Light – Reflection and Refraction — Notes

Laws of reflection: (1) the angle of incidence equals the angle of reflection; (2) the incident ray, the reflected ray and the normal at the point of incidence lie in the same plane. They apply to all reflecting surfaces. A plane mirror forms a virtual, erect, same-size, laterally inverted image as far behind the mirror as the object is in front.

A spherical mirror is part of a hollow sphere: concave if the reflecting surface curves inwards, convex if it bulges outwards.

• Pole (P): centre of the reflecting surface.
• Centre of curvature (C): centre of the sphere of which the mirror is part (in front of a concave mirror, behind a convex one). Radius of curvature (R) = PC.
• Principal axis: the line through P and C.
• Principal focus (F): the point where rays parallel to the axis meet after reflection (concave) or appear to diverge from (convex). Focal length f = PF.
• For mirrors of small aperture, R = 2f.

Ray rules: a ray parallel to the axis reflects through F; a ray through F reflects parallel; a ray through C reflects back along itself; a ray at P reflects at an equal angle on the other side of the axis. Any two rays locate the image. Virtual rays and images are drawn dashed.

The table below summarises all six object positions. Every real image from a single mirror is inverted; the only virtual image (object between F and P) is erect and enlarged.

A convex mirror always forms a virtual, erect, diminished image between P and F behind the mirror (at F, point-sized, for an object at infinity).

Uses: concave mirrors in torches, searchlights and headlights (bulb at F gives a parallel beam), shaving and make-up mirrors (enlarged erect image), dentists' mirrors, and solar furnaces (concentrate sunlight). Convex mirrors as rear-view (wing) mirrors of vehicles: always erect, diminished, giving a wider field of view.

New Cartesian sign convention: the pole is the origin and the principal axis is the x-axis; the object is placed to the left; distances in the direction of the incident light are positive and against it negative; heights above the axis are positive and below it negative.

For mirrors: u is negative; f is negative for concave and positive for convex; v is negative for a real image and positive for a virtual image.

Magnification m = h′/h = −v/u. Negative m → real, inverted; positive m → virtual, erect; |m| > 1 enlarged, |m| < 1 diminished. (Derivation of the formula is not required.)

Refraction is the change in direction of light as it passes obliquely from one medium to another, because its speed changes. From a rarer to a denser medium the ray bends towards the normal; from denser to rarer, away from the normal; along the normal it does not bend.

Glass slab: the ray bends towards the normal on entering and away on leaving; the emergent ray is parallel to the incident ray (angle of emergence = angle of incidence) but laterally shifted.

Laws of refraction: (1) the incident ray, the refracted ray and the normal at the point of incidence lie in the same plane; (2) for a given pair of media and a given colour of light, sin i / sin r is constant (Snell's law). The constant is the refractive index of the second medium with respect to the first.

The absolute refractive index of a medium is n = c/v, where c = 3 × 10⁸ m/s is the speed of light in vacuum and v its speed in the medium. The relative refractive index of medium 2 with respect to medium 1 is n₂₁ = v₁/v₂ = n₂/n₁.

A medium with a larger refractive index is optically denser (light travels more slowly in it). Optical density is not mass density: kerosene (n = 1.44) floats on water (n = 1.33) but is optically denser.

A convex (converging) lens is thicker in the middle; a concave (diverging) lens is thinner in the middle. Terms: optical centre O (rays through it pass undeviated), principal axis, two principal foci F₁ and F₂, focal length f, and the points 2F₁, 2F₂.

Ray rules: parallel to the axis → through F₂ (convex) or appearing to come from F₁ (concave); through F₁ (convex) or towards F₂ (concave) → emerges parallel; through O → straight on.

Convex-lens images follow the table below. A concave lens always gives a virtual, erect, diminished image between F₁ and O (at F₁ for an object at infinity).

With the optical centre as origin and the same sign convention: f is positive for a convex lens and negative for a concave lens. For a real image (convex lens) v is positive; for a virtual image v is negative.

Magnification for a lens: m = h′/h = v/u (no minus sign).

Power P = 1/f with f in metres; unit dioptre (D), 1 D = 1 m⁻¹. Convex lenses have positive power, concave lenses negative power. For lenses in contact, P = P₁ + P₂ + P₃ + …