Polarization Examples: 44 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Polarization.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Polarization is the restriction of a transverse wave's oscillations to one direction or plane.

A polarizing filter only lets through certain vibration directions.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Polarization starts by following rays or wavefronts through boundaries, materials, and image locations.

Common stuck point: Students often know a formula related to polarization but skip the recognition step: Am I tracking how light travels through space or materials, including boundary rules and image location when needed? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Am I tracking how light travels through space or materials, including boundary rules and image location when needed?

Worked Examples

Example 1

medium

Linearly polarized light of intensity

I_0

hits a polarizer at

\theta = 30^\circ

to its polarization. Find

I

Answer

I = 0.75 I_{0}

First step

Use Malus's law:

I = I_0 \cos^2\theta

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Example 2

medium

Unpolarized light

I_0

passes a polarizer and then a second polarizer rotated

45^\circ

. Find the output intensity.

Example 3

medium

Light reflects off glass at Brewster's angle. What is the polarization state of the reflected ray?

Example 4

medium

A quarter-wave plate converts linearly polarized light at

45^\circ

to its slow axis into which polarization state?

Example 5

medium

An LCD display passes linearly polarized light through a twisted liquid-crystal cell that rotates polarization by

90^\circ

when off. The light then meets a second polarizer crossed with the first. What does the viewer see?

Example 6

hard

Unpolarized light

I_0

passes three polarizers oriented at

0^\circ

45^\circ

, and

90^\circ

relative to the first. Find the final intensity.

Example 7

hard

Unpolarized light

I_0

passes four polarizers with successive angles

0^\circ

30^\circ

60^\circ

90^\circ

. Find the final intensity (each angle measured from the previous polarizer's axis).

Example 8

hard

Polarized light at intensity

I_0

hits two polarizers. The first is at

\theta

to the polarization; the second is crossed with the first. For what value of

\theta

is the final intensity maximized?

Example 9

challenge

Unpolarized light

I_0

passes N polarizers, each rotated by

90^\circ/N

from the previous. Derive the output intensity for

N = 5

and compare with the crossed case (

N = 2

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Polarization restricts a wave's oscillations to one direction. What type of wave can be polarized?

Example 2

easy

Unpolarized light of intensity

I_0

passes through one ideal polarizing filter. What intensity emerges?

Example 3

easy

Light is already polarized vertically and hits a polarizer with a vertical axis. How much passes?

Example 4

easy

Vertically polarized light hits a polarizer with a horizontal axis (

\theta = 90^\circ

). How much passes?

Example 5

easy

Is polarization the same phenomenon as reflection?

Example 6

easy

Polarized sunglasses reduce glare from a horizontal surface. What axis orientation do they use?

Example 7

easy

Can a longitudinal sound wave in air be polarized?

Example 8

easy

Two polarizers are crossed at

90^\circ

. Unpolarized light enters. What comes out?

Example 9

medium

Polarized light of intensity

I_0

passes a polarizer at

\theta = 60^\circ

to its polarization. Find the output intensity.

Example 10

medium

Unpolarized light

I_0

passes a polarizer, then a second polarizer at

30^\circ

to the first. Find the final intensity.

Example 11

medium

Polarized light at

I_0

passes a polarizer and emerges at

I_0/4

. Find the angle

\theta

between the light and the axis.

Example 12

medium

Light reflecting off water becomes partially polarized. In which plane is the reflected light mostly polarized?

Example 13

medium

Through one polarizer, unpolarized light becomes

I_0/2

. A student claims rotating that single polarizer changes the output. Is the student correct?

Example 14

medium

Polarized light

I_0

45^\circ

to a polarizer. Find the transmitted intensity.

Example 15

challenge

Three polarizers: first vertical, third horizontal (crossed), middle at

45^\circ

. Unpolarized

I_0

enters. Find the final intensity.

Example 16

challenge

Polarized light passes two polarizers. The first is aligned with the light; the second is at angle

\theta

. The final intensity is

3I_0/8

. Find

\theta

Example 17

challenge

Unpolarized light

I_0

passes three polarizers at

0^\circ

30^\circ

, and

60^\circ

(each angle relative to the previous). Find the final intensity.

Example 18

medium

Unpolarized light

I_0

passes a polarizer, then a second at

90^\circ

, then is measured. What is the output, and why?

Example 19

medium

Polarized light

I_0

passes a polarizer at

\theta=30^\circ

. Find the transmitted intensity.

Example 20

medium

Polarized light loses three-quarters of its intensity through a polarizer. Find the angle

\theta

Example 21

easy

Unpolarized light of intensity

80 \text{ W/m}^2

passes a single ideal polarizer. Find the transmitted intensity.

Example 22

easy

Linearly polarized light meets a polarizer whose axis is aligned with its polarization. What fraction of the intensity passes?

Example 23

easy

Vertically polarized light hits a polarizer rotated

90^\circ

from vertical. Find the transmitted intensity in terms of

I_0

Example 24

easy

Light from an incandescent bulb is unpolarized. What is the degree of polarization of the source?

Example 25

easy

Identify whether each is polarizable: (a) light, (b) sound in air, (c) waves on a stretched string.

Example 26

medium

Polarized light loses half its intensity through one polarizer. Find the angle

\theta

between the polarization and the axis.

Example 27

medium

Find Brewster's angle for light going from air (

n_1 = 1.00

) into water (

n_2 = 1.33

Example 28

medium

Light passes two polarizers and the output is

0.10\,I_0

when the input was polarized along the first polarizer's axis. Find the angle

\theta

between the polarizers.

Example 29

medium

Polarized light of intensity

400 \text{ W/m}^2

passes a polarizer at

\theta = 60^\circ

to its polarization. Find the output intensity.

Example 30

medium

Polarized light enters a polarizer and 1/3 of the intensity emerges. Find the angle

\theta

Example 31

medium

A 200 W/m^2 unpolarized beam passes through two ideal polarizers, the second at

60^\circ

to the first. Find the final intensity.

Example 32

hard

Polarized light of intensity

I_0

passes a single polarizer; the output is

0.4\,I_0

. Find the angle

\theta

to one decimal place.

Example 33

hard

Light from a partially polarized source has intensity

I_{\max} = 90

when measured through a polarizer aligned with its dominant axis and

I_{\min} = 10

when crossed. Find the degree of polarization

P = (I_{\max}-I_{\min})/(I_{\max}+I_{\min})

Example 34

hard

Find Brewster's angle for light passing from water (

n_1 = 1.33

) into glass (

n_2 = 1.52

Example 35

challenge

Polarized light of intensity

I_0

hits three polarizers at successive angles

\theta_1 = 20^\circ

\theta_2 = 40^\circ

\theta_3 = 70^\circ

measured from the original polarization direction. Find the final intensity.

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Related Concepts

Transverse Wave

Background Knowledge

These ideas may be useful before you work through the harder examples.

transverse wave