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PhysicsDifficulty 2-4

Light

Light is an electromagnetic wave that travels through vacuum at exactly 299,792,458 m/s and more slowly through materials. At a boundary it reflects (angle of reflection equals angle of incidence) and refracts according to Snell's law, n₁ sin θ₁ = n₂ sin θ₂. Visible light spans roughly 400–750 nm, and it also behaves as a stream of photons, each with energy E = hf.

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Light as an Electromagnetic Wave and a Photon

Maxwell's equations predict transverse electromagnetic waves in free space travelling at c = 1/√(ε₀μ₀). Light was already known to be a wave travelling at that speed, so it had to be electromagnetic radiation; from 1887 Hertz produced and detected such waves in the laboratory and confirmed their speed. Today c is fixed exactly at 299,792,458 m/s.

c=1ε0μ0,c=fλc = \dfrac{1}{\sqrt{\varepsilon_0\mu_0}}, \qquad c = f\lambda

Speed of electromagnetic waves in vacuum, and the wave relation linking frequency and wavelength.

In matter n(λ) varies with wavelength (dispersion): for water, n ≈ 1.331 at 660 nm (red) and 1.342 at 410 nm (violet). Snell's law then gives each colour a slightly different refraction angle, the origin of prism spectra and rainbows.

The photoelectric effect, explained by Einstein in 1905, shows light also delivers energy in quanta: E = hf = hc/λ. A 550 nm photon carries hc/λ = (6.626 × 10⁻³⁴)(3.00 × 10⁸)/(550 × 10⁻⁹) ≈ 3.61 × 10⁻¹⁹ J.

Info: Critical-angle check: θc = sin⁻¹(n₂/n₁) gives 48.6° for water–air and 24.4° for diamond–air. Diamond's small critical angle traps light by repeated internal reflection, part of why it sparkles.
Full explanation — the complete reference version every reading depth is based on

What light is

Light is the small slice of the electromagnetic spectrum that human eyes respond to, from about 400 nm (violet) to about 750 nm (red). Like radio waves and X-rays it is a transverse wave of oscillating electric and magnetic fields, so it needs no medium — sunlight crosses the vacuum of space to reach us. Red light has the longest wavelength and lowest frequency of the visible colours; violet has the shortest and highest.

c=299 792 458 m/s (exact)c = 299\,792\,458\ \text{m/s} \ (\text{exact})

The speed of light in vacuum is a defining SI constant.

c=fλc = f\lambda

It links every electromagnetic wave's frequency to its wavelength.

Light in materials: the refractive index

Light slows down inside matter because it interacts with atoms. The refractive index n compares the two speeds. For visible light in ordinary transparent materials it is never faster than in vacuum, so n is at least 1.

n=cvn = \dfrac{c}{v}

Refractive index: speed of light in vacuum divided by its speed in the material.

  • Air (0 °C): n ≈ 1.0003 — so close to 1 that it is usually taken as 1.
  • Fresh water (20 °C): n ≈ 1.333, so light travels at about c/1.333 ≈ 2.25 × 10⁸ m/s in water.
  • Crown glass: n ≈ 1.52.
  • Diamond: n ≈ 2.419.
  • These values are for yellow light of wavelength 589 nm; n changes slightly with wavelength.

Reflection and refraction

When light reaches a boundary, some reflects and some passes through. Angles are always measured from the normal — an imaginary line at right angles to the surface. The reflected ray leaves at the same angle it arrived. The transmitted ray changes direction (refracts): it bends towards the normal entering a higher-index material and away from it entering a lower-index one.

θr=θin1sin⁡θ1=n2sin⁡θ2\theta_r = \theta_i \qquad\qquad n_1\sin\theta_1 = n_2\sin\theta_2

The law of reflection (left) and Snell's law of refraction (right).

Worked example: air into water

A ray in air (n₁ ≈ 1.00) meets still water (n₂ = 1.333) at 30° to the normal. Snell's law gives sin θ₂ = (1.00 × sin 30°) / 1.333 = 0.375, so θ₂ ≈ 22.0°: the ray bends towards the normal. The ScienceVerse light-refraction simulation uses n = 1.0003 for air and n = 1.33 for water by default and so reports about 22.1° for the same ray — the small difference comes only from rounding the indices.

Total internal reflection

Going the other way, from water into air, the ray bends away from the normal. At one particular angle of incidence, the critical angle, the refracted ray would skim along the surface at 90°. Beyond it no light escapes: all of it reflects back. This can only happen when light travels towards a material with a lower refractive index.

θc=sin⁡−1 ⁣(n2n1),n1>n2\theta_c = \sin^{-1}\!\left(\dfrac{n_2}{n_1}\right), \quad n_1 > n_2

Critical angle for total internal reflection. For water to air it is 48.6°.

That textbook value uses n = 1.333 for water and takes air as exactly 1. The ScienceVerse light-refraction simulation uses n = 1.33 for water and 1.0003 for air, so it reports a critical angle of about 48.8° (sin⁻¹(1.0003/1.33), our calculation) — the same physics with slightly different inputs, which is why it describes the water–air critical angle as about 49°.

Optical fibres exploit this: light inside a thin glass or plastic fibre keeps hitting the wall beyond the critical angle and is guided along it, carrying telephone, internet and cable TV signals.

Colour, dispersion and photons

Because n depends slightly on wavelength — water's index is about 1.331 for red light and 1.342 for violet — each colour refracts by a slightly different angle. White light therefore fans out into a spectrum in a prism, and refraction and reflection inside raindrops produce rainbows. Light also arrives in packets called photons: each carries energy E = hf, so violet photons carry more energy than red ones.

Info: Assumptions and limits: the ray rules above (θr = θi and Snell's law) treat light as straight-line rays, which works while the wavelength is tiny compared with the mirrors, lenses and gaps involved. Quoted refractive indices are for one wavelength (589 nm) and room temperature. Interference and diffraction need the wave model, and the photoelectric effect needs photons. The light-refraction simulation computes directions only, for a flat boundary, and does not show how much light is reflected or transmitted.
E=hf=hcλE = hf = \dfrac{hc}{\lambda}

Photon energy, with h = 6.62607015 × 10⁻³⁴ J s exactly.

Where light connects

Light is a special case of Waves, so wavelength, frequency and interference all apply. It is produced and explained by the coupling of Electricity and Magnetism, and its photons carry Energy. Large telescopes use reflection to form images of stars, and fibre-optic internet depends on total internal reflection.

How we know

  • Ole Roemer made the first measurement of light's speed in 1675 by timing eclipses of Jupiter's moon Io, which arrived early or late depending on how far Earth was from Jupiter.
  • Snell discovered the law of refraction in 1621; Ibn Sahl had found it earlier, in 984.
  • Maxwell's equations predicted electromagnetic waves travelling at the speed of light, and from 1887 Heinrich Hertz produced and detected such waves and confirmed their speed.
  • Einstein's 1905 explanation of the photoelectric effect treated light as a stream of photons, each with energy E = hf.
Common misconception: Misconception: refraction happens because light 'wants' to take a shorter path, or because glass pushes it. Refraction happens because light changes speed when it enters a new material; the change of speed at an angle changes its direction, exactly as Snell's law describes.
Common misconception: Misconception: a pencil in a glass of water is really bent. The pencil is straight; light from the underwater part refracts at the water's surface on its way to your eye, so that part appears displaced.

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Content status: published 1 October 2026.

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