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Chapter 30: Problem 21

Find the refractive index of a material for which the polarizing angle in air is \(62^{\circ}\)

Short Answer

Expert verified
The refractive index of the material is equal to the tangent of the polarizing angle when the angle is in radians. So, after converting \(62^{\circ}\) to radians and applying the tangent function, the refractive index will be found.

Step by step solution

01

Convert the Angle to Radians

The trigonometric functions in calculators generally use radians, not degrees. Therefore, the polarizing angle of \(62^{\circ}\) must be converted to radians first. The formula to convert degrees to radians is \(\frac{\pi}{180}\) multiplied by the angle in degrees. Hence, the angle in radians will be \(62 \times \frac{\pi}{180}\).
02

Apply Brewster's law

Once having the angle in radians, use Brewster's law to calculate the refractive index. The law is expressed as \(n = tan(\Theta)\) where \(n\) is the refractive index and \(\Theta\) is the polarizing angle. Replace \(\Theta\) with the angle calculated in Step 1.
03

Calculate the refractive index

Finally, calculate \(\tan(\Theta)\) to get the refractive index. The tangent of the angle will give the refractive index of the material.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Brewster's Law
When light enters a medium at a certain angle, known as the Brewster angle or polarizing angle, it is refracted in such a way that the reflected light is completely polarized parallel to the surface. Brewster's Law mathematically establishes a relationship between the refractive index (\( n \)) of the material and the polarizing angle (\( \theta_B \)

It states that the refractive index (\( n \) is equal to the tangent of the polarizing angle, expressed as:\[ n = \tan(\theta_B) \]
Using this law, you can calculate the refractive index if the polarizing angle is known. It's significant in optics, as it helps in designing polarizing filters and understanding the behavior of light.
Polarizing Angle
The polarizing angle, also known as the Brewster angle, is the angle of incidence at which light with a particular polarization will be perfectly transmitted through a transparent dielectric surface, with no reflection. When light strikes the surface at this angle, the reflected light is perfectly polarized perpendicular to the plane of incidence.

In the given problem, the refractive index is found by utilizing the known polarizing angle in air for a material. This value is essential for producing polarized lenses and screens, and it broadly applies to technologies in photography, LCDs, and reducing glare.
Convert Degrees to Radians
Angles can be measured in degrees or radians. Since many mathematical calculations, especially those involving trigonometric functions, use radians, it's useful to know how to convert between these units.

The formula to change degrees to radians is simple:\[ radians = degrees \times \frac{\text{\textpi}}{180} \]
For example, converting the polarizing angle of \(62^{\text{\textdegree}}\) into radians involves multiplying 62 by \(\frac{\text{\textpi}}{180}\), as seen in the exercise.
Trigonometric Functions
Trigonometric functions are a cornerstone of mathematics, particularly in the study of triangles and the modeling of periodic phenomena. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan).

Each of these functions relates an angle of a right triangle to the ratios of two of its sides. The tangent of an angle, in particular, is the ratio of the opposite side to the adjacent side. Trigonometric functions are pivotal in calculations involving angles and have significant applications in physics, engineering, and navigation.
Tangent of an Angle
The tangent of an angle is one of the fundamental trigonometric functions and is defined as the ratio of the side opposite to the angle to the side adjacent to the angle in a right-angled triangle.

The notation for tangent is 'tan', and it is a function often used when we have the angle and need to find the relationship between the two sides of the triangle. The tangential value of the polarizing angle is especially important in the context of Brewster's Law, as it directly helps in calculating the refractive index of a material, which is exactly what we see in the given textbook problem.

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Most popular questions from this chapter

What is the critical angle for light propagating in glass with \(n=1.52\) when the glass is immersed in (a) water, (b) benzene, and (c) diiodomethane?

For the interface between air (refractive index 1 ) and a material with refractive index \(n\), show that the critical angle and the polarizing angle are related by \(\sin \theta_{c}=\cot \theta_{p}\)

You're an optometrist, mounting a projector at the back of your 4.2-m-long exam room, \(2.6 \mathrm{m}\) above the floor. It shines an eye test pattern on the opposite wall. Patients will sit with their eyes \(3.3 \mathrm{m}\) from the wall and \(1.4 \mathrm{m}\) above the floor to view the pattern. At what height should you center the pattern on the wall?

Total internal reflection occurs at an interface between plastic and air at incidence angles greater than \(37^{\circ} .\) Find the plastic's refractive index.

A compound lens is made from crown glass \((n=1.52)\) bonded to flint glass \((n=1.89) .\) What's the critical angle for light incident on the flint-crown interface?
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