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Waves - Dispersion

Grade 9IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Dispersion is defined as the splitting of white light into its constituent colors when it passes through a transparent medium such as a glass prism.

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The band of colors produced is called the visible spectrum, arranged in order of decreasing wavelength: Red, Orange, Yellow, Green, Blue, Indigo, and Violet (ROYGBIVROYGBIV).

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Dispersion occurs because the refractive index (nn) of a material depends on the wavelength (color) of the light. This is expressed as n(λ)n(\lambda).

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In a vacuum, all colors of light travel at the same speed (c≈3×108 m/sc \approx 3 \times 10^8 \text{ m/s}), but in a medium like glass or water, different colors travel at different speeds (vv).

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Red light has the longest wavelength and travels the fastest in the medium, resulting in the least deviation. Violet light has the shortest wavelength and travels the slowest, resulting in the greatest deviation.

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The relationship between speed, frequency, and wavelength is given by v=fλv = f \lambda. During refraction and dispersion, the frequency (ff) remains constant while speed (vv) and wavelength (λ\lambda) change.

📐Formulae

n=cvn = \frac{c}{v}

v=fλv = f \lambda

ncolor=sin⁡isin⁡rcolorn_{color} = \frac{\sin i}{\sin r_{color}}

💡Examples

Problem 1:

White light travels from air into a glass block. The refractive index of the glass for red light is 1.511.51 and for violet light is 1.531.53. If the speed of light in a vacuum is c=3.0×108 m/sc = 3.0 \times 10^8 \text{ m/s}, calculate the speed of red and violet light inside the glass.

Solution:

Using the formula n=cvn = \frac{c}{v}, we can rearrange it to find v=cnv = \frac{c}{n}.

For Red light: vred=3.0×108 m/s1.51≈1.987×108 m/sv_{red} = \frac{3.0 \times 10^8 \text{ m/s}}{1.51} \approx 1.987 \times 10^8 \text{ m/s}

For Violet light: vviolet=3.0×108 m/s1.53≈1.961×108 m/sv_{violet} = \frac{3.0 \times 10^8 \text{ m/s}}{1.53} \approx 1.961 \times 10^8 \text{ m/s}

Explanation:

The calculations show that violet light travels slower than red light in the glass block. This difference in speed is the fundamental reason why the two colors refract at different angles, leading to dispersion.

Problem 2:

A ray of white light hits a prism at an angle of incidence of 45∘45^{\circ}. If the angle of refraction for red light is 28∘28^{\circ} and for violet light is 26∘26^{\circ}, which color has a higher refractive index?

Solution:

Using Snell's Law: n=sin⁡isin⁡rn = \frac{\sin i}{\sin r}

For Red light: nred=sin⁡45∘sin⁡28∘≈0.7070.469≈1.507n_{red} = \frac{\sin 45^{\circ}}{\sin 28^{\circ}} \approx \frac{0.707}{0.469} \approx 1.507

For Violet light: nviolet=sin⁡45∘sin⁡26∘≈0.7070.438≈1.614n_{violet} = \frac{\sin 45^{\circ}}{\sin 26^{\circ}} \approx \frac{0.707}{0.438} \approx 1.614

Explanation:

Violet light has a smaller angle of refraction, meaning it bends more toward the normal. This results in a higher refractive index (nviolet>nredn_{violet} > n_{red}). This confirms that shorter wavelengths experience more deviation during dispersion.