Chapter 9: Light – Reflection and Refraction

Equation / formula : angle i = angle r ; R = 2f

In the figure : The focal length f is the distance PF, and the radius R is twice as long.

Rule 1 : Laws of reflection and the plane mirror — Angles are always measured from the normal, not from the surface. A plane mirror makes a virtual, erect image of the same size, as far behind the mirror as the object is in front; left and right appear interchanged (lateral inversion).

• Easy : A ray hits a mirror at 25 degrees to the normal and leaves at 25 degrees.
• Medium : A ray makes 35 degrees with the mirror surface, so its angle of incidence is 90 - 35 = 55 degrees; the angle between incident and reflected rays is 110 degrees.
• For exams : A girl 4 m from a plane mirror walks 1.5 m towards it. She is now 2.5 m away, so her image is 2.5 + 2.5 = 5 m from her.

Rule 2 : Terms of a spherical mirror — The focus F lies midway between P and C, so R = 2f. A concave mirror brings parallel rays together at F; a convex mirror spreads them so that they seem to come from a point F behind the mirror. Distances f and R are measured from the pole.

• Easy : Radius of curvature 30 cm gives focal length 15 cm.
• Medium : A concave mirror has f = 12 cm: its centre of curvature C is 2 x 12 = 24 cm in front. A convex mirror with f = 8 cm has its focus 8 cm behind it.
• For exams : Sunlight focused by a concave mirror makes a bright spot 9 cm away. So f = 9 cm and R = 18 cm.

Spot the difference

Equation / formula : Concave: beyond F real and inverted; inside F virtual, erect, enlarged. Convex: always virtual, erect, diminished.

In the figure : Blue arrow is the object, red arrow the image (dashed means virtual), orange lines are rays. Here the object height is drawn small so the rays fit.

Rule 1 : Image positions for a concave mirror — Object at infinity: image at F, point-sized. Beyond C: between F and C, diminished. At C: at C, same size. Between C and F: beyond C, enlarged. At F: image at infinity. Between P and F: behind the mirror, enlarged and erect. All real images are inverted.

• Easy : A candle 50 cm from a concave mirror of radius 40 cm is beyond C, so its image is between F and C, smaller and upside down.
• Medium : With f = 10 cm, a candle at 15 cm lies between F (10) and C (20), so the image is beyond C, enlarged, real and inverted.
• For exams : For a big real image on a screen, keep the object between F and C. Inside F the image is virtual and cannot be caught on a screen.

Rule 2 : Convex mirror and uses of mirrors — A convex mirror always forms a small, upright, virtual image between P and F, so it shows a wide area: rear-view mirrors, road-bend mirrors. Concave mirrors: torch and headlight (bulb at F gives a parallel beam), shaving and dentist mirrors (object inside F gives an enlarged erect image), solar furnace (sunlight focused at F).

• Easy : A scooter’s side mirror is convex and shows a wide view.
• Medium : Why is the headlight bulb at the focus? Rays from F leave the concave reflector parallel, so the beam travels far without spreading.
• For exams : Which mirror can never form a real image? The convex mirror. Which gives a magnified erect image? A concave mirror with the object inside F.

Spot the difference

Equation / formula : 1/v + 1/u = 1/f ; m = h'/h = -v/u

In the figure : Always write the signs first and only then put the numbers in the formula.

Rule 1 : Using the mirror formula — Write u and f with signs, then 1/v = 1/f - 1/u. A negative v means a real image in front of the mirror; a positive v means a virtual image behind it.

• Easy : Concave, f = -10 cm, u = -30 cm: 1/v = -1/10 + 1/30 = -2/30, so v = -15 cm (real, in front).
• Medium : Concave, f = -10 cm, u = -6 cm: 1/v = -1/10 + 1/6 = 2/30, so v = +15 cm (virtual, behind the mirror).
• For exams : Convex, f = +24 cm, u = -24 cm: 1/v = 1/24 + 1/24 = 1/12, so v = +12 cm and m = +0.5.

Rule 2 : Reading the magnification — m = h'/h = -v/u. A negative m means a real, inverted image; a positive m means a virtual, erect image. If the number part is more than 1 the image is enlarged; less than 1, diminished; exactly 1, same size.

• Easy : Object 4 cm tall and m = -2: the image is 8 cm tall and inverted.
• Medium : u = -20 cm and v = +10 cm: m = -10/(-20) = +0.5, a virtual, erect image half the size; 1/f = 1/10 - 1/20 gives f = +20 cm, a convex mirror.
• For exams : Which mirror can give m = +2? A concave mirror with the object inside F; a convex mirror only gives m between 0 and 1.

Spot the difference

Equation / formula : sin i / sin r = n ; n = c / v

In the figure : The change of speed at the boundary is the only reason light bends.

Rule 1 : Direction of bending and the glass slab — Rarer to denser: towards the normal, angle of refraction smaller than angle of incidence. Denser to rarer: away from the normal. A ray along the normal does not bend. In a rectangular glass slab the ray bends towards the normal on entering and away on leaving by the same amount, so the emergent ray is parallel to the incident ray but shifted sideways (lateral displacement); a thicker slab shifts it more.

• Easy : Light going from air into water bends towards the normal.
• Medium : A ray enters a glass slab at 40 degrees and leaves at 40 degrees, shifted sideways but parallel to the first ray.
• For exams : A ray entering a slab along the normal passes straight, with no bending and no lateral shift.

Rule 2 : Refractive index and optical density — n = c / v, so a larger n means slower light and an optically denser medium. n is never less than 1. For two media, n21 = v1 / v2 = n2 / n1. Optical density is not the same as mass density, and the frequency of light does not change on refraction.

• Easy : n = 1.5 gives v = 3 x 10^8 / 1.5 = 2 x 10^8 m/s.
• Medium : Light moves at 2.5 x 10^8 m/s in a liquid, so n = 3 x 10^8 / 2.5 x 10^8 = 1.2.
• For exams : Kerosene (n = 1.44) floats on water (n = 1.33) yet is optically denser: light travels at about 2.08 x 10^8 m/s in kerosene and 2.26 x 10^8 m/s in water.

Spot the difference

Equation / formula : 1/v - 1/u = 1/f ; m = v/u ; P = 1/f

In the figure : The ray through O goes straight; the ray parallel to the axis bends through F2 (convex) or seems to come from F1 (concave).

Rule 1 : Image positions for a convex lens — Object at infinity: image at F2, point-sized. Beyond 2F1: between F2 and 2F2, diminished. At 2F1: at 2F2, same size. Between F1 and 2F1: beyond 2F2, enlarged. At F1: image at infinity. Between F1 and O: virtual, erect, enlarged on the same side (magnifying glass). A concave lens always gives a virtual, erect, smaller image between O and F1.

• Easy : An object at 2F1 gives an image at 2F2, real, inverted and the same size.
• Medium : f = 12 cm and u = -36 cm: 1/v = 1/12 - 1/36 = 2/36, so v = +18 cm and m = 18/(-36) = -0.5 (diminished, real, inverted).
• For exams : For an enlarged real image keep the object between F1 and 2F1. To use the lens as a magnifying glass, keep the object between F1 and O.

Rule 2 : Lens formula and power — Signs: u is negative, f is positive for a convex and negative for a concave lens. Use 1/v = 1/f + 1/u. Power P = 1/f (f in metres): positive for convex, negative for concave; 1 dioptre is the power of a lens of focal length 1 m. The powers of lenses in contact add.

• Easy : f = 0.25 m gives P = 1/0.25 = +4 D.
• Medium : Lenses of +3.5 D and -1.5 D in contact give +2 D, so f = 1/2 = 0.5 m = 50 cm.
• For exams : Concave lens, f = -10 cm, u = -30 cm: 1/v = -1/10 - 1/30 = -4/30, so v = -7.5 cm and m = (-7.5)/(-30) = +0.25: virtual, erect, diminished.

Spot the difference