Problem 4 - Physics

Question

The refractive indices of materials \(A\) and \(B\) have a ratio of \(n_{\mathrm{A}} / n_{\mathrm{B}}=1.33 .\) The speed of light in material \(A\) is \(1.25 \times 10^{8} \mathrm{~m} / \mathrm{s}\). What is the speed of light in material \(B ?\)

Step-by-Step Solution

Answer

The speed of light in material B is \( 1.6625 \times 10^8 \text{ m/s} \).

Step 1: Understanding the Problem

We are given the ratio of the refractive indices of materials A and B and the speed of light in material A. We need to find the speed of light in material B.

Step 2: Formula for Refractive Index

Remember that the refractive index of a material is given by the formula \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum \( (3 \times 10^8 \text{ m/s}) \) and \( v \) is the speed of light in the material.

Step 3: Express Refractive Index Ratio

We know \( \frac{n_A}{n_B} = 1.33 \). We can express \( \frac{c/v_A}{c/v_B} = 1.33 \), which simplifies to \( \frac{v_B}{v_A} = 1.33 \) since the speed of light in a vacuum \( c \) cancels out.

Step 4: Rearrange for Speed of Light in Material B

From the equation \( \frac{v_B}{v_A} = 1.33 \), we can solve for \( v_B \) by multiplying both sides of the equation by \( v_A \): \( v_B = 1.33 \times v_A \).

Step 5: Substitute Known Values

We know \( v_A = 1.25 \times 10^8 \text{ m/s} \). Substitute this into the equation for \( v_B \): \( v_B = 1.33 \times 1.25 \times 10^8 \text{ m/s} \).

Step 6: Calculate the Speed of Light in Material B

Perform the multiplication: \( v_B = 1.6625 \times 10^8 \text{ m/s} \).

Key Concepts

Speed of Light

The speed of light is a fundamental constant of nature that greatly impacts how we observe the universe. It is denoted by the symbol \( c \) and in a vacuum, light travels at an incredible speed of \( 3 \times 10^8 \) meters per second (approximately 300,000 km/s). This speed is the ultimate speed limit in the universe, meaning no information or matter can travel faster than light in a vacuum.

When light enters a medium like glass, water, or any other material, it slows down due to interactions with the atoms in the medium. This change in speed is described by the material's refractive index \( n \), which is calculated using the formula:

• \( n = \frac{c}{v} \)

where \( v \) is the speed of light in the material.

Understanding the speed of light and how it changes in different materials is crucial for fields such as optics and telecommunications.

Optics

Optics is the branch of physics that studies the behavior and properties of light. It deals with how light interacts with matter and how it can be manipulated. Light is a form of electromagnetic radiation, and optics helps us understand phenomena like reflection, refraction, diffraction, and interference.

At its core, optics seeks to explain and predict how light will behave under various conditions.

• Reflection: Light bounces off a surface.
• Refraction: Light bends when entering a different medium because of a change in speed.

For instance, if light travels from air into water, it changes speed, causing the light path to bend. This bending of light is described using Snell's law, which relates the angles and refractive indices of the different media.

By understanding the principles of optics, we can design lenses, microscopes, telescopes, and many other optical devices that enhance our ability to see and measure the world around us.

Light Propagation

Light propagation refers to the transmission of light through space and various media. It is vital in understanding how light travels, changes direction, and slows down or speeds up.

In a vacuum, light moves in straight lines and maintains its speed of \( 3 \times 10^8 \) m/s. But in any other medium, its path and speed are affected by the medium's refractive index.

• Straight Line Propagation: In space or a uniform medium, light travels in straight lines.
• Refraction and Bending: When crossing media of different refractive indices, light bends.

Consider a ray of light entering water from air. The light slows down due to the water's higher refractive index compared to air, causing the ray to bend. This is called refraction and is crucial in technologies like fiber optics, where light must bend and bounce to transmit signals over long distances.

By mastering the principles of light propagation, scientists and engineers can manipulate light for various applications, from lasers to optical fibers and beyond.