Reflection by spherical mirrors
The laws of reflection hold for curved mirrors too: the angle of incidence equals the angle of reflection, measured from the normal at that point.
Cartesian sign convention: all distances are measured from the pole of the mirror (or the optical centre of a lens). Distances in the direction of the incident light are positive; distances against it are negative. Heights above the principal axis are positive; heights below are negative.
With light coming from the left, a concave mirror's focal length is negative and a convex mirror's is positive.
The focal length is half the radius of curvature: f = R/2.
Mirror equation: 1/v + 1/u = 1/f
Magnification: m = h′/h = −v/u
A negative m means the image is real and inverted; a positive m means it is virtual and erect.
Example. An object is 30 cm in front of a concave mirror of focal length 10 cm. With u = −30 and f = −10: 1/v = 1/f − 1/u = −1/10 + 1/30 = −2/30, so v = −15 cm (real, in front of the mirror). m = −(−15)/(−30) = −0.5: inverted and half the size.
A convex mirror always forms a virtual, erect, diminished image, giving a wide field of view. That is why it is used as a rear-view mirror.
Refraction
When light passes from one medium to another, it bends. Snell's law:
n₁ sin i = n₂ sin r
The refractive index n = c/v, where c is the speed of light in vacuum and v in the medium. Light bends towards the normal when entering a denser medium.
Apparent depth: an object at real depth d under water appears at depth d/n. A pool looks shallower than it is.
Total internal reflection
When light goes from a denser to a rarer medium, the refracted ray bends away from the normal. At a particular angle of incidence, the critical angle i_c, the refracted ray grazes the surface (r = 90°):
sin i_c = 1/n (for a medium in air)
For angles of incidence greater than i_c, no light is refracted; it is all reflected back. This is total internal reflection.
For water (n = 1.33), i_c ≈ 49°; for glass (n = 1.5), i_c ≈ 42°; for diamond (n = 2.42), only about 24°, which is why diamonds sparkle.
Applications: optical fibres carry light signals for long distances by repeated total internal reflection; mirages form in deserts; totally reflecting prisms are used in binoculars.
Refraction at a spherical surface and lenses
For refraction at a single spherical surface of radius R from medium n₁ to n₂: n₂/v − n₁/u = (n₂ − n₁)/R.
Applying this to both surfaces of a thin lens gives the lens maker's formula:
1/f = (n − 1) (1/R₁ − 1/R₂)
Lens formula: 1/v − 1/u = 1/f
Magnification: m = v/u
A convex (converging) lens has positive f; a concave (diverging) lens has negative f.
Example. An object is 20 cm from a convex lens of focal length 15 cm. With u = −20, f = +15: 1/v = 1/15 − 1/20 = 1/60, so v = +60 cm (real image on the other side). m = 60/(−20) = −3: inverted and three times larger.
Power of a lens
P = 1/f, with f in metres. The unit is the dioptre (D). A converging lens has positive power; a diverging lens negative.
For thin lenses in contact: P = P₁ + P₂ + …, or 1/f = 1/f₁ + 1/f₂ + ….
Example. A lens of +2.5 D and one of −1.0 D in contact give +1.5 D, a converging combination with f = 1/1.5 ≈ 0.67 m.
Prism and dispersion
Light passing through a prism of angle A is deviated. At minimum deviation D, the ray passes symmetrically through the prism, and
n = sin((A + D)/2) ⁄ sin(A/2)
White light splits into its colours (dispersion) because the refractive index depends on wavelength: violet light bends the most and red the least.
Natural phenomena
Rainbows: sunlight is refracted, dispersed and internally reflected inside raindrops. Primary rainbows have red on the outside.
Blue sky: air molecules scatter short (blue) wavelengths much more than long (red) ones, since scattering ∝ 1/λ⁴.
Red sunrise and sunset: light travels through more atmosphere, so most blue is scattered away and red reaches us.
Optical instruments
The eye. The eye lens focuses images on the retina. The ciliary muscles change the lens's focal length (accommodation). The near point is about 25 cm.
Defect | Cause | Correction |
|---|---|---|
Myopia (short sight) | image forms in front of the retina | concave lens |
Hypermetropia (long sight) | image forms behind the retina | convex lens |
Astigmatism | cornea not spherical | cylindrical lens |
Simple microscope: a convex lens of short focal length. Magnification m = 1 + D/f when the image is at the near point (D = 25 cm).
Compound microscope: a short-focus objective forms a real, magnified image, which the eyepiece magnifies further. Total magnification ≈ (L/fₒ)(D/fₑ), where L is the tube length.
Astronomical telescope: a long-focus objective and a short-focus eyepiece. Magnifying power m = fₒ/fₑ (normal adjustment), and the tube length is fₒ + fₑ. Large telescopes use concave mirrors as objectives (reflecting telescopes, such as the Cassegrain design), which avoid chromatic aberration and are easier to support.
Summary
Mirrors: 1/v + 1/u = 1/f, m = −v/u, f = R/2.
Snell's law n₁ sin i = n₂ sin r; total internal reflection beyond sin i_c = 1/n.
Lenses: 1/v − 1/u = 1/f, m = v/u; lens maker's formula; P = 1/f in dioptres.
Prisms disperse light; violet deviates most.
Telescope magnification fₒ/fₑ; myopia is corrected with a concave lens.