Polarization Examples in Physics

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.

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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 I0I_0 hits a polarizer at ฮธ=30โˆ˜\theta = 30^\circ to its polarization. Find II.

Answer

I=0.75โ€‰I0I = 0.75\,I_0

First step

1
Use Malus's law: I=I0cosโก2ฮธI = I_0 \cos^2\theta.

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Example 2

medium
Unpolarized light I0I_0 passes a polarizer and then a second polarizer rotated 45โˆ˜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 45โˆ˜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 90โˆ˜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 I0I_0 passes three polarizers oriented at 0โˆ˜0^\circ, 45โˆ˜45^\circ, and 90โˆ˜90^\circ relative to the first. Find the final intensity.

Example 7

hard
Unpolarized light I0I_0 passes four polarizers with successive angles 0โˆ˜0^\circ, 30โˆ˜30^\circ, 60โˆ˜60^\circ, 90โˆ˜90^\circ. Find the final intensity (each angle measured from the previous polarizer's axis).

Example 8

hard
Polarized light at intensity I0I_0 hits two polarizers. The first is at ฮธ\theta to the polarization; the second is crossed with the first. For what value of ฮธ\theta is the final intensity maximized?

Example 9

challenge
Unpolarized light I0I_0 passes N polarizers, each rotated by 90โˆ˜/N90^\circ/N from the previous. Derive the output intensity for N=5N = 5 and compare with the crossed case (N=2N = 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 I0I_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 (ฮธ=90โˆ˜\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 90โˆ˜90^\circ. Unpolarized light enters. What comes out?

Example 9

medium
Polarized light of intensity I0I_0 passes a polarizer at ฮธ=60โˆ˜\theta = 60^\circ to its polarization. Find the output intensity.

Example 10

medium
Unpolarized light I0I_0 passes a polarizer, then a second polarizer at 30โˆ˜30^\circ to the first. Find the final intensity.

Example 11

medium
Polarized light at I0I_0 passes a polarizer and emerges at I0/4I_0/4. Find the angle ฮธ\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 I0/2I_0/2. A student claims rotating that single polarizer changes the output. Is the student correct?

Example 14

medium
Polarized light I0I_0 at 45โˆ˜45^\circ to a polarizer. Find the transmitted intensity.

Example 15

challenge
Three polarizers: first vertical, third horizontal (crossed), middle at 45โˆ˜45^\circ. Unpolarized I0I_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 ฮธ\theta. The final intensity is 3I0/83I_0/8. Find ฮธ\theta.

Example 17

challenge
Unpolarized light I0I_0 passes three polarizers at 0โˆ˜0^\circ, 30โˆ˜30^\circ, and 60โˆ˜60^\circ (each angle relative to the previous). Find the final intensity.

Example 18

medium
Unpolarized light I0I_0 passes a polarizer, then a second at 90โˆ˜90^\circ, then is measured. What is the output, and why?

Example 19

medium
Polarized light I0I_0 passes a polarizer at ฮธ=30โˆ˜\theta=30^\circ. Find the transmitted intensity.

Example 20

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

Example 21

easy
Unpolarized light of intensity 80ย W/m280 \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 90โˆ˜90^\circ from vertical. Find the transmitted intensity in terms of I0I_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 ฮธ\theta between the polarization and the axis.

Example 27

medium
Find Brewster's angle for light going from air (n1=1.00n_1 = 1.00) into water (n2=1.33n_2 = 1.33).

Example 28

medium
Light passes two polarizers and the output is 0.10โ€‰I00.10\,I_0 when the input was polarized along the first polarizer's axis. Find the angle ฮธ\theta between the polarizers.

Example 29

medium
Polarized light of intensity 400ย W/m2400 \text{ W/m}^2 passes a polarizer at ฮธ=60โˆ˜\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 ฮธ\theta.

Example 31

medium
A 200 W/m^2 unpolarized beam passes through two ideal polarizers, the second at 60โˆ˜60^\circ to the first. Find the final intensity.

Example 32

hard
Polarized light of intensity I0I_0 passes a single polarizer; the output is 0.4โ€‰I00.4\,I_0. Find the angle ฮธ\theta to one decimal place.

Example 33

hard
Light from a partially polarized source has intensity Imaxโก=90I_{\max} = 90 when measured through a polarizer aligned with its dominant axis and Iminโก=10I_{\min} = 10 when crossed. Find the degree of polarization P=(Imaxโกโˆ’Iminโก)/(Imaxโก+Iminโก)P = (I_{\max}-I_{\min})/(I_{\max}+I_{\min}).

Example 34

hard
Find Brewster's angle for light passing from water (n1=1.33n_1 = 1.33) into glass (n2=1.52n_2 = 1.52).

Example 35

challenge
Polarized light of intensity I0I_0 hits three polarizers at successive angles ฮธ1=20โˆ˜\theta_1 = 20^\circ, ฮธ2=40โˆ˜\theta_2 = 40^\circ, ฮธ3=70โˆ˜\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