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Chapter 22: Problem 71

An unpolarized beam of light (intensity \(I_{0}\) ) is moving in the \(x\) -direction. The light passes through three ideal polarizers whose transmission axes are (in order) at angles \(0.0^{\circ}, 45.0^{\circ},\) and \(30.0^{\circ}\) counterclockwise from the \(y\) -axis in the \(y z\) -plane. (a) What is the intensity and polarization of the light that is transmitted by the last polarizer? (b) If the polarizer in the middle is removed, what is the intensity and polarization of the light transmitted by the last polarizer?

Short Answer

Expert verified
Answer: a) With three polarizers: The final intensity is \(I_3 = I_2 \cdot \cos^{2}(30^{\circ})\) and the polarization angle is \(30^{\circ}\) to the y-axis. b) With only the first and third polarizers: The final intensity is \(I_3 = I_1 \cdot \cos^{2}(30^{\circ})\) and the polarization angle is also \(30^{\circ}\) to the y-axis.

Step by step solution

01

Intensity and polarization after first polarizer

Since the first polarizer has a transmission axis at \(0.0^{\circ}\), the light passing through it will become linearly polarized parallel to the y-axis. As it is an ideal polarizer, the intensity of the light remains the same. So, the intensity of light after passing through the first polarizer is \(I_{1} = I_{0}\). #Step 2: Light passing through the second polarizer#
02

Intensity and polarization after second polarizer

The angle between the transmission axis of the first and second polarizers is \(45.0^{\circ}\). Using Malus' law, the intensity of light after passing through the second polarizer is \(I_{2} = I_{1} \cdot \cos^{2}(45^{\circ})\). As the second polarizer is at \(45^{\circ}\) counterclockwise from the y-axis, the transmitted light will be linearly polarized with a polarization angle of \(45^{\circ}\) to the y-axis. #Step 3: Light passing through the third polarizer#
03

Intensity and polarization after the third polarizer

The angle between the transmission axis of the second and third polarizers is \(30.0^{\circ}\). Using Malus' law again, the intensity of light after passing through the third polarizer is \(I_{3} = I_{2} \cdot \cos^{2}(30^{\circ})\). The transmitted light is now linearly polarized with a polarization angle of \(30^{\circ}\) to the y-axis. #b) Case with only the first and third polarizers# #Step 4: Light passing through the first polarizer#
04

Intensity and polarization after first polarizer (removing the middle polarizer)

Same as before, the first polarizer polarizes the light parallel to the y-axis and does not change the intensity \(I_{1} = I_{0}\). #Step 5: Light passing through the third polarizer#
05

Intensity and polarization after the third polarizer (removing the middle polarizer)

Since there is no middle polarizer, the angle between the first and third polarizers' transmission axes is \(30.0^{\circ}\). Using Malus' law, the intensity of light after passing through the third polarizer is \(I_{3} = I_{1} \cdot \cos^{2}(30^{\circ})\). Again, the transmitted light is linearly polarized with a polarization angle of \(30^{\circ}\) to the y-axis.

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