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Etienne-Louis Malus found in 1809 that reflected light can be polarised. David Brewster found the rule that says at which angle. Our post on Malus and polarisation by reflection covers the basic glass example. This one is about Brewster: how a tarnished glass surface nearly hid his law, and what his paper proved about light inside glass and at the boundary of two media.
Who was David Brewster?
The Royal Society of Edinburgh's index of former Fellows records that Brewster was born on 11 December 1781 in Jedburgh, where his father was rector of the grammar school, and died on 10 February 1868 at Allerly, near Melrose. He took an MA at Edinburgh in 1800, was elected to the Royal Society of Edinburgh in 1808 and to the Royal Society of London in 1815. He was Principal of St Andrews University from 1837 to 1859 and of Edinburgh University from 1859 until his death.
The Linda Hall Library notes that he invented the kaleidoscope around 1816 and the stereoscope in the 1840s, and helped found the British Association for the Advancement of Science in 1831.
How did a tarnished piece of glass hide the law?
Brewster's paper, On the laws which regulate the polarisation of light by reflexion from transparent bodies, was read to the Royal Society on 16 March 1815. In it he tells the story honestly. In the summer of 1811 he repeated Malus's experiments and measured the polarising angles of many transparent bodies. For water and precious stones the tangent of the polarising angle came surprisingly close to the refractive index. Glass refused to fit, and he gave up for more than twelve months.
Measuring topaz brought him back. He then looked hard at his glass sample. One polished face agreed almost exactly with the tangent rule; the other was off by no less than two degrees. That face had suffered a chemical change from exposure to air, which altered its polarising angle without changing the glass inside. With the anomaly explained, all his measurements fitted one rule, and he tabulated eighteen substances against it.
He then tested the extreme case. Arago had told him that air polarises light at about 45 degrees. With a refractive index for air of 1.00031, the rule gives 45° 0' 32", which Brewster called a striking agreement.
Why are the reflected and refracted rays at right angles?
Brewster did not stop at the law. He turned it into a set of numbered propositions. One states that at the polarising angle the angles of incidence and refraction add up to a right angle; the next, that the reflected ray is perpendicular to the refracted ray.
The proof takes two lines. Snell's law gives sin θB = n sin r. Brewster's law gives n = sin θB / cos θB. Put them together and sin r = cos θB, so r = 90° − θB.
Worked example 1: light inside a glass plate
Malus had noticed something puzzling: the reflections from the front and back faces of a glass plate become polarised at the same time. Brewster's paper explains it. For light inside the glass meeting the back surface, the polarising angle has a tangent equal to 1/n.
Take glass with n = 1.50.
From air onto the front face: tan θB = 1.50, so θB = 56.3°.
Refracted angle: r = 90° − 56.3° = 33.7°.
Back face, from inside: tan θ' = 1/1.50 = 0.667, so θ' = 33.7°.
The ray reaches the parallel back face at 33.7°, which is exactly its polarising angle. Both reflections are polarised together.
Worked example 2: water lying on glass
Malus had searched for the polarising angle at the boundary between water and glass and never found it. Brewster's numbers show why. He used n = 1.525 for glass and n = 1.336 for water.
Light in water meeting glass: tan θB = 1.525 / 1.336 = 1.141, so θB = 48.8° (Brewster gives 48° 47').
To meet that boundary at 48.8°, light must enter the water from air at angle x with sin x = 1.336 × sin 48.8° = 1.336 × 0.752 = 1.005.
No angle has a sine greater than 1.
So when light from air passes through a flat layer of water onto glass, it can never reach the boundary at the polarising angle. The experiment Malus tried was impossible, not badly done.
Which mistakes should you avoid?
Using one refractive index at a boundary of two media. Use the ratio n2/n1, with n1 for the medium the light is travelling in.
Measuring angles from the surface. The polarising angle is measured from the normal.
Forgetting the right-angle check. If your θB and r do not add to 90°, one of them is wrong.
The JEE (Main) 2026 syllabus lists plane-polarised light, Brewster's law and Polaroids under wave optics. Check the syllabus for your own exam year.
What did Brewster find in squeezed glass?
A year later Brewster reported, in a paper read on 29 February 1816, that ordinary plate glass squeezed edgewise between two screws starts acting on polarised light like a crystal. A strip bent edgewise showed different coloured fringes on its compressed inner edge and stretched outer edge, and the colours climbed Newton's scale as the force increased. The same effect is now called photoelasticity, and it lets engineers see stress in transparent models.
If it is hard to picture why the reflected and refracted rays end up at right angles, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
What can a student take from Brewster's way of working?
Suspect the apparatus before the law. Brewster's two-degree error came from a tarnished surface, not from nature.
Come back after giving up. A year away and one new measurement on topaz turned failure into a law.
Test the extreme case. Air, with n barely above 1, was a sharp check.
Work out the consequences. The right-angle rule and the back-face result both fell out of one equation.
Try it now: find the polarising angle for light inside glass of n = 1.60 meeting air, then check it against 90° minus the Brewster angle from outside.
