Ch. 33 - Lenses and Optical Instruments

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 33 - Lenses and Optical Instruments

Problem 63

Chapter 32, Problem 63

A series of polarizers are each rotated 10° from the previous polarizer. Unpolarized light is incident on this series of polarizers. How many polarizers does the light have to go through before it is 1/6 of its original intensity?

Verified step by step guidance

Start by recalling Malus's Law, which states that the intensity of light passing through a polarizer is given by:

I = I_0 \(\cos\)^2(\(\theta\))

, where

I_0

is the initial intensity,

\(\theta\)

is the angle between the light's polarization direction and the polarizer's axis, and

I

is the transmitted intensity.

For unpolarized light incident on the first polarizer, the intensity is reduced by half. Thus, after the first polarizer, the intensity becomes

I_1 = \(\frac{I_0}{2}\)

For each subsequent polarizer, the transmitted intensity is reduced according to Malus's Law. If the angle between each polarizer is

10^\(\circ\)

, the intensity after the second polarizer is

I_2 = I_1 \(\cos\)^2(10^\(\circ\))

. Similarly, after the third polarizer, the intensity is

I_3 = I_2 \(\cos\)^2(10^\(\circ\))

, and so on.

Generalize the formula for the intensity after

n

polarizers. The intensity after the

n

-th polarizer is given by:

I_n = \(\frac{I_0}{2}\) \(\cdot\) \(\cos\)^2(10^\(\circ\))^{n-1}

. Here,

n-1

accounts for the number of additional polarizers after the first one.

Set the final intensity

I_n

equal to

\(\frac{I_0}{6}\)

and solve for

n

. This gives the equation:

\(\frac{I_0}{6}\) = \(\frac{I_0}{2}\) \(\cdot\) \(\cos\)^2(10^\(\circ\))^{n-1}

. Simplify and solve for

n

by taking the logarithm of both sides to isolate

n

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Malus's Law

Malus's Law states that when polarized light passes through a polarizer, the intensity of the transmitted light is proportional to the cosine square of the angle between the light's polarization direction and the polarizer's axis. Mathematically, it is expressed as I = I0 * cos²(θ), where I0 is the initial intensity, I is the transmitted intensity, and θ is the angle between the light's polarization direction and the polarizer.

Unpolarized Light

Unpolarized light consists of waves vibrating in multiple planes perpendicular to the direction of propagation. When unpolarized light encounters a polarizer, it becomes polarized, with its intensity reduced according to the angle of the polarizer. The first polarizer will reduce the intensity of unpolarized light to half of its original value, regardless of the angle.

Intensity Reduction through Multiple Polarizers

When light passes through multiple polarizers, the intensity reduction is cumulative. Each polarizer further reduces the intensity based on the angle between the light's polarization direction and the polarizer's axis. For a series of polarizers rotated by a constant angle, the intensity after n polarizers can be calculated using the formula I = I0 * (1/2) * cos²(θ)^(n-1), where θ is the angle between successive polarizers.

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09:47

Multiple Polarizers & Malus's Law

Related Practice

Textbook Question

(II) An aquarium filled with water has flat glass sides whose index of refraction is 1.51. A beam of light from outside the aquarium strikes the glass at a 43.5° angle to the perpendicular (Fig. 32–52). What is the angle of this light ray when it enters (a) the glass, and then (b) the water? (c) What would be the refracted angle if the ray entered the water directly?

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Textbook Question

As early morning passed toward midday, and the sunlight got more intense, a photographer noted that, if she kept her shutter speed constant, she had to change the f-number from f/5.6 to f/16. By what factor had the sunlight intensity increased during that time?

Textbook Question

"(II) Two plane mirrors meet at a 135° angle, Fig. 32–47. If light rays strike one mirror at 32° as shown, at what angle θ do they leave the second mirror?

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Textbook Question

(II) (a) What is the minimum index of refraction for a glass or plastic prism to be used in binoculars (Fig. 32–34) so that total internal reflection occurs at 45°? (b) Will binoculars work if their prisms (assume n = 1.58) are immersed in water? (c) What minimum n is needed if the prisms are immersed in water?

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Textbook Question

Figure 33–49 is a photograph of an eyeball with the image of a boy in a doorway. (a) Is the eye here acting as a lens or as a mirror? (b) Is the eye being viewed right side up or is the camera taking this photo upside down? (c) Explain, based on all possible images made by a convex mirror or lens.

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Textbook Question

(II) A planoconvex lens (Fig. 33–2a) has one flat surface and the other has R = 15.3 cm. This lens is used to view a red and yellow object which is 62.0 cm away from the lens. The index of refraction of the glass is 1.5106 for red light and 1.5226 for yellow light. What are the locations of the red and yellow images formed by the lens?

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