Reflection of light
When light falls on a polished surface it bounces back. This is reflection. Every reflection obeys two laws:
The angle of incidence equals the angle of reflection.
The incident ray, the reflected ray and the normal at the point of incidence all lie in the same plane.
A plane mirror forms an image that is virtual, erect, the same size as the object, and as far behind the mirror as the object is in front. The image is also laterally inverted: left and right appear swapped.
Spherical mirrors
A spherical mirror is part of a hollow sphere.
Concave mirror: the reflecting surface curves inwards. It converges light.
Convex mirror: the reflecting surface bulges outwards. It diverges light.
Key terms: the pole (P) is the centre of the mirror's surface, the centre of curvature (C) is the centre of the sphere, and the principal focus (F) is where rays parallel to the principal axis meet after reflection, or appear to come from. The focal length f is half the radius of curvature:
f = R / 2
Uses. Concave mirrors are used in torches and headlights (to produce a parallel beam), as shaving mirrors and by dentists (to give an enlarged image). Convex mirrors are used as rear-view mirrors, because they always give an erect image and a wider field of view.
Sign convention and the mirror formula
Distances are measured from the pole. Distances in the direction of incident light are positive and those against it are negative. Heights above the principal axis are positive and heights below are negative. With an object placed on the left, the object distance u is therefore always negative.
Mirror formula: 1/v + 1/u = 1/f, and magnification m = h′/h = −v/u
A negative m means a real, inverted image. A positive m means a virtual, erect image.
Example. An object is 30 cm in front of a concave mirror of focal length 10 cm. Where is the image?
u = −30, f = −10. 1/v = 1/f − 1/u = −1/10 + 1/30 = −2/30, so v = −15 cm. The image is real, 15 cm in front of the mirror, and m = −v/u = −0.5: inverted and half the size.
Refraction of light
When light passes from one transparent medium into another, it changes speed and bends. This is refraction. Light bends towards the normal when it enters an optically denser medium (for example, from air into glass) and away from the normal when it leaves it.
The laws of refraction:
The incident ray, the refracted ray and the normal lie in the same plane.
Snell's law: sin i / sin r is a constant for a given pair of media.
The refractive index of a medium is
n = speed of light in vacuum / speed of light in the medium = c / v
Water has n ≈ 1.33 and crown glass about 1.52, so light travels more slowly in glass than in water.
Spherical lenses
A convex lens is thicker in the middle and converges light.
A concave lens is thinner in the middle and diverges light.
Lens formula: 1/v − 1/u = 1/f, and magnification m = v/u
The same sign convention applies, measured from the optical centre. The focal length is positive for a convex lens and negative for a concave lens.
Example. An object is placed 15 cm from a convex lens of focal length 10 cm. Find the image position.
u = −15, f = +10. 1/v = 1/f + 1/u = 1/10 − 1/15 = 1/30, so v = +30 cm. The image is real and inverted, on the other side of the lens, and m = v/u = −2 (twice the size).
Power of a lens
P = 1 / f, with f in metres. The unit is the dioptre (D).
A convex lens has positive power and a concave lens negative power. A lens of focal length −50 cm has power 1 / (−0.5) = −2 D. For lenses in contact, the powers simply add.
Quick summary
Mirror | Lens | |
|---|---|---|
Formula | 1/v + 1/u = 1/f | 1/v − 1/u = 1/f |
Magnification | m = −v/u | m = v/u |
Converging type | Concave | Convex |