Problem 31 - College Physics

Question

\(\bullet$$\bullet\) Using a fast-pulsed laser and electronic timing circuitry, you find that light travels 2.50 \(\mathrm{m}\) within a plastic rod in 11.5 \(\mathrm{ns} .\) What is the refractive index of the plastic?

Step-by-Step Solution

Answer

The refractive index of the plastic is approximately 1.30.

Step 1: Understand the Problem

We need to find the refractive index of the plastic material. The refractive index () is calculated using the formula: \(n = \frac{c}{v} \), where \(c\) is the speed of light in vacuum and \(v\) is the speed of light in the plastic.

Step 2: Identify Constants and Given Values

The speed of light in vacuum, \(c\), is approximately \(3.00 \times 10^8\) m/s. The distance light travels in the plastic rod is \(2.50\) m and the time it takes is \(11.5\) ns (nanoseconds).

Step 3: Convert Time to Compatible Units

Since the speed of light is given in meters per second, convert the time from nanoseconds to seconds: \(11.5 \text{ ns} = 11.5 \times 10^{-9} \text{ s}\).

Step 4: Calculate the Speed of Light in Plastic

Use the formula for speed: \(v = \frac{\text{distance}}{\text{time}} = \frac{2.50\, \text{m}}{11.5 \times 10^{-9}\, \text{s}}\)Calculate \(v\).

Step 5: Calculate the Refractive Index

Now, calculate the refractive index using the formula: \(n = \frac{c}{v}\), where \(v\) is the speed of light in the plastic from Step 4.

Key Concepts

Understanding the Speed of Light

The speed of light, usually denoted by the symbol \(c\), is a fundamental constant in physics, representing how fast light travels in a vacuum. It is incredibly fast, approximately \(3.00 \times 10^8\) meters per second. This speed is constant and is used as a baseline for various calculations in physics, especially in optics and other related fields.

When light travels through different materials, its speed changes due to interactions with the material’s atoms. This change in speed affects how we perceive light, and it is critical to solving many optics problems, like the one in this exercise. By comparing the speed of light in a vacuum and in a material, we can learn how the material alters the speed of light. This understanding helps solve complex physics problems.

The Laser Experiment

In optical experiments, lasers play a crucial role due to their coherent and highly focused light beams. They enable precise measurements required in experiments like the one in this exercise. Specifically, a fast-pulsed laser can emit extremely short bursts of light, which are ideal for timing experiments.

For this problem, the laser helps measure how long it takes light to travel across a certain distance within a material. The accurate timing is facilitated by electronic circuitry that can register nanosecond-level changes. With these precise measurements, the speed of light in different materials can be determined with great accuracy, which is essential in calculating the refractive index.

The World of Optics

Optics is the branch of physics that studies the behavior of light and its interactions with different materials. It encompasses various phenomena, like reflection, refraction, and diffraction. The core element in optics is understanding how light behaves in different media and how it bends when transitioning between materials of different densities.

The refractive index is an essential concept in optics. It quantifies how much light bends and slows down when passing through a medium, compared to its speed in a vacuum. This index helps in designing lenses, microscopes, and numerous optical instruments. It is calculated by comparing the speed of light in a vacuum to the speed in another medium, as highlighted in this exercise.

Solving Physics Problems Step-by-Step

Problem-solving in physics often involves a structured approach to break down complex scenarios. This involves understanding the problem, identifying known and unknown factors, and applying relevant mathematical and physical principles.

• **Understand the Problem**: It starts with a clear grip of what is being asked and what information is provided.

• **Identify Constants and Values**: Like in this example, knowing the speed of light in vacuum and the provided measurements are critical.

• **Unit Conversion**: Ensures all quantities are in compatible units for calculation.

• **Apply Formulas**: Use the identified values within the necessary formulas, like calculating speed and refractive index in this exercise.

• **Solve Step-by-Step**: Gradually break down the problem into smaller, manageable steps, checking each part before moving on.

This structured approach helps ensure accuracy and a deep understanding of physics concepts, making it easier to solve complex problems.