Question

The minimum deviation produced by a glass prism of angle \(60^{\circ}\) is \(30^{\circ}\). If the velocity of light in vaccum is \(3 \times 10^{8} \mathrm{~m} / \mathrm{s}\) Then what is the velocity of light in glass in \(\mathrm{m} / \mathrm{s}\) ? (A) \(2.72 \times 10^{8}\) (B) \(3.1 \times 10^{8}\) (C) \(2.9 \times 10^{8}\) (D) \(2.121 \times 10^{8}\)

Short answer

The speed of light in the glass is approximately \(2.4 \times 10^{8}\) m/s. None of the provided options match this value, but the closest answer among them would be (A) \(2.72 \times 10^{8}\) m/s.

Step-by-step solution

Calculate the refractive index of the glass.

Using the minimum angle of deviation, we can determine the refractive index of the glass. The formula for the angle of minimum deviation (δ) is: \( \delta = 2(\mu - 1)A \) where \( \delta \) is the minimum angle of deviation, μ is the refractive index, and A is the angle of prism. Rearrange the formula to calculate the refractive index: \( \mu = \frac{\delta}{2A} + 1 \) Substituting the given values: \( \mu = \frac{30^{\circ}}{2 \times 60^{\circ}} + 1 \)

Refractive index calculation.

With the given values, calculate the refractive index μ: \( \mu = \frac{30}{120} + 1 \) \( \mu = \frac{1}{4} + 1 \) \( \mu = 1.25 \) The refractive index of the glass is 1.25.

Use Snell's law to find the speed of light in the glass.

Snell's law relates the speed of light in vacuum (c) to the speed of light in a medium with a refractive index μ (v): \( \mu = \frac{c}{v} \) Rearrange the formula to solve for the speed of light in the glass (v): \( v = \frac{c}{\mu} \) Substitute the known values: \( v = \frac{3 \times 10^{8} \mathrm{m/s}}{1.25} \)

Calculate the speed of light in the glass.

With the provided data, calculate the speed of light (v) in the glass: \( v = \frac{3 \times 10^{8}}{1.25} \) \( v = 2.4 \times 10^{8} \mathrm{m/s} \) None of the presented options is equal to the speed of light we calculated. There may be an error in the given options or the provided data. However, the closest answer to our result would be: (A) \(2.72 \times 10^{8}\) m/s

Key concepts

Snell's Law

Snell's Law forms the foundational principle for understanding how light behaves when moving through different mediums. At its core, Snell's Law links the angle of incidence and refraction to the speed of light in the respective mediums. The law is expressed as:
\[\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2} = \mu\]where:

• \(\theta_1\) is the angle of incidence.
• \(\theta_2\) is the angle of refraction.
• \(v_1\) and \(v_2\) are the speeds of light in the first and second mediums, respectively.
• \(\mu\) is the refractive index of the second medium relative to the first.

In simpler terms, Snell's Law helps us determine how swiftly light will travel through a particular medium by factoring in this refractive index. It provides a way to compare speed efficiencies between mediums. Primarily, when light enters a denser medium like glass, its speed reduces, indicating its refractive properties.

Angle of Deviation

The angle of deviation is a measurement that describes how much a light ray changes direction when it passes through a prism or another optical device. It’s crucial in optics because it helps us understand the bending of light, especially in devices like prisms. The name gives away its purpose: deviation from the initial path.
In a prism, this deviation occurs because light enters and exits the prism at angles that are influenced by the prism’s shape and material. The equation to find the minimum angle of deviation \(\delta\) is:
\[ \delta = 2(\mu - 1)A \]where:

• \(\delta\) is the deviation angle.
• \(\mu\) is the refractive index of the prism material.
• \(A\) is the prism's angle.

This concept is essential not just for classroom problems but also in correcting lenses and in dispersive elements in scientific equipment.

Speed of Light in Glass

Understanding the speed of light in glass begins with the refractive index. The speed of light in glass is slower than in a vacuum. This slowdown results from interaction at a molecular level within the glass, causing light rays to 'bend'. Snell’s Law comes into action here as it informs us how much the speed changes.
Given Snell's Law:
\[ \mu = \frac{c}{v} \]where:

• \(\mu\) is the refractive index.
• \(c\) is the speed of light in a vacuum \((3 \times 10^8 \, \text{m/s})\).
• \(v\) is the speed of light in glass.

By rearranging the equation, you can solve for the speed of light in the glass:
\[ v = \frac{c}{\mu} \]This allows us to find how fast light travels within a specific material given its refractive index, which for glass, results in approximately \(2.4 \times 10^8\, \text{m/s}\), signifying much slower travel compared to a vacuum. This concept is pivotal for designing lenses and other optical elements.