A full lesson: a cartoon movie to start, then a wave on a string through a slot, polaroids and Malus' law, chains of three polaroids, why the blue sky is polarised, Brewster's angle for reflected glare, NEET shortcuts, problems to solve yourself, and 12 MCQs.
Glare from water vibrates mostly sideways. A polaroid lets through only one direction of vibration, so it removes the glare. Two crossed polaroids block all light. That only makes sense if light is a transverse wave.
Priya and Arjun are at the lake on a sunny Sunday. The water throws so much glare into Priya's eyes that she cannot see the fish. Raju bhaiya at the stall hands her a pair of "polaroid" sunglasses for ₹299. She puts them on and the glare is gone, while the trees and the sky look almost the same, just a little darker.
Then he does a trick: he holds two lenses one behind the other and slowly turns one. The overlap gets darker and darker and at 90° becomes completely black. Arjun turns his phone while wearing the glasses: the screen goes dark too. Ordinary tinted glass never does this. What is special about these lenses, and what does it tell us about light?
Try at home: look at your phone or laptop screen through polarised sunglasses (or 3D-cinema glasses) and slowly tilt your head. Also look at the glare on a wet road or a table top and tilt the glasses.
"A polaroid is like a fence with vertical gaps. It lets through only the vibration that fits the gaps. If light can be blocked like this, its vibration must be sideways: light is a transverse wave."
"Could you make 'polaroid earplugs' that block sound the same way?"
No. Sound in air is longitudinal: the air moves back and forth along the direction of travel. There is no sideways direction to select, so sound cannot be polarised.
Shake the end of a long string. The wave y(x, t) = a sin(kx − ωt) travels along x, while each bit of string moves along y: a transverse wave. Because the vibration stays in the x–y plane, it is linearly (plane) polarised. Now thread the string through a slot in a board and turn the slot.
y-polarised: vibrates in the x–y plane. z-polarised: in the x–z plane. Also called linearly or plane polarised.
Light from the sun or a bulb: the vibration direction changes about 10⁸ times a second, with no favourite.
Sound in air passes any slot unchanged. This is how polarisation proves light is transverse.
"For a light wave, the thing that vibrates is the electric field E. When we say 'the direction of polarisation', we mean the direction in which E oscillates."
A polaroid is a plastic sheet with long-chain molecules all lined up in one direction. Electric fields along the chains drive electrons along them and are absorbed. The component perpendicular to the chains passes: that direction is the pass-axis. Drag the orange analyser P₂ (or use the slider) to turn it.
The direction of E is random, and the average of cos²θ over all directions is ½. Turning P₁ changes nothing.
θ is the angle between the polarisation (P₁'s axis) and P₂'s pass-axis.
θ = 0°: all passes. θ = 90° (crossed): nothing passes. Two dark positions per full turn of P₂.
"Malus' law is for light that is ALREADY polarised. For unpolarised light, the first polaroid always gives one half. Apply the half only once, then cos² for every polaroid after it."
"Through which angle must the analyser be turned from the bright position to cut the light to one quarter?"
cos²θ = 1/4, so cos θ = 1/2 and θ = 60°.
P₁ and P₃ are crossed, so no light gets through. Now slide a third polaroid P₂ between them at angle α. Surprisingly, light comes back! Turn P₂ and watch the intensity bars.
The one and only "half".
Malus with the angle between P₁ and P₂.
The angle between P₂ and P₃ is 90° − α.
Maximum I₀/8 at α = 45°. Zero at α = 0° or 90° (P₂ lines up with P₁ or P₃).
"Each polaroid only cares about the polaroid just before it. Write the angle between neighbours, square the cosine, and multiply along the chain."
Syllabus note: polarisation by scattering is enrichment; it is not in the rationalised NCERT text.
Sunlight shakes the electrons in air molecules, and these re-radiate (scatter) light in all directions. An oscillating charge sends no light along its own line of vibration. Drag Priya around the molecule to change the scattering angle ψ, and turn her polaroid.
Unpolarised: both directions of vibration, equally.
The in-page oscillation points straight at the observer, and a charge does not radiate along its own line of motion.
Turning the polaroid makes that part of the sky darker and brighter. Bees and some birds navigate using this.
"Scattering explains two facts at once: the sky is blue because short wavelengths scatter much more, and it is polarised because of the 90° rule."
Syllabus note: Brewster’s law is in the NEET syllabus but no longer in the rationalised NCERT text. Learn it for NEET.
When light reflects from water or glass, the vibration in the plane of incidence (blue arrows ↔) reflects less than the vibration perpendicular to it (red dots •). At one special angle, the Brewster angle, the ↔ part is not reflected at all: the reflected light is completely plane polarised. Drag the lamp or use the slider.
Glass 56.3°, water 53.1°, diamond 67.5°. (Not the critical angle: that is sin i_c = 1/n.)
Reflected and refracted rays are perpendicular. From Snell: sin i_B = n cos i_B ⇒ tan i_B = n.
Polaroid glasses have a vertical pass-axis, which blocks the horizontally vibrating glare from roads and water.
Only at the first polaroid. Every later polaroid: × cos² (angle from the previous one).
cos²30° = 3/4, cos²45° = 1/2, cos²60° = 1/4, cos²90° = 0.
Maximum I₀/8 at 45°. For a middle polaroid rotating at ω: output varies as sin²2ωt.
A question asking for the amplitude ratio wants cos θ, not cos²θ.
If the refracted angle is given, i_B = 90° − r first, then n = tan i_B.
Along the beam (ψ = 0°) the scattered light is unpolarised.
"In polaroid chains, NEET hides the angle: they give the angle of each polaroid from the FIRST one. Always convert to the angle between neighbours before using cos squared."
Cover the solution, write the formula, and solve on paper. Open the hint only if you are stuck for more than a minute.
Plane polarised light of intensity 32 W/m² falls on an analyser. Find the transmitted intensity when the angle between the polarisation and the pass-axis is 30° and when it is 60°.
The light is already polarised: no factor ½. I = I₀ cos²θ.
Unpolarised light of intensity 64 W/m² passes through two polaroids whose pass-axes are at 60°. Find the final intensity.
½ at the first polaroid, then cos²60° at the second.
Unpolarised light of intensity 80 W/m² falls on two crossed polaroids. A third polaroid is placed between them with its pass-axis at 30° to the first. Find the final intensity. At what angle would it be largest, and how large?
Angles between neighbours: 30° and 60°. Or use (I₀/8) sin²2α.
Find the polarising (Brewster) angle for glass of refractive index 1.5 and the angle of refraction at this angle of incidence.
tan i_B = n, and i_B + r = 90°.
At what height of the sun above the horizon is the sunlight reflected from a calm lake (n = 4/3) completely polarised?
i_B is measured from the vertical normal; the sun's height is measured from the horizontal.
First decide: is the incoming light polarised or unpolarised? Then click.
Score: 0 / 12
| If the question says… | Write this first | The trap |
|---|---|---|
| "unpolarised light through a polaroid" | I = I₀/2 | Using cos²θ for unpolarised light |
| "plane polarised light", "analyser at θ" | I = I₀ cos²θ | Also dividing by 2 |
| "two crossed polaroids" | I = 0 | Answering I₀/2 or I₀/4 |
| "third polaroid between crossed ones" | (I₀/8) sin²2α, max I₀/8 | Using the angle from P₁ for both steps |
| "polaroid rotated with angular speed ω" | θ = ωt, I ∝ cos²ωt (or sin²2ωt between crossed) | Forgetting the square |
| "polarising angle", "Brewster angle" | tan i_B = n | Using sin i = 1/n (critical angle) |
| "reflected and refracted rays" | perpendicular at i_B: i_B + r = 90° | Answering 180° or 0° |
| "which waves cannot be polarised" | longitudinal (sound in air) | Picking radio or X-rays (they are EM, transverse) |
| "sky light at 90° to the sun" | plane polarised | Saying unpolarised because sunlight is |
This completes Wave Optics: Huygens wavefronts, interference, diffraction and polarisation all treat light as a transverse electromagnetic wave. The next chapter, Dual Nature of Radiation and Matter, shows where the wave picture fails and light behaves like particles (photons).
Values used: n(water) = 4/3 ≈ 1.33, n(glass) = 1.5, n(diamond) = 2.42; cos²30° = 3/4, cos²45° = 1/2, cos²60° = 1/4.
Follow the four steps: understand it, remember it, try it, then prove it with past papers.