Intuition The one core ideaLight is a wiggle that has a direction to its wiggling, and both a polarizer and a shiny surface can filter light by that direction. Everything in this topic is just careful bookkeeping of how much wiggle survives — measured by projecting a vector onto an axis (Malus) or by lining up geometry with Snell's law (Brewster).
Before you can read the parent note, you need to earn every symbol it throws at you. Below, each item is built from the one before it: plain words → the picture → why the topic needs it. Nothing is used before it is defined.
Definition Transverse waveA wave where the thing that wiggles moves perpendicular (at a right angle) to the direction the wave travels. Picture a jump-rope: the wave runs along the rope, but each bit of rope moves up-and-down , sideways to that run.
Why the topic needs it. Polarization is only meaningful for transverse waves. If the wiggle were along the travel direction (like sound), there'd be no "sideways choice" to filter — no up-down vs left-right. See Wave Nature of Light and Electromagnetic Waves for why light qualifies.
Definition Electric fieldE E E length is how strong that push is. The plain letter E length (magnitude), a positive number.
Intuition Arrow vs. numberE E direction is the polarization.
Why the topic needs it. The direction of E is the polarization. Malus's law asks "how much of E Electromagnetic Waves , light is an E
E 0 0 E 0 largest length the arrow E
Picture the rope: the rope crosses the middle line many times, but E 0 to this maximum swing .
Why the topic needs it. Malus's law starts with incoming amplitude E 0
This is the heart of Malus's law, so we build it slowly.
θ θ angle — the amount of tilt between two directions, measured in degrees (∘ 36 0 ∘ E
Now: when an arrow of length E 0 θ projection .
cos θ E 0 hypotenuse ), and drop a straight line down onto the axis. The piece lying along the axis is the adjacent side. Then
cos θ = hypotenuse adjacent = E 0 length along axis . E 0 cos θ
Intuition Why cosine and not something else?We need a tool that answers "what fraction of a tilted arrow lies along a chosen line?" Cosine is exactly that fraction — it's 1 θ = 0 0 θ = 9 0 ∘
Why the topic needs it. The surviving amplitude through a polarizer is E 0 cos θ is Malus's law once we square it.
Worked example Check the extremes
θ = 0 ∘ cos 0 = 1 E 0 θ = 9 0 ∘ cos 9 0 ∘ = 0 θ = 3 0 ∘ cos 3 0 ∘ = 2 3 ≈ 0.866
I I brightness a detector actually reads — the energy the light delivers per second per area. Your eye and a light meter feel I not the raw arrow length.
Intuition Why the square?The energy carried by an E E 2 Electromagnetic Waves — energy density goes as the field squared, just like a spring stores energy as displacement squared ). So when the surviving amplitude is E 0 cos θ brightness is ( E 0 cos θ ) 2 = E 0 2 cos 2 θ cos 2 θ squared because brightness is amplitude squared.
Why the topic needs it. Malus's law is written in intensity (I = I 0 cos 2 θ E I
cos 2 θ ( cos θ ) 0 1 2 1
Intuition Why the average is exactly2 1 Unpolarized vs Polarized Light ) is an even mix of all tilt angles. To get its surviving fraction we average cos 2 θ cos 2 θ 2 1 2 1
Why the topic needs it. The 2 1 not a separate rule to memorize — it is the average height of this curve.
Definition Refractive indexn slows and bends light. Vacuum/air: n ≈ 1 n = 1.33 n = 1.5 n which medium: n 1 n 2
Why the topic needs it. Brewster's angle depends entirely on the ratio n 2 / n 1 Reflection and Refraction at Interfaces .
Definition The normal, and sinesin θ normal is the dashed line drawn perpendicular to the surface at the point light hits — all these angles are measured from it, never from the surface itself.
Sine , sin θ = hypotenuse opposite facing the angle over the slanted side. It is the natural partner of cosine and is what governs bending.
Why the topic needs it. Brewster's derivation is Snell's law with one extra geometric fact plugged in. Full details in Snell's Law and Refraction .
arctan question-undoer : arctan ( x ) "which angle has tangent equal to x If tan θ B = 1.5 θ B = arctan ( 1.5 ) = 56. 3 ∘
Intuition Why tangent shows up in Brewster's lawThe Brewster derivation divides one Snell equation by cos θ B sin / cos tan "given the ratio n 2 / n 1 — and arctan turns that ratio back into the actual angle.
Angle theta and cosine projection
Malus amplitude E0 cos theta
Intensity goes as E squared
Average of cos2 is one half
Unpolarized halves at first filter
Brewster tan theta B equals n2 over n1
Return to the parent: Polarization — Malus's law, Brewster's angle derivation .
Cover the right side and test yourself — you are ready when every line comes instantly.
What makes a wave "transverse"? The wiggle is perpendicular to the travel direction.
What is E E E is the wiggle arrow (direction + strength);
E is just its length.
What does E 0 The peak amplitude — the biggest swing of the arrow.
Define cos θ adjacent over hypotenuse — the fraction of a tilted arrow lying along the axis.
Why does the surviving amplitude through a polarizer equal E 0 cos θ It is the projection of the tilted field onto the transmission axis.
What is intensity I E The brightness a detector reads; I ∝ E 2
Why does Malus's law have cos 2 cos Cosine from projection of amplitude, then squared because I ∝ E 2
Why is the average of cos 2 θ 2 1 It spends equal time above and below 2 1
What does the refractive index n How much a medium slows and bends light; n = 1 1.33 1.5
State Snell's law and what the angles are measured from. n 1 sin θ 1 = n 2 sin θ 2
What is tan θ sin θ / cos θ
What does arctan ( x ) "Which angle has tangent equal to x
