Source: Volume 3, Chapters 1 and 2
Tags: optics, light, index of refraction, Snell's law, total internal reflection, critical angle, dispersion, polarisation, Malus's law, Brewster's angle, mirrors, lenses, thin lens equation, magnification, focal length, real image, virtual image
Difficulty: Intermediate
Prerequisites: Chapters 15-16 notes (electromagnetic waves, wave properties). Basic geometry and trigonometry.
This is where the abstract electromagnetic wave from Chapter 16 becomes the light you actually see and work with. These chapters cover geometric optics: how light travels in straight lines (rays), how it bounces off mirrors, bends through lenses, and gets polarised. This material is essential for understanding optical instruments (cameras, microscopes, telescopes, the human eye) and appears in applications from fibre optics to corrective lenses. It is also the physics behind everything in photography, vision science, and laser design.
Light travels at c ≈ 3 x 10⁸ m/s in vacuum and slows down in materials by a factor called the index of refraction. Snell's law governs how light bends at interfaces. Mirrors and thin lenses form images described by a single equation (1/f = 1/d_o + 1/d_i), with sign conventions telling you whether the image is real or virtual, upright or inverted, magnified or reduced.
Speed of light
c = 3.00 x 10⁸ m/s in vacuum.
Index of refraction (n)
A dimensionless number describing how much a medium slows light: n = c/v, where v is the speed of light in the medium. Always ≥ 1. For vacuum, n = 1.
Law of reflection
The angle of reflection equals the angle of incidence: θ_r = θ_i. Both angles are measured from the normal to the surface.
Snell's law (law of refraction)
n₁ sin θ₁ = n₂ sin θ₂. Governs how light bends when crossing from one medium to another. Think of it as: light bends toward the normal when entering a denser medium and away from the normal when entering a less dense one.
Total internal reflection
When light travels from a denser to a less dense medium at an angle greater than the critical angle, all light is reflected and none is transmitted. Critical angle: θ_c = sin⁻¹(n₂/n₁), valid only when n₁ > n₂. In simple terms, this is why fibre-optic cables work: light bounces along inside the fibre without escaping.
Dispersion
The spreading of white light into its component wavelengths (colours) because the index of refraction varies slightly with wavelength. This is what prisms do.
Polarisation
Light is polarised when its electric field oscillates in a definite direction. Unpolarised light has electric field oscillations in all directions perpendicular to propagation.
Malus's law
When polarised light passes through a polariser (analyser), the transmitted intensity is I = I₀ cos² θ, where θ is the angle between the polarisation direction and the polariser axis.
Brewster's angle
The angle of incidence at which reflected light is completely polarised: tan θ_B = n₂/n₁.
Huygens' principle
Every point on a wavefront acts as a source of tiny new wavelets. The new wavefront is the surface tangent to these wavelets.
Plane mirror
A flat reflecting surface. It produces virtual images that are the same size as the object, at the same distance behind the mirror as the object is in front.
Concave mirror
A mirror whose reflecting surface is on the inner side of a sphere. It can form both real and virtual images.
Convex mirror
A mirror whose reflecting surface is on the outer side of a sphere. It always forms virtual, upright, reduced images.
Focal point (f)
The point where parallel rays converge (concave mirror/converging lens) or appear to diverge from (convex mirror/diverging lens). Focal length: f = R/2 for a spherical mirror.
Converging lens
A lens (typically convex) that brings parallel rays to a focus. f is positive.
Diverging lens
A lens (typically concave) that spreads parallel rays apart so they appear to come from a focal point on the same side as the incoming light. f is negative.
Real image
An image formed where light rays actually converge. Can be projected onto a screen. Image distance d_i is positive.
Virtual image
An image formed where light rays appear to come from but do not actually pass through. Cannot be projected. d_i is negative.
Magnification (m)
The ratio of image height to object height: m = -d_i/d_o. Positive m means upright; negative means inverted. |m| > 1 means enlarged; |m| < 1 means reduced.
Thin lens equation
1/f = 1/d_o + 1/d_i. Same form as the mirror equation. Combined with the lensmaker's equation for the relationship between f and the lens geometry.
Optical power
P = 1/f. Measured in dioptres (D = m⁻¹). For multiple thin lenses in contact: P_total = P₁ + P₂ + ...
Light is an electromagnetic wave. In vacuum it travels at c = 3 x 10⁸ m/s.
In a medium with index n, its speed is v = c/n.
Light can be modelled as a straight-line ray in geometric optics (valid when objects are much larger than the wavelength).
Law of reflection: θ_r = θ_i (angles from the normal).
A corner reflector (three mutually perpendicular surfaces) sends any incoming ray back parallel to its original direction. This is used in retroreflectors on roads and on the Moon.
When light passes from one medium to another, it changes speed and bends.
Snell's law: n₁ sin θ₁ = n₂ sin θ₂.
Moving into a denser medium (higher n): the ray bends toward the normal.
Moving into a less dense medium (lower n): the ray bends away from the normal.
Occurs only when going from a higher-n medium to a lower-n medium (n₁ > n₂).
Critical angle: θ_c = sin⁻¹(n₂/n₁).
At angles greater than θ_c, all light is reflected. No refracted ray exists.
The index of refraction depends slightly on wavelength: shorter wavelengths (blue/violet) have a higher n and bend more.
A prism separates white light into a spectrum because of this wavelength dependence.
Still governed by Snell's law, applied separately for each wavelength.
Polarised light: electric field oscillates in one plane. Unpolarised: oscillations in all perpendicular directions.
A polariser transmits only the component aligned with its axis.
Malus's law: I = I₀ cos² θ.
When unpolarised light passes through a polariser (not at an angle to a prior polarisation), the intensity is halved: I = I₀/2.
Brewster's angle: tan θ_B = n₂/n₁. At this angle of incidence, the reflected light is completely polarised.
Scattering partially polarises light (this is why the sky is polarised).
Image is virtual, upright, same size, and at the same distance behind the mirror as the object is in front.
d_o = -d_i (with appropriate sign conventions).
Concave (converging): reflecting surface on the inside of the sphere. f is positive.
Convex (diverging): reflecting surface on the outside. f is negative.
Focal length: f = R/2, where R is the radius of curvature.
Mirror equation: 1/f = 1/d_o + 1/d_i.
Magnification: m = -d_i/d_o.
Ray tracing rules for concave mirrors:
A ray parallel to the axis reflects through the focal point.
A ray through the focal point reflects parallel to the axis.
A ray through the centre of curvature reflects back on itself.
For a single refracting surface: n₁/d_o + n₂/d_i = (n₂ - n₁)/R.
Apparent depth: d_i = d_o × (n_i/n_o), where the image appears closer to the surface than the object actually is.
Converging lenses (biconvex, plano-convex, converging meniscus): bring parallel rays to a focus. f > 0.
Diverging lenses (biconcave, plano-concave, diverging meniscus): spread parallel rays. f < 0.
A lens is "thin" if its thickness is much less than the radii of curvature of its surfaces.
Thin lens equation: 1/f = 1/d_o + 1/d_i.
Lensmaker's equation: 1/f = (n - 1)(1/R₁ - 1/R₂).
Magnification: m = -d_i/d_o.
Converging lens:
A ray parallel to the axis passes through the far focal point.
A ray through the centre of the lens continues straight.
A ray through the near focal point exits parallel to the axis.
Diverging lens:
A ray parallel to the axis exits along the line from the near focal point.
A ray through the centre continues straight.
A ray heading toward the far focal point exits parallel to the axis.
d_i positive: real image (opposite side from object for lenses, same side for mirrors).
d_i negative: virtual image.
f positive: converging (concave mirror, convex lens).
f negative: diverging (convex mirror, concave lens).
m positive: upright. m negative: inverted.
R positive: surface is convex toward the object. R negative: concave toward the object.
The near point is the closest distance at which the eye can focus clearly (about 25 cm for a normal eye).
The far point is the farthest distance for clear vision (infinity for a normal eye).
Optical power: P = 1/f, in dioptres. For lenses in contact: P_total = P₁ + P₂ + ...
Simple magnifier (magnifying glass): angular magnification M ≈ 25 cm / f (for the image at infinity), or M = 1 + 25 cm / f (for the image at the near point).
Compound microscope net magnification: M_net = m_objective × M_eyepiece.
Telescope magnification: M = -f_objective / f_eyepiece.
Quantity | Formula |
|---|---|
Index of refraction | n = c/v |
Snell's law | n₁ sin θ₁ = n₂ sin θ₂ |
Critical angle | θ_c = sin⁻¹(n₂/n₁) |
Malus's law | I = I₀ cos² θ |
Brewster's angle | tan θ_B = n₂/n₁ |
Mirror / thin lens equation | 1/f = 1/d_o + 1/d_i |
Magnification | m = -d_i/d_o |
Focal length (mirror) | f = R/2 |
Lensmaker's equation | 1/f = (n-1)(1/R₁ - 1/R₂) |
Optical power | P = 1/f (dioptres) |
Simple magnifier | M ≈ 25 cm / f |
Microscope | M_net = m_obj × M_eye |
Telescope | M = -f_obj / f_eye |
Total internal reflection is the basis of fibre-optic communications, which carry nearly all long-distance internet traffic. Dispersion through a prism is how Newton first demonstrated that white light is a mixture of colours, and it is exploited in spectrometers used in chemistry and astronomy. Polarising sunglasses work by filtering out horizontally polarised glare reflected from roads and water (using Brewster's angle).
Students often confuse the angle of incidence with the angle measured from the surface. Both the angle of incidence and the angle of refraction are measured from the normal, not from the surface.
Total internal reflection can only happen when going from a higher-n medium to a lower-n medium. Students sometimes try to apply it the wrong way round.
A virtual image is not "imaginary" or "fake." It simply means the light rays diverge and your eye (or a lens) traces them back to an apparent location. You see virtual images every time you look in a flat mirror.
The thin lens equation works the same way as the mirror equation, but the sign conventions for d_i differ slightly (real images are on the opposite side of a lens from the object, but on the same side of a mirror).
⚠️ Snell's law and the critical angle are nearly guaranteed exam topics. Practise calculating angles and identifying when total internal reflection occurs.
⚠️ Know the sign conventions for mirrors and lenses cold. Most errors in optics problems come from incorrect signs, not from wrong formulas.
⚠️ Ray tracing: be able to draw three principal rays for both converging and diverging lenses, and for concave and convex mirrors.
⚠️ Polarisation: know that unpolarised light through a polariser becomes I₀/2, and subsequent polarisers use Malus's law with the angle between the transmission axes.
Fill in the blank: The critical angle for total internal reflection is θ_c = ________.
sin⁻¹(n₂/n₁), where n₁ > n₂.
True or false: A convex mirror can form a real image.
False. Convex mirrors always form virtual, upright, reduced images.
True or false: When unpolarised light passes through a single ideal polariser, the transmitted intensity is half the incident intensity.
True.
Fill in the blank: The focal length of a spherical mirror is ________ the radius of curvature.
Half (f = R/2).
True or false: A diverging lens has a positive focal length.
False. Diverging lenses have a negative focal length.
Q: Light travels from water (n = 1.33) into air (n = 1.00). What is the critical angle for total internal reflection?
A: θ_c = sin⁻¹(1.00/1.33) = sin⁻¹(0.7519) ≈ 48.8°.
Q: A concave mirror has a radius of curvature of 40 cm. An object is placed 30 cm from the mirror. Where is the image?
A: f = R/2 = 20 cm. 1/d_i = 1/f - 1/d_o = 1/20 - 1/30 = 1/60. d_i = 60 cm. The image is real (positive d_i) and 60 cm from the mirror.
Q: Polarised light of intensity 100 W/m² passes through a polariser whose axis is at 30° to the polarisation direction. What is the transmitted intensity?
A: I = I₀ cos² θ = 100 × cos²(30°) = 100 × 0.75 = 75 W/m².
Q: A thin converging lens has a focal length of 10 cm. An object is placed 15 cm from the lens. What is the magnification?
A: 1/d_i = 1/f - 1/d_o = 1/10 - 1/15 = 1/30. d_i = 30 cm. m = -d_i/d_o = -30/15 = -2. The image is real, inverted, and twice the size.
Optics ties back to EM waves (Ch. 16): light is an electromagnetic wave, and its speed in a medium depends on the material's electromagnetic properties. Polarisation is a direct consequence of light being a transverse wave. The wave nature of light also leads to diffraction and interference, which go beyond geometric optics. Lenses and mirrors connect to practical applications like cameras, corrective eyewear, microscopes, and telescopes.
optics, geometric optics, light, speed of light, index of refraction, Snell's law, refraction, reflection, law of reflection, total internal reflection, critical angle, fibre optics, dispersion, prism, polarisation, Malus's law, Brewster's angle, Huygens' principle, plane mirror, concave mirror, convex mirror, focal point, focal length, real image, virtual image, magnification, thin lens, converging lens, diverging lens, lensmaker's equation, mirror equation, optical power, dioptre, near point, far point, magnifying glass, microscope, telescope, PHYS 212