Polarisation and Malus's Law

Polarisation is arguably the subtlest of the basic wave phenomena. It is also one of the most powerful pieces of experimental evidence for the transverse nature of light, and it has genuine practical applications — from sunglasses and camera filters to LCD screens and 3D cinema. The OCR A-Level Physics A specification (H556) asks you to understand what polarisation is, why only transverse waves can be polarised, how polarising filters (like Polaroid) work, and — in the OCR context uniquely — to apply Malus's law, I = I₀ cos²θ.

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What Does It Mean for a Wave to Be Polarised?

In an unpolarised transverse wave, the direction of oscillation changes rapidly and randomly — there is no preferred plane. Natural light from the Sun or from a filament bulb is unpolarised: the electric field vector oscillates in all directions perpendicular to the direction of propagation, changing direction millions of times per second.

A polarised transverse wave is one in which the oscillation is restricted to a single plane containing the direction of propagation. That plane is called the plane of polarisation (OCR's convention is to define this as the plane containing the electric field vector and the direction of propagation).

Only transverse waves can be polarised, because only transverse waves have oscillations in multiple possible directions perpendicular to propagation. Longitudinal waves (like sound) oscillate only along the direction of travel — there is nothing left to restrict, so polarisation is meaningless for them.

This is the crucial experimental evidence that light is transverse: by 1808, Étienne-Louis Malus had observed that a beam of light reflected off glass at certain angles could be partially blocked by a second piece of glass oriented perpendicular to the first. Something about the reflected light made it asymmetric about the direction of travel — and that could only happen for a transverse wave.

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Polarising Filters (Polaroid)

A polarising filter (commonly called a Polaroid filter, after the commercial product invented by Edwin Land in 1929) contains long-chain conducting molecules all aligned in the same direction. These chains absorb the component of the electric field parallel to them, transmitting only the component perpendicular to them.

The direction along which the filter transmits is called its transmission axis.

flowchart LR
    U[Unpolarised light: E vibrates in all directions] --> F1[Polariser: transmission axis vertical]
    F1 --> P[Vertically polarised light: amplitude reduced by sqrt 2]
    P --> F2[Analyser at angle theta]
    F2 --> O[Transmitted light: amplitude A cos theta, intensity I cos squared theta]

What Happens to Unpolarised Light?

When unpolarised light of intensity I₀ passes through a single polariser, the filter transmits only the component along its transmission axis. Since the incoming light is random in orientation, only half the intensity is transmitted:

I₁ = I₀ / 2

This is the starting point of any polarisation problem.

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Malus's Law

Once light has been polarised by a first filter (the polariser), you can analyse it with a second filter (the analyser). If the analyser's transmission axis makes an angle θ with the plane of polarisation of the incoming light, the transmitted intensity is given by Malus's law:

I = I₀ cos²θ

where:

• I₀ is the intensity of the already polarised light incident on the analyser,
• I is the intensity of the light transmitted through the analyser,
• θ is the angle between the analyser's transmission axis and the plane of polarisation of the incoming polarised light.

Why the cos² Factor?

The analyser transmits the component of the electric field along its transmission axis. If the incoming polarised light has electric field amplitude E₀ at angle θ to the transmission axis, then the transmitted amplitude is:

E = E₀ cos θ

But intensity is proportional to the square of the amplitude:

I ∝ E²

so:

I = I₀ cos²θ

The squaring is why cos, not cos², appears in amplitude equations but cos² appears in intensity equations. Be vigilant about this distinction.

Exam Tip: OCR is one of the few A-Level Physics specifications that names Malus's law explicitly and asks you to apply it. Be ready for a worked calculation in which you must identify θ correctly and compute I = I₀ cos²θ.

Special Angles

θ	cos²θ	I / I₀	Interpretation
0°	1	1	Full transmission — analyser aligned with polariser
30°	0.75	3/4
45°	0.5	1/2
60°	0.25	1/4
90°	0	0	Crossed — no light transmitted

The last row is worth dwelling on: when the analyser is oriented perpendicular to the polariser (crossed polarisers), no light gets through at all. This is the definitive test for whether light is polarised — if rotating the second filter through 90° extinguishes the light, the light was polarised.

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Worked Example 1 — Single Polariser

Q. Unpolarised light of intensity 200 W m⁻² is incident on a polarising filter. Calculate the transmitted intensity.

A. For unpolarised incident light, a single polariser transmits half the intensity:

I = I₀ / 2 = 200 / 2 = 100 W m⁻²

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Worked Example 2 — Two Polarisers with an Angle Between Them