Problem 44 - College Physics

Question

The index of refraction of crown glass is 1.515 for red light and 1.523 for blue light. (a) If light of both colors is incident on crown glass from air, the blue color will be refracted (1) more, (2) less, or (3) the same amount as the red color. Explain. (b) Find the angle separating rays of the two colors in a piece of crown glass if their angle of incidence is \(37^{\circ}\).

Step-by-Step Solution

Answer

(a) Blue light is refracted more. (b) Angle separating the rays is approximately 0.29 degrees.

Step 1: Understanding Refraction Index Differences

The index of refraction for red light in crown glass is 1.515, and for blue light, it is 1.523. The refraction index indicates how much the light slows and bends when entering a medium. A higher index means more bending of light.

Step 2: Applying Snell's Law

Snell's Law is given by \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \), where \( n_1 \) and \( n_2 \) are the indices of refraction for the initial and new mediums, and \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction, respectively. Since the light enters from air (\( n_1 = 1 \)) into crown glass \( n_2 = 1.515 \text{ (red)} \) and \( n_2 = 1.523 \text{ (blue)} \), the blue light will have a smaller angle of refraction compared to red light, meaning the blue light bends more.

Step 3: Calculating Refraction Angles for Both Colors

Using Snell's Law and an incidence angle of \( \theta_1 = 37^{\circ} \): 1. For red light: \[ \sin \theta_{red} = \frac{\sin 37^{\circ}}{1.515} \] Calculate \( \theta_{red} \) using this equation.2. For blue light: \[ \sin \theta_{blue} = \frac{\sin 37^{\circ}}{1.523} \] Calculate \( \theta_{blue} \) using this equation.

Step 4: Finding Angle of Separation

Once \( \theta_{red} \) and \( \theta_{blue} \) are calculated, find the angle of separation by subtracting the angles: \[ \Delta \theta = \theta_{red} - \theta_{blue} \], where \( \Delta \theta \) is the angle separating the two rays in the glass.

Key Concepts

Index of Refraction

The index of refraction is a crucial concept in optics that describes how much light bends, or refracts, as it enters a different medium. This index is represented by the symbol "n" and is a ratio that compares the speed of light in a vacuum to its speed in another medium. For example, when light enters a material like crown glass, it slows down and bends—a change of direction that depends highly on the material's index of refraction.

The formula is straightforward: \[ n = \frac{c}{v} \] where:

• \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \) meters/second.
• \( v \) is the speed of light in the material.

Different colors of light, such as red and blue, have slightly different indices of refraction due to their wavelengths. In crown glass, the index of refraction for red light is 1.515 and for blue light, it's 1.523. The higher the index, the more the light bends. This difference in indices is why we see light separating into different colors when it passes through a prism. So, if blue light has a higher index than red, it will bend more when entering the glass.

Crown Glass

Crown glass is a type of glass with relatively low dispersion and a moderate index of refraction, making it ideal for lenses and optical instruments. It is particularly transparent and lightweight, which helps in transmitting light efficiently.

In optical terms, low dispersion means that crown glass doesn't spread incoming light into its different color components as much as other materials might. This property is advantageous for applications needing clarity and precision, such as prescription glasses and some camera lenses.

The differences in the refraction indices of crown glass for different colors result in varied bending of light; it's this property of crown glass that causes phenomena like chromatic aberration when not corrected in lenses. With crown glass, blue light (higher refraction index) bends slightly more than red light (lower refraction index). This ability to refract light differently based on color is crucial for understanding and designing optical instruments.

Angle of Incidence

The angle of incidence is an essential factor in understanding how light travels between different media. It is the angle that a light ray makes with a line perpendicular to the surface it encounters, known as the normal. This angle determines how much a light ray will bend as it enters a new medium like crown glass.

Snell's Law is the key to understanding this bending process, and it is given by:\[ n_1 \sin \theta_1 = n_2 \sin \theta_2 \] where:

• \( n_1 \) and \( n_2 \) are the indices of refraction for the first and second mediums.
• \( \theta_1 \) is the angle of incidence.
• \( \theta_2 \) is the angle of refraction.

When light hits crown glass at an incidence angle, such as \(37^\circ\), the indices of refraction for red and blue light mean they will have different refraction angles once they enter the glass. Blue light, having a higher index of refraction, will bend more than red light. Calculating these angles precisely allows us to determine how far apart the light rays of different colors will be when passing through the glass. This separation between angles is what causes the diverging of colors in lenses and can be used to correct or exploit optical properties.