Problem 26 - University Physics with Modern Physics

Question

A beam of light strikes a sheet of glass at an angle of \(57.0^{\circ}\) with the normal in air. You observe that red light makes an angle of \(38.1^{\circ}\) with the normal in the glass, while violet light makes a \(36.7^{\circ}\) angle. (a) What are the indexes of refraction of this glass for these colors of light? (b) What are the speeds of red and violet light in the glass?

Step-by-Step Solution

Answer

Red light index: 1.52, speed: \(1.97 \times 10^8\) m/s. Violet light index: 1.54, speed: \(1.95 \times 10^8\) m/s.

Step 1: Identify Known Values

First, identify all the information provided in the problem. We have the angle of incidence (in air) of the light beam, which is \(57.0^{\circ}\). The angles of refraction in glass are \(38.1^{\circ}\) for red light and \(36.7^{\circ}\) for violet light.

Step 2: Apply Snell's Law for Red Light

Use Snell's Law: \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \), where \( n_1 \) is the index of refraction of air (approximately 1), \( \theta_1 = 57.0^{\circ} \), and \( \theta_2 = 38.1^{\circ} \) for red light. Solve for \( n_2 \) (index of refraction for red light):\[ n_2 = \frac{\sin 57.0^{\circ}}{\sin 38.1^{\circ}} \approx 1.52 \]

Step 3: Apply Snell's Law for Violet Light

Similarly for violet light, using Snell's Law with \( \theta_2 = 36.7^{\circ} \):\[ n_2 = \frac{\sin 57.0^{\circ}}{\sin 36.7^{\circ}} \approx 1.54 \]

Step 4: Calculate Speed of Red Light in Glass

The speed of light in a medium is given by \( v = \frac{c}{n} \), where \( c \) is the speed of light in vacuum (approximately \(3.00 \times 10^8 \) m/s) and \( n \) is the index of refraction. For red light, \( n \approx 1.52 \):\[ v_{\text{red}} = \frac{3.00 \times 10^8}{1.52} \approx 1.97 \times 10^8 \text{ m/s} \]

Step 5: Calculate Speed of Violet Light in Glass

Using the same formula for violet light with \( n \approx 1.54 \):\[ v_{\text{violet}} = \frac{3.00 \times 10^8}{1.54} \approx 1.95 \times 10^8 \text{ m/s} \]

Key Concepts

Index of Refraction

The index of refraction, often denoted as "n," is a crucial concept in optics. It tells us how much the speed of light decreases when it passes through a material. When light travels from one medium to another, its speed changes, and this change affects how light bends, which is known as refraction.
The index of refraction can be calculated using Snell's Law, where:

• \( n_1 \) is the index of refraction of the first medium (such as air).
• \( \theta_1 \) is the angle of incidence, the angle at which the light hits the surface.
• \( n_2 \) is the index of refraction of the second medium (such as glass).
• \( \theta_2 \) is the angle of refraction within the second medium.

Using the formula \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \), we can solve for \( n_2 \). For example, if light travels from air (\( n \approx 1.0 \)) into glass, the index of refraction tells us how much slower light moves in the glass. For red light in glass, this was found to be approximately 1.52, while for violet light, it is about 1.54.

Speed of Light in Glass

The speed of light in different materials is a key factor in optics, dictating how light propagates through the medium. In a vacuum, light travels at its maximum speed of approximately \(3.00 \times 10^8 \) meters per second. However, when light enters different materials like glass, its speed is reduced. This reduction in speed is directly related to the index of refraction of the material.
To find the speed of light in a material, you can use 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,\(3.00 \times 10^8 \),
• \( n \) is the index of refraction of the material.

For red light in a glass with \( n = 1.52 \), the calculated speed of light is approximately \(1.97 \times 10^8 \text{ m/s}\). Similarly, for violet light, using an index of 1.54, the speed becomes roughly \(1.95 \times 10^8 \text{ m/s}\). This change in speed affects how the light refracts within the material.

Angle of Incidence

The angle of incidence is an essential concept in understanding how light behaves when it strikes a surface. It is defined as the angle between the incoming light ray and the normal (an imaginary line perpendicular to the surface) at the point of contact.
In our example, the angle of incidence for both colors (red and violet light) was given as \(57.0^{\circ}\). This angle serves as a starting point in applying Snell's Law, helping to determine how much the light will bend as it enters a different medium, like glass.

• The larger the angle of incidence, the more significant the refraction effect might be, depending on the mediums involved.
• The relationship between the angle of incidence and angle of refraction is what Snell's Law clearly defines.

Thus, understanding the angle of incidence is essential for calculating the resulting path of the light as it passes from air to glass, helping to determine the index of refraction and how it changes with different wavelengths of light (such as red and violet).