Problem 36 - Algebra 2

Question

OPTICS If a glass prism has an apex angle of measure \(\alpha\) and an angle of deviation of measure \(\beta,\) then the index of refraction \(n\) of the prism is given by \(n=\frac{\sin \left[\frac{1}{2}(\alpha+\beta)\right]}{\sin \frac{\alpha}{2}}\) . What is the angle of deviation of a prism with an apex angle of \(40^{\circ}\) and an index of refraction of 2\(?\)

Step-by-Step Solution

Answer

The angle of deviation \( \beta \) can be found using: \( \beta = 2 \sin^{-1}(2 \times \sin 20^{\circ}) - 40^{\circ} \).

Step 1: Understand the given formula

The formula provided for the index of refraction is: \( n = \frac{\sin \left[\frac{1}{2}(\alpha + \beta)\right]}{\sin \frac{\alpha}{2}} \). Here, \( \alpha \) is the apex angle, \( \beta \) is the angle of deviation, and \( n \) is the index of refraction.

Step 2: Identify known quantities

From the exercise, we know that \( \alpha = 40^{\circ} \) and \( n = 2 \). We need to find the angle of deviation \( \beta \).

Step 3: Substitute known values into the formula

Substituting the known values into the index of refraction formula: \[ 2 = \frac{\sin \left[\frac{1}{2} (40^{\circ} + \beta)\right]}{\sin \frac{40^{\circ}}{2}} \].

Step 4: Calculate \( \sin \frac{\alpha}{2} \)

Calculate the sine of half the apex angle: \[ \sin \frac{40^{\circ}}{2} = \sin 20^{\circ} \].

Step 5: Rearrange the equation to find \( \sin \left[ \frac{1}{2} (\alpha + \beta) \right] \)

Multiply both sides of the equation by \( \sin 20^{\circ} \):\[ \sin \left[ \frac{1}{2} (40^{\circ} + \beta) \right] = 2 \times \sin 20^{\circ} \].

Step 6: Solve for \( \frac{1}{2} (\alpha + \beta) \)

Use the inverse sine function: \[ \frac{1}{2} (40^{\circ} + \beta) = \sin^{-1}(2 \times \sin 20^{\circ}) \].

Step 7: Calculate \( \alpha + \beta \)

Multiply both sides by 2 to solve for \( \alpha + \beta \): \[ 40^{\circ} + \beta = 2 \sin^{-1}(2 \times \sin 20^{\circ}) \].

Step 8: Find \( \beta \)

Subtract \( 40^{\circ} \) from both sides: \[ \beta = 2 \sin^{-1}(2 \times \sin 20^{\circ}) - 40^{\circ} \]. Calculate the specific value to find the angle of deviation.

Key Concepts

Apex Angle

The apex angle in optics refers to the angle formed by the two plane faces of a prism at its peak. This is an integral part of a prism's geometry, as it impacts how light will refract when it passes through the prism. With the apex angle denoted by \(\alpha\), it directly influences the behavior of light. A larger apex angle generally increases the path the light must travel inside the prism before exiting, leading to greater refraction.
A measured apex angle of \(40^{\circ}\), as given in the example problem, tells us about the internal geometry of the prism. Smaller apex angles are common in more slender prisms, while larger ones are seen in bulkier shapes. Understanding the apex angle is essential for calculations involving light refraction and prism design.

Angle of Deviation

The angle of deviation, symbolized as \(\beta\), highlights the amount by which a light ray is bent or deviated as it exits the prism. In the context of this exercise, it shows how much the light path is altered by both the prism's shape and material. To find this angle, we typically need more information about the prism and the incident angle.
For instance, here we use the given values of apex angle and index of refraction, combined in the formula:

• \( n = \frac{\sin \left[\frac{1}{2}(\alpha+\beta)\right]}{\sin \frac{\alpha}{2}} \)

This equation results from trigonometric principles and helps us solve for \(\beta\) when other values are known. Grasping the concept of deviation allows us to predict how lenses and prisms can be designed to achieve desired optical effects.

Index of Refraction

The index of refraction, denoted as \(n\), is a fundamental property of materials that describes how light propagates through the medium. This dimensionless number indicates the bending of light as it enters a material from another medium. It’s a ratio of the speed of light in a vacuum to its speed in the specific medium.
In the provided exercise, the index of refraction for the prism is given as \(n=2\). This tells us that light travels twice as slow in this prism material compared to a vacuum. The importance of the finders like the formula used in the exercise,

• \( n = \frac{\sin \left[\frac{1}{2}(\alpha+\beta)\right]}{\sin \frac{\alpha}{2}} \)

permits calculation involving light behavior across different media using the material's refractive index. This insight is crucial for the design of lenses, glasses, and optical devices since the performance strongly depends on the refractive behaviors of the materials used.