(θ₁) = n₂sin(θ₂)), a lower refractive index results in less bending (a smaller angle of deviation) for the same angle of incidence.

This leads to the familiar order of the spectrum, often remembered by the acronym VIBGYOR: Violet, Indigo, Blue, Green, Yellow, Orange, Red.

• Violet light (shortest wavelength, λ ≈ 400 nm) has the highest refractive index (nᵥ), so it bends the most.
• Red light (longest wavelength, λ ≈ 700 nm) has the lowest refractive index (nᵣ), so it bends the least.

{{VISUAL: diagram: A beam of white light entering a triangular glass prism. The light splits inside the prism, with the red ray on top (deviated least) and the violet ray on the bottom (deviated most), emerging as a full VIBGYOR spectrum.}}

Quantifying Dispersion: Angular Dispersion & Dispersive Power

We can put numbers to this phenomenon using two key concepts: angular dispersion and dispersive power.

1. Angular Dispersion (θ)

This is the angle between the two extreme colours of the spectrum (violet and red) after they emerge from the prism. It tells you how "spread out" the spectrum is.

For a prism with a small refracting angle A, the angle of deviation δ for any colour is given by δ ≈ (n - 1)A. The angular dispersion is the difference in the deviations of violet and red light.

θ = δᵥ - δᵣ

Substituting the formula for deviation:

θ = (nᵥ - 1)A - (nᵣ - 1)A

θ = (nᵥ - nᵣ)A

Where:

• θ is the angular dispersion.
• nᵥ is the refractive index for violet light.
• nᵣ is the refractive index for red light.
• A is the refracting angle of the prism.

2. Dispersive Power (ω)

This is a property of the material of the prism, not its shape. It measures the ability of the material to disperse light relative to its ability to deviate it. It's defined as the ratio of angular dispersion to the mean deviation. The "mean deviation" is usually taken for yellow light, as it's roughly in the middle of the visible spectrum.

The formula for dispersive power is:

ω = (Angular Dispersion) / (Mean Deviation)

ω = (δᵥ - δᵣ) / δᵧ

Substituting the expressions for deviation:

ω = (nᵥ - nᵣ)A / (nᵧ - 1)A

The prism angle A cancels out, leaving us with a property purely of the material:

ω = (nᵥ - nᵣ) / (nᵧ - 1)

A material with a high dispersive power (like flint glass) creates a wide spectrum, while one with low dispersive power (like crown glass) creates a narrow spectrum. This is crucial for designing lenses.

{CALLOUT: type=tip | text=Achromatic Lenses: To correct for chromatic aberration (where different colours focus at different points), opticians combine lenses made of materials with different dispersive powers, like crown glass and flint glass. The combination is designed to bring all colours to a single focus point.}

Solved Example 1: Basic Dispersion in a Prism (Easy)

Given: A thin prism with a refracting angle A = 4°. The refractive indices of its glass for red and violet light are nᵣ = 1.50 and nᵥ = 1.54 respectively.

To Find: The angular dispersion produced by the prism.

Approach: We can directly use the formula for angular dispersion, θ = (nᵥ - nᵣ)A.

Working:

1. Identify the given values:

   • A = 4°
   • nᵥ = 1.54
   • nᵣ = 1.50

2. Substitute these values into the formula:

   θ = (1.54 - 1.50) × 4°
   

3. Calculate the difference in refractive indices:

   θ = (0.04) × 4°
   

4. Calculate the final angular dispersion:

   θ = 0.16°
   

Final Answer: The angular dispersion produced by the prism is 0.16°.

Solved Example 2: Dispersive Power (Medium)

Given: For a certain glass, the refractive indices for red, yellow, and violet light are nᵣ = 1.61, nᵧ = 1.63, and nᵥ = 1.66.

To Find: The dispersive power of the glass.

Approach: Use the formula for dispersive power, ω = (nᵥ - nᵣ) / (nᵧ - 1), which depends only on the material's refractive indices.

Working:

1. List the given refractive indices:

   • nᵥ = 1.66
   • nᵣ = 1.61
   • nᵧ = 1.63

2. Substitute these into the formula for dispersive power:

   ω = (1.66 - 1.61) / (1.63 - 1)
   

3. Calculate the numerator and the denominator:

   ω = 0.05 / 0.63
   

4. Perform the final division:

   ω ≈ 0.07936
   

   Dispersive power is a dimensionless quantity.

Final Answer: The dispersive power of the glass is approximately 0.0794.

Atmospheric Refraction: The Sky's Optical Illusions

Now, let's shift our focus from a solid prism to the vast expanse of our atmosphere. The air around us is not a uniform medium. Its density, and therefore its refractive index, changes with altitude, temperature, and pressure. Generally, air is densest near the Earth's surface and becomes progressively rarer (less dense) as we go up.

Atmospheric refraction is the bending of light from a celestial object (like the Sun or a star) as it passes through the Earth's atmosphere. Since the light is entering a denser medium as it travels downwards, it continuously bends towards the normal. For an observer on the ground, this makes the object appear to be at a higher position in the sky than it actually is.

{{VISUAL: diagram: Cross-section of the Earth with its atmosphere shown in layers of decreasing density. A light ray from a distant star enters the atmosphere. The ray is shown bending continuously downwards as it passes through the layers. An observer on the surface sees the star at an apparent position which is higher in the sky than its actual position.}}

Phenomena Caused by Atmospheric Refraction

This simple bending of light is responsible for several beautiful and intriguing astronomical phenomena.

1. Twinkling of Stars

Stars are extremely far away, so they act as point sources of light. As the light from a star travels through our turbulent atmosphere, it passes through layers of air with constantly changing temperatures and densities. This causes the light path to waver and bend randomly from moment to moment.

Sometimes more light is refracted towards our eyes, making the star appear bright, and sometimes less light reaches us, making it appear dim. This rapid fluctuation in brightness is what we perceive as twinkling. Planets, being much closer, appear as extended sources (tiny discs). The light from different points on the disc averages out the fluctuations, which is why planets generally do not twinkle.

2. Advanced Sunrise and Delayed Sunset

This is a classic and testable consequence of atmospheric refraction.

• Advanced Sunrise: In the morning, the Sun is visible to us about 2 minutes before it actually crosses the horizon. This is because its light rays, upon entering the atmosphere, bend downwards, allowing us to see it while it's still physically below our line of sight.
• Delayed Sunset: Similarly, in the evening, we can see the Sun for about 2 minutes after it has already dipped below the horizon.

The combined effect is that the day is approximately 4 minutes longer than it would be without an atmosphere! The apparent shift in the Sun's position near the horizon is about 0.5°.

{CALLOUT: type=exam | title=Key Exam Fact | text=The time difference for both advanced sunrise and delayed sunset is approximately 2 minutes. This is due to an apparent vertical shift of about 0.5 degrees when the object is at the horizon.}

3. Apparent Flattening (Oval Shape) of the Sun

At sunrise or sunset, the Sun is near the horizon. The light rays from the bottom edge of the Sun have to travel through a greater thickness of the atmosphere than the rays from the top edge. Consequently, the rays from the bottom edge are refracted more than the rays from the top edge.

This differential refraction causes the bottom edge to be lifted up more than the top edge, making the vertical diameter of the Sun appear smaller than its horizontal diameter. This gives the Sun its characteristic flattened or oval shape when it's low in the sky.

Solved Example 3: Apparent Depth due to Atmosphere (Hard)

Given: An astronaut in a satellite orbiting at an altitude of 360 km looks down at the Earth. Assume the Earth's atmosphere has a uniform refractive index of n = 1.0003 and extends to a height of 80 km. Treat the Earth as a flat surface for this problem.

To Find: The apparent depth of the Earth's surface as seen by the astronaut.

Approach: This is an apparent depth problem. The formula for apparent depth is d_app = d_real / n, where d_real is the real depth of the object inside the medium, and n is the refractive index of the medium, viewed from a rarer medium (vacuum/space, n_space ≈ 1). Here, the "depth" is the thickness of the atmosphere.

Working:

1. Identify the relevant depths and refractive indices. The astronaut is looking from space (n₁ = 1) into the atmosphere (n₂ = 1.0003). The object is the Earth's surface, and the medium causing the shift is the atmosphere itself.

   • Real depth (thickness of the atmosphere) d_real = 80 km.
   • Refractive index of the medium n = 1.0003.

2. The astronaut is at a height of 360 km. The total distance to the surface is 360 km. However, the refractive medium only exists for the first 80 km from the surface. The shift in position only happens due to this 80 km thick layer.

3. Calculate the apparent depth of this 80 km layer of atmosphere.

   d_app = d_real / n
   

   d_app = 80 km / 1.0003
   

4. Perform the division:

   d_app ≈ 79.976 km
   

5. This means the 80 km thick atmosphere appears to be only 79.976 km thick. The apparent shift is Δd = d_real - d_app.

   Δd = 80 km - 79.976 km = 0.024 km
   

6. The astronaut sees the surface lifted up by this amount. The total apparent distance from the astronaut to the surface is the distance in vacuum plus the apparent depth of the atmosphere.

   • Distance in vacuum = 360 km - 80 km = 280 km.
   • Total apparent distance = 280 km + 79.976 km = 359.976 km.

Final Answer: The apparent depth of the Earth's surface from the astronaut's position is 359.976 km. The surface appears to be lifted by about 24 metres.

Common Numerical Traps

Students often make predictable errors in exams. Here’s a table to help you avoid them.

Exam-Style MCQ Bank

Test your understanding with these questions.

1. A beam of white light is incident on a hollow prism of glass. The light emerging from the prism will show: a) Dispersion b) No deviation c) Deviation but no dispersion d) Interference

💡 Answer: b) No deviation Explanation: A hollow prism is essentially two parallel glass plates (the walls) with air in between. The light enters the first glass surface, then air, then the second glass surface, and finally emerges. The net effect is that the emergent ray is parallel to the incident ray, with only a slight lateral shift. Since the effective refracting angle is zero (air inside has the same refractive index as air outside), there is no deviation and hence no dispersion.

2. The dispersive power of a prism material is 0.04. If the refractive index for the mean colour (yellow) is 1.60, what is the difference between the refractive indices for violet and red light? a) 0.024 b) 0.064 c) 0.032 d) 0.048

💡 Answer: a) 0.024 Solution: We know ω = (nᵥ - nᵣ) / (nᵧ - 1). We need to find (nᵥ - nᵣ). Rearranging the formula: (nᵥ - nᵣ) = ω × (nᵧ - 1). Given: ω = 0.04 and nᵧ = 1.60. (nᵥ - nᵣ) = 0.04 × (1.60 - 1) = 0.04 × 0.60 = 0.024.

3. The twinkling of stars is primarily due to: a) The vast distance of stars b) Dispersion of light in the atmosphere c) Fluctuations in the refractive index of the atmosphere d) Total internal reflection in the upper atmosphere

💡 Answer: c) Fluctuations in the refractive index of the atmosphere Explanation: The continuous and random change in the density and temperature of air layers causes the light path from the star to change continuously. This leads to fluctuations in the amount of light reaching the observer's eye, which is perceived as twinkling.

4. If a thin prism of angle 5° gives a deviation of 2.5° for a certain colour, the refractive index of the prism material is: a) 1.5 b) 2.0 c) 1.25 d) 1.75

💡 Answer: a) 1.5 Solution: For a thin prism, δ = (n - 1)A. Given: δ = 2.5° and A = 5°. 2.5° = (n - 1) × 5° (n - 1) = 2.5 / 5 = 0.5 n = 0.5 + 1 = 1.5.

5. Due to atmospheric refraction, the length of the day: a) Increases by approximately 4 minutes b) Decreases by approximately 4 minutes c) Increases by approximately 2 minutes d) Remains unchanged

💡 Answer: a) Increases by approximately 4 minutes Explanation: Sunrise is advanced by about 2 minutes and sunset is delayed by about 2 minutes. The total effect is an increase in the duration of daylight by approximately 4 minutes.

Practice Problems

Solve these to master the concepts. Answers are provided below.

1. (Easy) Find the angle of deviation for a ray of light passing through a prism of angle 6° if the refractive index of the material is 1.62.
2. (Easy) The refractive indices for flint glass are 1.644 for red light and 1.664 for violet light. If a thin prism of this glass has a refracting angle of 5°, what is the angular dispersion?
3. (Medium) A crown glass prism (nᵥ = 1.523, nᵣ = 1.514) and a flint glass prism (nᵥ = 1.664, nᵣ = 1.644) are combined to produce dispersion without deviation. If the crown glass prism has an angle of 6°, what must be the angle of the flint glass prism? (Hint: Set the mean deviations equal and opposite).
4. (Medium) Calculate the dispersive power for crown glass using the data from the previous question. Assume nᵧ = (nᵥ + nᵣ) / 2.
5. (Hard) A star is observed to be at an angle of elevation of 60° from the horizon. The light from the star passes through an atmosphere with an effective refractive index of 1.00029. Using Snell's law at the boundary of the atmosphere (assume it's flat for simplicity), calculate the true angle of elevation of the star.

💡 Answers:

1. 3.72°
2. 0.1°
3. 3.0° (The prism must be inverted)
4. 0.0173
5. 59.98° (The apparent shift is very small at high angles)

Formula & Concepts Cheatsheet

A quick summary for your last-minute revision.

{FLASHCARD: q=Why don't planets twinkle? | a=Planets are much closer to Earth, so they appear as extended sources (discs) rather than point sources. The light from different points on the disc undergoes different refractions, but these effects average out, cancelling the twinkling effect.}