Brewster’s law

What is Brewster’s law?

In plain words: the glare from a lake or a wet road is light reflected at a slant. At one particular slant the reflected light’s electric field vibrates only side to side, parallel to the surface. That angle depends only on the refractive index, through tan i_B = μ, and it is the angle at which the reflected and refracted rays make a right angle.

Scope: a smooth boundary between two isotropic, transparent, non-magnetic, effectively lossless media with different refractive indices. Statements that reflection produces complete or partial polarisation assume initially unpolarised light. If the incident light is purely in-plane polarised at i_B, the reflected intensity is zero, not a nonzero fully polarised beam. Index-matched media have no ordinary interface reflection. Real absorption, coatings, roughness and wavelength spread can prevent perfect extinction.

Reflected light fully plane-polarised at the Brewster angle

• Unpolarised light has its electric field in all directions across the ray (NCERT Summary, page 272). Any such field can be split into two parts: one in the plane of incidence (the plane holding the incident ray and the normal), and one perpendicular to it, along the surface (reasoned).
• On reflection the in-plane part is reflected less than the perpendicular part, so reflected light is usually partly polarised. At the Brewster angle the in-plane part is not reflected at all: the reflected light is completely plane-polarised, with its field perpendicular to the plane of incidence (beyond the NCERT text).
• At the Brewster angle, the refracted light carries all the incident in-plane power and a fraction of the perpendicular power, so it is only partly polarised (beyond the NCERT text).
• Why (beyond the NCERT text): the light entering the glass sets the electrons in it oscillating along the refracted wave’s field, and these accelerated charges send out the reflected light. An oscillating charge sends out no light along its own line of oscillation. At the Brewster angle the reflected ray is perpendicular to the refracted ray, so it lies along the in-plane oscillation, and that part cannot appear in the reflected light.

Brewster angle: tan i_B = μ

• At the Brewster angle the reflected and refracted rays are at right angles, so i_B + r = 90° and sin r = cos i_B. Snell’s law, sin i_B = μ sin r, then gives sin i_B = μ cos i_B (reasoned):

• μ is the refractive index of the second medium relative to the first. For glass of μ = 1.5 in air, i_B = tan⁻¹ 1.5 ≈ 56.3°; for water (1.33), about 53.1° (reasoned).
• NCERT’s answer section still gives answer 10.8 as tan⁻¹(1.5) ≈ 56.3°, but the 2026–27 book has only six Chapter 10 exercises, so there is no question for it (NCERT answers, page 350).
• Since μ depends on the wavelength, so does the Brewster angle, slightly (beyond the NCERT text).
• The Brewster angle is also called the polarising angle, i_p. From index μ₁ into index μ₂, tan i_B = μ₂/μ₁: from water into glass, tan⁻¹(1.5/1.33) ≈ 48.4°. For incidence from air into a denser material, μ = 1/sin i_c, where i_c is for the reverse material-to-air interface; then tan i_B = 1/sin i_c (reasoned).
• Inside the glass, going towards air, the Brewster angle is tan⁻¹(1/μ) ≈ 33.7° for μ = 1.5. This is less than the critical angle, sin⁻¹(1/1.5) ≈ 41.8°. Here tan i_B = 1/μ = sin i_c, and tan x > sin x for any acute angle, so the Brewster angle is always the smaller (reasoned).

Reflected and refracted rays at right angles

• At the Brewster angle the angle of refraction is r = 90° − i_B, and the reflected ray makes 90° with the refracted ray (reasoned from Snell’s law above).
• The reverse holds too: if the reflected and refracted rays are at right angles, the light is meeting the surface at the Brewster angle, and the reflected light is fully polarised.
• Uses (beyond the NCERT text): glare from water, wet roads and other horizontal surfaces is partly polarised at most angles, and strongly so near the Brewster angle, with its field mostly horizontal. Polaroid sunglasses, with a vertical pass axis, and camera polaroid filters block much of it. NCERT (page 271) names polaroids in sunglasses and cameras, but not this reason.
• A pile of glass plates at the Brewster angle uses this. The in-plane part passes every surface without reflection, because inside each plate it meets the far face at 90° − i_B, the glass-to-air Brewster angle. For ordinary glass of index 1.5 in air, each surface reflects about 15% of the perpendicular power at this angle. After many plates, the light that gets through is nearly plane-polarised in the plane of incidence (reasoned).

Solved examples

Common mistakes to avoid

Quick revision summary

• At the Brewster angle, light reflected from a transparent surface is completely plane-polarised, with its field perpendicular to the plane of incidence; the refracted light is partly polarised.
• Brewster’s law: tan i_B = μ; about 56.3° for glass of μ = 1.5 and 53.1° for water.
• At i_B the reflected and refracted rays are at right angles: i_B + r = 90°.
• From glass to air the Brewster angle, tan⁻¹(1/μ), is smaller than the critical angle.
• Polaroid sunglasses cut glare because reflected light is mostly polarised parallel to the surface.

Questions students ask

What is the difference between Brewster’s law and Malus’ law?

Brewster’s law (beyond the NCERT text) says when reflection polarises light: at the angle of incidence i_B with tan i_B = μ, reflected light is fully plane-polarised; Malus’ law says how much plane-polarised light a polaroid then passes, I₀ cos² θ (NCERT Eq. (10.18), page 270). So Brewster’s law is about producing polarised light and depends on the refractive index, while Malus’ law is about analysing it and depends only on an angle. The angles differ in kind too: i_B is measured from the normal to the surface, θ between two directions across the beam, the field and the pass axis.

Is Brewster’s law in the JEE Main and NEET syllabus if it is not in NCERT?

Yes, Brewster’s law is in both syllabuses: Unit 16 of the JEE Main syllabus lists “plane-polarized light, Brewster’s law, uses of plane-polarized light and Polaroid”, and NEET (UG) 2026 Unit 16 has “Polarization, plane-polarized light: Brewster’s law”. The 2026–27 NCERT Class 12 Physics Part II lacks it because NCERT’s 2022 rationalisation dropped Section 10.7.2, Polarisation by Reflection, with Section 10.7.1 (scattering) and Exercises 10.7–10.21 (NCERT rationalised-content list, Class XII, page 51). NCERT Exemplar Problems for Class XII Physics still asks about polarisation by reflection (Chapter 10, questions 10.1 and 10.16), and this page covers what the dropped section did.

Is light reflected from a surface always polarised?

For initially unpolarised light at an ordinary lossless dielectric interface, complete linear polarisation by reflection occurs at the Brewster angle; at other angles glass or water reflects partly polarised light, and at normal incidence it is not polarised at all, as both components then meet the surface alike (beyond the NCERT text). For glass of μ = 1.5 at 30°, the Fresnel equations give about 5.8% of the light polarised along the surface reflected but only 2.5% of the light polarised in the plane of incidence, and at grazing incidence both near 100%, so the polarisation fades again. Ordinary absorbing metal mirrors generally have no exact zero of the in-plane reflectance, so they do not give complete polarisation by reflection of initially unpolarised light. Their reflectance and phase shifts depend on material, wavelength and angle.

What is the Brewster angle used for?

The Brewster angle is used where light must cross glass without reflection loss or be polarised by reflection: the end windows of many gas lasers, such as helium–neon lasers, are set at it, so light polarised in the plane of incidence crosses them with no reflection loss while the other polarisation keeps losing light, and the beam comes out plane-polarised (beyond the NCERT text; Thorlabs, Optics Selection Guide: Brewster Windows). It also gives the refractive index of a transparent material from reflected light alone: find the angle of incidence at which a turning polaroid can make the reflection vanish; then μ = tan i_B.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Find the Brewster angle for a glass of refractive index 1.6.

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   i_B = tan⁻¹ 1.6 ≈ 58.0° (Brewster’s law; beyond the NCERT text).

2. Q2. In which direction does the electric field of light reflected at the Brewster angle vibrate?

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   Perpendicular to the plane of incidence, parallel to the reflecting surface (beyond the NCERT text).

3. Q3. Light falls on a liquid at 53°, and the reflected and refracted rays are at right angles. Find the refractive index of the liquid.

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   The rays at right angles mean 53° is the Brewster angle, so μ = tan 53° ≈ 1.33 (Brewster’s law; beyond the NCERT text).