Learn all of Chapter 8 from zero.
Every lesson explains the idea in plain words, gives you the exact lines to write in the exam, and ends with questions so you know it stuck. Pages 167 to 184 of your book are covered, including every numerical.
Light is a transverse wave, so its vibrations can be forced into one plane. That is polarization. How much light survives a polarizer follows Malus's law and the angle where reflection polarizes light follows Brewster's law. The second half is about gravitational waves, ripples in space-time that LIGO detects with laser interferometry.
If you have 2 hours
- Lessons 1 to 8 (polarization). About 50 minutes. Do the mini quiz at the end of each.
- Lessons 9 and 10 (gravitational waves, LIGO). About 20 minutes.
- Lesson 11. Do the numericals by hand first, then open the solution. About 25 minutes.
- Lesson 12. Read the model answers out loud once. About 15 minutes.
- Lesson 13, the mock test, then Lesson 14 as your last revision.
If you only have 45 minutes
Read the blue "In one line" and orange "Write this in the exam" boxes in every lesson, do Lesson 11, then Lesson 14. These boxes hold most of the marks.
MCQs and short questions ask for definitions and one-line reasons. Long questions want: definition, a labelled figure, the formula with where it comes from, and one example. Numericals want: given, formula, substitution, answer with unit. Write those four things in that order and you will not lose easy marks.
Waves in two minutes
You cannot understand polarization without one picture: a wave has a direction it travels and a direction it vibrates. These two can be related in two ways.
- Transverse wave: particles (or fields) vibrate perpendicular to the direction of propagation. Examples: waves on a rope, light, all electromagnetic waves.
- Longitudinal wave: particles vibrate parallel to the direction of propagation. Examples: sound waves, a pushed spring.
- Light is an oscillating electric field vector E accompanied by a magnetic field vector B at right angles to it. Both are at right angles to the direction of travel. We always follow E and ignore B.
Shake a rope through a vertical slit in a fence. An up-down shake passes. A side-to-side shake is blocked. Light going through a polarizer is that rope. Sound cannot be blocked like this because it vibrates along its travel, so a slit makes no difference to it.
Only transverse waves can be polarized, because only they have a sideways vibration that can be restricted.
A stadium wave moves around the crowd but nobody travels with it, people only move up and down. That is a transverse wave. Sound from a speaker pushes air back and forth along its path, which is longitudinal.
Predict first: Can sound in air be polarized?
What is polarization?
A bulb or the sun sends out light whose E vector points in every possible direction, all around the line of travel. This is unpolarized (ordinary) light. Now put a filter in the way that only lets one direction through. What comes out has E vibrating in one plane only. That light is polarized.
Polarization is the process by which the electric and magnetic vibrations of light waves are restricted to a single plane of vibration. It is a property shown only by transverse waves such as light. It does not occur for longitudinal waves such as sound.
Key terms
- Plane of polarization: the plane containing the direction of vibration of the particles of the medium and the direction of propagation of the wave.
- Unpolarized light: vibrations in all possible planes. Sources: sunlight, incandescent bulbs, fluorescent bulbs, light from a candle or fire.
- Polarized (plane polarized) light: vibrations confined to one plane only.
- Polarizing filter / polaroid sheet: a material with a transmission axis. Only the part of E parallel to that axis passes. All other vectors are blocked.
How can unpolarized light be polarized? (3 methods)
- Passing it through a polarizing filter (for example a polaroid sheet).
- Using a polarizing beam splitter.
- Using certain optical crystals such as calcite or quartz.
The most common method is the polarizing filter or polaroid sheet. Light can also be polarized naturally by reflection, refraction and scattering.
PBC: Polaroid, Beam splitter, Crystals.
Polarization squeezes light's vibrations from every direction down to one plane.
Wear polarized sunglasses and look at a phone or laptop screen. Tilt your head or the phone and the screen goes dark at one angle. LCD screens already give out polarized light.
Predict first: Why does the screen go dark when you turn it 90°?
Types of polarization
Look straight down the beam, toward the source. The tip of the E vector draws a shape. The shape gives the name.
1. Linear polarization
The electric field vector oscillates in a single plane. Example: light passing through a polarizing filter, like sunglasses (Fig. 8.4a).
2. Circular polarization
The electric field rotates in a circle. Clockwise is right-handed. Counterclockwise is left-handed. The two components of E differ in phase by 90° (Fig. 8.4b). Example: light reflected off a CD or a DVD.
3. Elliptical polarization
A combination of linear and circular. The tip of E traces an ellipse. This happens when the two components Eₓ and E_y are not equal, or they differ in phase by an arbitrary angle θ (Fig. 8.4c). Example: light passing through a stress plate or a waveplate.
| Type | E vector | Phase between Eₓ and E_y | Example |
|---|---|---|---|
| Linear | stays in one plane | none (one direction) | polarizing filter, sunglasses |
| Circular | rotates, clockwise or counterclockwise | 90° | CD / DVD |
| Elliptical | traces an ellipse | arbitrary θ, or unequal components | stress plate, waveplate |
LCE in the order of the shapes: Line, Circle, Ellipse. CD gives Circle. Waveplate gives Ellipse.
Linear, circular and elliptical describe the path the E vector traces when seen end-on.
Cinema 3D glasses use circular polarization: one lens passes clockwise light, the other anticlockwise, so each eye gets its own picture. It still works when you tilt your head, which linear glasses cannot do.
Predict first: In the 3D model, which setting makes a circle?
Polarizer and analyzer: the experiment that proves light is transverse
An ordinary bulb gives unpolarized light. Two polaroid sheets are placed one behind the other.
- Polaroid 1 is the polarizer. It makes the light plane polarized.
- Polaroid 2 is the analyzer. It tests that light. An analyzer is just a polarizer placed after a polarizer.
What you see as you slowly rotate the analyzer
- Axes parallel: light passes and is brightest.
- While rotating: the light gets dimmer and dimmer.
- Axes perpendicular (crossed): light disappears completely.
- Keep rotating: it reappears and is brightest again when the axes are parallel again.
This experiment proves that light waves are transverse. If light waves were longitudinal, they would never disappear even if the two polaroids were mutually perpendicular.
Light vanishing at crossed polaroids proves it is transverse.
Hold two pairs of polarized sunglasses in front of a light and turn one. Bright, dim, then black. Turn it further and the light comes back.
Predict first: What happens at 90° between the two lenses?
Polarization by reflection and Brewster's law
In 1808, Malus found that light reflected from a plane sheet of glass is polarized. If you look at that reflected light through a polaroid and slowly rotate it, the light is almost extinguished at one certain orientation.
- Best angle of incidence for glass: about 57°. At that angle the reflected ray is practically plane polarized.
- Light reflected from a table top gets darker when seen through a rotating polaroid. So it is only partially plane polarized.
- Light reflected from a smooth water surface is partially polarized parallel to the surface.
Brewster's law
The particular angle of incidence on a transparent medium for which the reflected light is almost plane polarized is the polarizing angle θₚ. At that angle the reflected and refracted beams are at right angles to each other.
Derivation (learn this, it is asked in long questions)
- Reflected and refracted rays are at 90°, so θₚ + θᵣ = 90°, which gives θᵣ = 90° − θₚ.
- Snell's law: n₁ sinθₚ = n₂ sinθᵣ.
- Put in θᵣ: n₁ sinθₚ = n₂ sin(90° − θₚ) = n₂ cosθₚ.
- Divide both sides by n₁ cosθₚ: sinθₚ / cosθₚ = n₂ / n₁.
- If medium 1 is air, then n₁ = 1 and n₂ = n, so tan θₚ = n.
- The tangent of the polarizing angle equals the refractive index of the medium. The angle is also called the Brewster angle θ_B, so tan θ_B = n.
Do not write sin or cos. It is tan. And it is the angle of incidence that is θₚ, while θᵣ is found as 90° minus it.
"Tan is n." and "Brewster means 90° between the two rays."
Glare on a lake hides the fish. A polarizing filter removes it, because light reflected near the Brewster angle is strongly polarized. For water, n = 1.33, so θ = tan⁻¹(1.33) ≈ 53°.
Predict first: Light hits glass (n = 1.5). At what angle is reflected light fully polarized?
Malus's law
Polarized light of intensity I₀ reaches an analyzer. The axis of the analyzer makes an angle θ with the polarization direction of the light. How much comes through?
Malus's law states that the intensity I of plane polarized light after passing through an analyzer is directly proportional to the square of the cosine of the angle θ between the transmission axes of the analyzer and the polarizer.
I₀ is the intensity of the incident polarized light. I is the intensity that comes out of the analyzer.
Where it comes from (derivation)
- Polarized light of amplitude A₀ strikes the analyzer at angle θ. It is resolved into two components: A₀cosθ (along the axis) and A₀sinθ (across it).
- The A₀sinθ part is absorbed. Only A₀cosθ passes. So the transmitted amplitude is A = A₀ cosθ (8.2).
- Intensity is proportional to amplitude squared: I ∝ A², or I = kA². Taking k = 1, I = A².
- So I = A₀² cos²θ. Since I₀ = A₀², we get I = I₀ cos²θ.
The two extreme cases
| Angle | cos²θ | Result |
|---|---|---|
| θ = 0° (axes parallel) | 1 | I = I₀. Full intensity passes. |
| θ = 90° (crossed) | 0 | I = 0. Light is completely extinguished. |
Malus's law needs polarized light going in. If the light entering the first polaroid is unpolarized, the first sheet always cuts it to I₀/2 first (the average of cos² over all directions is ½). Only then use cos²θ for the next sheet. Look at the word "unpolarized" in the question.
"Cos² = Common sense." Parallel (0°) means everything passes, and cos 0° = 1. Crossed (90°) means nothing passes, and cos 90° = 0. If you wrote sin², the logic would flip.
A camera polarizing filter darkens the blue sky as you rotate it. Rotating changes the angle θ between the filter axis and the sky's polarization, and I = I₀cos²θ.
Predict first: Polarized light, analyzer at 60°. What fraction passes?
Optical activity
Optical activity is the ability of a substance to rotate the plane of polarization of light passing through it. The rotation is detected with a polarizer or analyzer (Fig. 8.8).
Many crystals and solutions rotate the plane of polarization. Such substances are said to be optically active.
- Examples: quartz crystals, cinnabar (HgS), sugar water, insulin and collagen. Organic substances such as sugar and tartaric acid show optical rotation in a solution.
- Why it happens: the asymmetric shape of the molecules, such as a helical shape.
- Use: because the rotation depends on how much substance is present, it is used to determine the concentration of optically active solutions. A few millimetres of such crystal can rotate the plane by many degrees.
The amount and direction of rotation depend on four things
- The type of substance.
- The concentration (amount of substance in a given quantity of mixture or solution).
- The distance the light travels through it.
- The wavelength of light.
TCDW: Type, Concentration, Distance, Wavelength.
Picture: the sugar solution in Fig. 8.8 turns the vertical polarization, so the analyzer set to block the original direction now lets some light through. You turn the analyzer by angle θ to get darkness again. That angle tells you the concentration.
Optically active substances twist the plane of polarization, and the twist measures concentration.
Sugar solutions rotate the plane of polarized light. Labs and the food industry use a polarimeter to find how much sugar is in syrup or juice from the angle the analyzer must be turned.
Predict first: If the tube is twice as long, what happens to the rotation?
Why polarization matters (uses)
Polaroid (also called a polarizing filter): certain transparent crystalline materials like tourmaline and calcite confine light vibrations to one plane. Such materials are called polaroids. They have high directionality in their crystal structure.
The two main applications
1. Sky photography
- A camera that photographs clouds has a polaroid fitted. Light from the sky is polarized by it.
- It reduces glare and haze, which are produced by scattering of light from small particles and molecules in the atmosphere.
- It improves contrast by blocking excess bright white light and letting other colours through. The result is a brighter, more detailed image.
- Glare means unwanted light that interferes with vision.
2. Stress analysis of materials
- When a material (plastic, metal, glass) is stressed, its molecular structure changes and affects how light interacts with it.
- Polarized light shone through a transparent stressed material forms interference fringes. They give information about the material.
- Researchers can find the stress patterns, identify weaknesses or defects, study optical properties, and see how it will behave under different conditions.
- This technique is called photoelasticity. It is used in engineering, materials science and quality control.
Other fields (the five headings in the book)
| Field | Use |
|---|---|
| Optics and photonics | polarized sunglasses, LCD screens, optical communication |
| Imaging and microscopy | less glare, better contrast |
| Medical | cancer diagnosis, tissue imaging, laser surgery (tell tissue types apart) |
| Astronomy | polarization of light from distant stars, cosmic microwave background |
| Miscellaneous | biology, chemistry, communication systems |
Why polaroid sunglasses beat ordinary ones
Glare from a road or a lake is mostly polarized parallel to the surface (horizontal). Polaroid lenses have a vertical axis, so they block light vibrating parallel to the highway while letting other light through. Ordinary sunglasses only darken everything equally.
The two main applications: Sky and Stress (Sky photography, Stress analysis).
Polarizers cut glare and haze for cameras and sunglasses, and reveal stress in transparent materials.
Put a clear plastic ruler or tape between two crossed polarizers: coloured stress patterns appear. Engineers use the same idea (photoelasticity) to see where a plastic or glass part is under strain.
Predict first: Why do stressed plastics show colours?
Gravitational waves
A gravitational wave is a stretching and compressing of space-time. It can be observed by measuring the change in length between two objects. Gravitational waves are ripples in the fabric of space-time, produced by violent cosmic events like colliding black holes or neutron stars, that travel at the speed of light carrying information about their source.
Start with the pond
Throw a stone into a pond. Ripples travel outward on the water and carry information about the stone. Gravitational waves are the same idea. The pond is space-time, the stone is a cosmic event, the shore is a gravitational wave observatory.
Why do they exist? Einstein's picture of gravity
Gravitational waves are a prediction of Einstein's general theory of relativity (predicted in 1916). According to it, gravity is not a force but a curvature of space-time caused by massive bodies. Think of a heavy ball making a dent in a mattress. Heavy things wrap space around them, and that is why we feel gravity. When massive bodies accelerate, the dent ripples outward.
| Fact | Value |
|---|---|
| Predicted by | Albert Einstein, 1916 (German-born theoretical physicist, 1879 to 1955) |
| First detected | 2015 (14 September), announced in 2016 |
| Source of that first signal | collision of two black holes, waves travelled 1.3 billion years |
| Detector | LIGO, USA |
| Nobel Prize | 2017: Rainer Weiss, Barry C. Barish, Kip S. Thorne |
| Pakistani link | Dr. Nergis Mavalvala, Pakistani-American astrophysicist at MIT, member of the team |
| Speed | speed of light, 3 × 10⁸ m/s |
| Amplitude | about 10⁻²¹ to 10⁻²² m, incredibly tiny |
Binary systems: the main source
A binary system (BS) is a system of two compact objects such as black holes, neutron stars or white dwarfs orbiting each other and emitting gravitational waves. As the masses orbit and accelerate, their gravitational intensity fluctuates and waves radiate outward in all directions. The waves are not bound by the system's gravity. They travel freely through space-time at the speed of light and weaken in intensity with distance from the source. Mergers of binary systems (two massive objects colliding and uniting into one more massive object) are among the most intense cosmic events.
Every accelerating object makes gravitational waves, even cars and aeroplanes, but their masses and accelerations are too small to detect with our instruments.
The four basic types
| Type | Source | Key point |
|---|---|---|
| Continuous | a single massive spinning object such as a neutron star; white dwarf binaries | constant frequency and amplitude |
| Compact binary inspirational | binary neutron stars, binary black holes, or neutron star and black hole orbiting each other | frequency and amplitude increase gradually. This is the 2015 detection type (colliding bodies) |
| Burst | violent events: supernovae, gamma-ray bursts, cosmic strings | short and violent |
| Stochastic | superposition of many weak sources, such as distant binary systems | weak random signals, the most difficult to detect |
CCBS. "Continuous Constant, Compact Climbs, Burst Blasts, Stochastic Scattered."
What a gravitational wave depends on
- Masses of the objects
- Orbital period and frequency
- Eccentricity of the orbit, e (how much an orbit differs from a perfect circle)
| e | Orbit |
|---|---|
| e = 0 | circular |
| 0 < e < 1 | elliptical |
| e = 1 | parabolic trajectory |
| e > 1 | hyperbolic trajectory |
Space-time distortion and tidal forces
A gravitational wave passing through a body with mass makes the body periodically stretch and compress, in the direction perpendicular to the wave's propagation. This is "space-time distortion". The effect is called tidal forces and it is similar to how tides on Earth are caused by the gravitational pull of the Moon and Sun. The amount depends on the strength of the wave and on the mass and size of the body. It is an important prediction of general relativity.
The distortion is only about 10⁻²¹ to 10⁻²² m, so it needs extremely sensitive instruments. Despite that, gravitational waves give a unique window into the universe: they let us study strong-field gravity, test general relativity and explore the universe in ways that were impossible before.
Page 181 prints 1 light year as "9.5 billion km". The real value is about 9.5 trillion km (9.46 × 10¹² km). If a question asks, write what your teacher taught, but do not use that figure in any calculation.
Gravitational waves are ripples in space-time from accelerating massive objects, and they stretch and squeeze anything they pass through by a tiny amount.
Put a bowling ball on a trampoline and roll a marble nearby. The marble curves, not because of a force, but because the sheet is bent. That is Einstein's picture of gravity. GPS satellites even need relativity corrections to keep their clocks right.
Predict first: In the 3D model, what happens to the ripples as the masses get closer?
Interferometer and LIGO
An interferometer is an optical tool used in detecting gravitational waves. It is a very sensitive detection device that uses the interference of LASER beams. An interferometer that detects gravitational waves is a highly sensitive instrument that uses laser light to measure tiny changes in the distance between mirrors caused by gravitational waves passing through the detectors.
LIGO = Laser Interferometer Gravitational-wave Observatory (USA). LASER = Light Amplification by Stimulated Emission of Radiation.
LIGO versus a conventional interferometer
- LIGO is 1000 times larger than a conventional device.
- LIGO uses a laser. A conventional interferometer has a normal light source.
The picture (Fig. 8.9)
The 7 basic components
- Laser produces a stable, high intensity beam of light.
- Power recycling mirror reflects laser light that has already travelled through the instrument back into the interferometer. This is why it is called "recycling".
- Beam splitter divides the laser beam into two perpendicular beams.
- Mirrors reflect the beams, creating two perpendicular arms.
- Fabry-Perot cavity is two mirrors facing each other. Its purpose is to enhance (lengthen) the path length.
- Photodetectors measure the returning beams and detect tiny phase shifts (if any).
- Arm cavities enhance the laser light, increasing sensitivity.
"Laser Power Beam Mirrors Fabry Photo Arms" = Let Physics Be My Favourite Personal Art.
How it works
- A laser beam is split into two perpendicular beams. Each travels down one of two identical arms.
- The beams bounce off mirrors at the ends and return to the starting point, where they are recombined.
- If a gravitational wave passes through, it causes a tiny disturbance in the distance between the mirrors. This produces a phase shift between the two beams.
- When the beams recombine they create an interference pattern, measured by a photodetector. The gravitational wave changes this pattern, so the detector senses the wave.
Virgo, the second detector
- Located at Cascina, near Pisa, Italy, run by the European Gravitational Observatory (EGO).
- Also an interferometer, with two arms of 3 km. LIGO's arms are 4 km.
- Named after the Virgo constellation (visible in March, April and May). The Virgo cluster is a group of about 1,500 galaxies about 50 million light years away.
- Involved in detecting gravitational wave events, with the first detection in 2017.
The first LIGO detection (14 September 2015) came from a collision of two black holes. The waves took 1.3 billion years to arrive. The 2017 Nobel Prize went to Rainer Weiss, Barry C. Barish and Kip S. Thorne.
LIGO splits a laser into two arms, and a passing gravitational wave changes the arm lengths slightly, which shifts the interference pattern.
Soap bubble colours come from light waves interfering. LIGO uses the same idea with lasers: two beams are recombined and a tiny change in arm length changes the brightness at the detector. LIGO detects length changes far smaller than a proton.
Predict first: In the 3D model, why does one arm stretch while the other shrinks?
Numericals: the recipe, then every solved example
- Given with units and what is to find.
- Formula.
- Substitute the numbers.
- Answer with the unit.
| If the question has | Use |
|---|---|
| intensity, angle between axes | I = I₀cos²θ |
| amplitude, angle | A = A₀cosθ |
| polarizing angle, refractive index | tanθₚ = n |
| find refraction angle at polarizing angle | θᵣ = 90° − θₚ |
| wavelength or frequency of gravitational waves | f = c/λ, λ = c/f, c = 3×10⁸ m/s |
| "reduced by x%" | transmitted fraction = 1 − x/100 |
| "unpolarized" light into a polarizer | first halve: I = I₀/2 |
Practice generator
Solved examples from the book
In Example 8.3 the book writes the unit as "N m⁻²" in the working. Intensity is in W m⁻². Write W m⁻² in your answer.
Sunglasses stacked on a phone screen is a numerical you can see. Unpolarized 100 W/m² goes through two polaroids at 60°.
Predict first: What intensity comes out?
Model answers for every exercise question
Tap a question to open its answer. First try to answer in your head, then check. For long questions also draw the figure named in the answer.
Short answer questions
Constructed response questions
Comprehensive questions
Mock test
25 random questions from everything above plus your book's MCQs. Tap the answer. Aim for 23 or more.
Q4 (angle between light wave and analyzer) and Q5 (purpose of analyzer) can be marked differently in some answer keys. Ask your teacher. Q1 in your book has (c) circled, but the correct answer is (b).
Last 10 minutes: everything on one page
Definitions
Polarization: restricting E and B vibrations of light to one plane. Only transverse waves.
Plane of polarization: contains vibration direction and propagation direction.
Optical activity: rotation of the plane of polarization by a substance.
Gravitational wave: stretching and compressing of space-time, ripples at the speed of light.
Interferometer: laser-interference tool to detect tiny length changes.
Formulas
I = I₀cos²θ
A = A₀cosθ
tanθₚ = n₂/n₁ (air: tanθₚ = n)
θₚ + θᵣ = 90°
f = c/λ, c = 3×10⁸ m/s
Unpolarized in: I₀/2 first
Lists
3 polarizing methods (PBC): Polaroid, Beam splitter, Crystals.
3 types (LCE): Linear, Circular (90°, CD), Elliptical (waveplate).
Optical activity (TCDW): Type, Concentration, Distance, Wavelength.
2 main uses: Sky photography, Stress analysis.
4 GW types (CCBS): Continuous, Compact binary, Burst, Stochastic (hardest).
Numbers and names
Malus: 1808. Brewster angle for glass ≈ 57°. Water ≈ 53°.
Einstein predicted GWs: 1916. Detected: 2015 (14 Sept). Announced: 2016.
Source: 2 black holes, 1.3 billion years away. Nobel 2017: Weiss, Barish, Thorne.
LIGO arms 4 km (USA). Virgo arms 3 km (Cascina, Italy). LIGO is 1000× larger than a conventional one.
GW amplitude ≈ 10⁻²¹ to 10⁻²² m. Dr. Nergis Mavalvala (MIT).
e = 0 circle, 0<e<1 ellipse, e = 1 parabola, e > 1 hyperbola.
Answers to book numericals
8.1: I₀/2
8.2: 50 W/m²
8.3: 8.66 units
8.4: 67.5° (≈ 67.2° exact)
8.5: 7.5 and 5.6 W/m²
8.6: 45°
8.7: 56.3° and 33.7°
8.8: 53°
8.9: 21%
8.10: 0.25
8.11: 100 Hz
Three lines to win marks
1. Parallel gives bright, crossed gives dark. This proves light is transverse.
2. At Brewster's angle, reflected and refracted rays are at 90°.
3. LIGO: split laser, two arms, mirrors, recombine, phase shift means a gravitational wave.
You now have the full chapter. Take a breath, read the question twice, write the definition and the formula first, and show your steps.