Problem 13 - Physics

Question

A beam of light is traveling in air and strikes a material. The angles of incidence and refraction are \(63.0^{\circ}\) and \(47.0^{\circ}\), respectively. Obtain the speed of light in the material.

Step-by-Step Solution

Answer

The speed of light in the material is approximately \(2.46 \times 10^8\) m/s.

Step 1: Understanding the Given Values

We are given that the beam of light travels from air (with an incident angle \( \theta_1 = 63.0^{\circ} \)) into a material (with a refraction angle \( \theta_2 = 47.0^{\circ} \)). Our objective is to find the speed of light in the material. We will use Snell's Law to help with this task.

Step 2: Applying Snell's Law

Snell's Law is given by the equation \( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \), where \( n_1 \) and \( n_2 \) are the refractive indices of the initial and secondary mediums, respectively. In this case, \( n_1 = 1.0 \) because the light is initially traveling in air, and we will solve for \( n_2 \).

Step 3: Solving for Refractive Index of the Material

Rearranging Snell's Law to solve for \( n_2 \), we get:\[n_2 = \frac{n_1 \sin(\theta_1)}{\sin(\theta_2)}\]Substituting the values, we have:\[n_2 = \frac{1.0 \sin(63.0^{\circ})}{\sin(47.0^{\circ})}\]Calculate \( \sin(63.0^{\circ}) \approx 0.891 \) and \( \sin(47.0^{\circ}) \approx 0.731 \), giving:\[n_2 = \frac{0.891}{0.731} \approx 1.218\]

Step 4: Calculating the Speed of Light in the Material

The speed of light in a material \( v \) can be calculated using the formula:\[v = \frac{c}{n_2}\]where \( c \) is the speed of light in a vacuum (approximately \( 3 \times 10^8 \text{ m/s} \)). Using \( n_2 = 1.218 \), we get:\[v = \frac{3 \times 10^8}{1.218} \approx 2.46 \times 10^8 \text{ m/s}\]

Step 5: Final Result

The speed of light in the material is approximately \( 2.46 \times 10^8 \text{ m/s} \). This result reflects how the light slows down as it enters the denser medium from air due to the higher refractive index.

Key Concepts

Refractive Index

The refractive index, often denoted as "n," is a fundamental concept in optics. It describes how much light bends, or refracts, when entering a different medium. Simply put, it is the ratio of the speed of light in a vacuum to the speed of light in that medium. This idea is elementary to understanding how materials bend light.
The refractive index formula is:

• The refractive index of a medium, denoted as \( n \), is calculated by \( n = \frac{c}{v} \), where:
  • \( c \) is the speed of light in a vacuum (\( 3 \times 10^8~ \text{m/s} \)).
  • \( v \) is the speed of light in the medium.

For example, if light travels through air (considered \( n_1 = 1.0 \)) and enters a glass, the refractive index of glass (\( n_2 \)) might be around 1.5. This means light travels faster in air than in glass. A high refractive index signifies that light moves slower in the medium.
When solving problems using refractive index, it's crucial to clearly understand what mediums you are considering, as this affects how much the light will bend.

Speed of Light in a Medium

The speed of light dramatically changes once it transitions between different media. Generally, light travels fastest in a vacuum, with a speed of approximately \( 3 \times 10^8 \text{ m/s} \). However, when light travels through a material, its speed reduces due to the refractive index of the medium.
The speed of light in a particular medium can be calculated using the formula:

• \( v = \frac{c}{n} \)

where:

• \( v \) is the speed of light in the material.
• \( c \) is the speed of light in a vacuum.
• \( n \) is the refractive index of the medium.

For instance, if the refractive index \( n \) of a material is 1.5, the light's speed in that material will be lower, calculated as \( v = \frac{3 \times 10^8}{1.5} \approx 2.0 \times 10^8 \text{ m/s}\).
An understanding of the speed of light in different media helps explain various optical phenomena and is essential in technologies like fiber optics that rely on manipulating light paths through various substances.

Angles of Incidence and Refraction

The angles of incidence and refraction are key to understanding how light behaves at the boundary between two different media. The angle of incidence is the angle between the incoming light ray and the normal to the surface (a line perpendicular to the surface). Similarly, the angle of refraction is the angle between the refracted light ray and the normal.
Snell's Law provides the backbone for understanding how these angles relate. It is given by:

• \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \)

where:

• \( n_1 \) and \( n_2 \) are the refractive indices of the initial and final mediums, respectively.
• \( \theta_1 \) is the angle of incidence.
• \( \theta_2 \) is the angle of refraction.

For example, if a light beam hits a glass at an incidence angle of \( 30^{\circ} \), and the refracted angle is \( 19.5^{\circ} \), we can determine how the refractive index affects light direction changes.
This law helps visualize the bending of light, which is foundational for understanding lenses, prisms, and even natural phenomena like rainbows. Managing these angles is also critical in many practical applications, including designing optical instruments and improving visibility through corrective lenses.