Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

In Fig. 33-76, unpolarized light is sent into a system of three polarizing sheets with polarizing directions at angles , $θ_{1} = 20 °$ , $θ_{2} = 60 °$ and $θ_{3} = 40 °$ . What fraction of the initial light intensity emerges from the system?

The fraction of the initial light intensity that emerges from the system is

$\frac{I}{I_{0}} = 0.034$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the given quantities

The values of the angle is:

$θ_{1} = 20 °$

$θ_{2} = 60 °$

$θ_{3} = 40 °$

Step 2: Determine the formulas for polarization

If the incident light is un-polarized, then the intensity of the merging light using one-half rule is given by:

$I = 0.5 I_{0}$ …… (i)

If the incident light is already polarized, then the intensity of the emerging light is cosine –squared of the intensity of incident light is as follows:

$I = I_{0} c o s^{2} θ$ ……. (ii)

Here, $θ$ is the angle between polarization of the incident light and the polarization axis of the sheet.

Step 3: Determine thefraction of the initial light intensity that emerges from the system

If the original light is initially unpolarized,the transmitted intensity is

$I = \frac{I_{0}}{2}$

As the original light is initially unpolarized here. Therefore,

$I_{1} = \frac{1}{2} I_{0}$

If the original light is initially polarized, the transmitted intensity is

$I = I_{0} \cos^{2} θ$

The intensity from the second sheet is as follows:

$I_{2} = I_{1} \cos^{2} θ^{'}$

And the intensity from the third sheet is

$I_{3} = I_{2} \cos^{2} θ^{''}$

Thus,

$I_{3} = I_{1} \cos^{2} θ^{'} \cos^{2} θ^{''}$

$I_{3} = \frac{I_{0}}{2} \cos^{2} θ^{'} \cos^{2} θ^{''}$ (1)

Here,the $θ^{'}$ is the relative angle between the first and the second polarizing sheet and the $θ^{'}$ is the relative angle between the second and the third polarizing sheet.

Thus, from the figure,

$θ^{'} = (θ_{2} - θ_{1})$

$\begin{matrix} θ^{'} = (60 - 20) \\ = 40 ° \end{matrix}$

Similarly for the angle, $θ^{''}$

$\begin{matrix} θ^{''} = 90 - θ_{2} + θ_{3} \\ = 90 - 60 + 40 \\ = 70 ° \end{matrix}$

Substitute the value of $θ^{'}$ and $θ^{'}$ in equation (1), we get,

$\begin{matrix} I_{3} = \frac{I_{0}}{2} \cos^{2} (40 °) \cos^{2} (70 °) \\ \frac{I_{3}}{I_{0}} = \frac{1}{2} \cos^{2} (40 °) \cos^{2} (70 °) \\ = \frac{1}{2} (0.5868) (0.117) \\ = 0.034 \end{matrix}$

Therefore, the fraction of the initial light intensity that emerges from the system is $\frac{I}{I_{0}} = 0.034$

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Fundamentals Of Physics

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Short Answer

In Fig. 33-76, unpolarized light is sent into a system of three polarizing sheets with polarizing directions at angles , $θ_{1} = 20 °$ , $θ_{2} = 60 °$ and $θ_{3} = 40 °$ . What fraction of the initial light intensity emerges from the system?

Step by Step Solution

Step 1: Determine the given quantities

Step 2: Determine the formulas for polarization

Step 3: Determine thefraction of the initial light intensity that emerges from the system

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Fundamentals Of Physics

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Short Answer

In Fig. 33-76, unpolarized light is sent into a system of three polarizing sheets with polarizing directions at angles , θ1=20°, θ2=60°andθ3=40°. What fraction of the initial light intensity emerges from the system?

Step by Step Solution

Step 1: Determine the given quantities

Step 2: Determine the formulas for polarization

Step 3: Determine thefraction of the initial light intensity that emerges from the system

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In Fig. 33-76, unpolarized light is sent into a system of three polarizing sheets with polarizing directions at angles , $θ_{1} = 20 °$ , $θ_{2} = 60 °$ and $θ_{3} = 40 °$ . What fraction of the initial light intensity emerges from the system?