Light — Reflection & Refraction

Reflection of light is the bouncing back of light into the same medium when it strikes a surface. Reflection is clearest at a smooth, polished surface such as a mirror, but it happens at every surface. The ray falling on the surface is the incident ray, and the ray that bounces back is the reflected ray. The line drawn at right angles (90°) to the surface, at the point where the ray strikes, is called the normal.

The angle between the incident ray and the normal is the angle of incidence (i), and the angle between the reflected ray and the normal is the angle of reflection (r). Both angles are always measured from the normal, never from the surface.

The two laws of reflection are:

1. The angle of incidence is equal to the angle of reflection, i.e. i = r.
2. The incident ray, the reflected ray and the normal at the point of incidence all lie in the same plane.

These laws are obeyed by every reflecting surface, whether it is smooth or rough.

• Regular reflection: takes place at a smooth, polished surface (like a plane mirror). Parallel incident rays stay parallel after reflection, so a clear image is formed.
• Diffuse (irregular) reflection: takes place at a rough surface (like paper or a wall). Parallel rays are scattered in different directions, so no clear image is formed. This happens because the surface is uneven, not because the laws fail — each single ray still obeys i = r.

Image formed by a plane mirror. When you stand in front of a plane mirror, the image has these features:

• Virtual and erect — it cannot be caught on a screen and stands the right way up.
• Same size as the object.
• As far behind the mirror as the object is in front of it.
• Laterally inverted — the left and right sides appear interchanged (your right hand looks like the left hand of the image).

A spherical mirror is a mirror whose reflecting surface is a part of a hollow sphere. Imagine cutting a thin piece from a hollow glass ball and silvering one side — you get a spherical mirror. There are two kinds:

• A concave mirror has its reflecting surface curved inward (like the inside of a spoon). It brings parallel rays of light together, so it is a converging mirror.
• A convex mirror has its reflecting surface curved outward (like the back of a spoon). It spreads parallel rays apart, so it is a diverging mirror.

Learn these five terms — every ray diagram is built from them:

• Pole (P): the centre point of the mirror's reflecting surface. It lies on the mirror itself.
• Centre of curvature (C): the centre of the sphere of which the mirror is a part. It is not a part of the mirror. For a concave mirror it lies in front of the mirror; for a convex mirror it lies behind it.
• Radius of curvature (R): the radius of that sphere, that is, the distance PC.
• Principal axis: the straight line passing through the pole P and the centre of curvature C. It is normal (perpendicular) to the mirror at the pole.
• Aperture: the diameter of the reflecting surface. We always assume a small aperture — much smaller than R — so that all rays focus at one sharp point.

Now the focus. When rays travelling parallel to the principal axis strike the mirror:

• On a concave mirror the reflected rays actually meet at one point on the principal axis. This point is the principal focus (F), and it is real — it lies in front of the mirror.
• On a convex mirror the reflected rays spread out, but they appear to come from one point behind the mirror. That point is its principal focus (F), and it is virtual — it lies behind the mirror.

The distance from the pole P to the principal focus F is the focal length (f). For a mirror of small aperture, the focus lies exactly midway between the pole and the centre of curvature, so f = R/2 (equivalently, R = 2f).

Quick example: if a concave mirror has a radius of curvature of magnitude R = 20 cm, then its focal length has magnitude f = R/2 = 10 cm. Note: in the New Cartesian sign convention, distances are measured from the pole and the direction of the incident light is taken as positive; so for a concave mirror both R and f come out negative (here R = −20 cm and f = −10 cm, because C and F lie in front of the mirror), while for a convex mirror both are positive.

To find where a mirror forms an image, we draw a ray diagram. From the top of the object we draw any two of the standard rays below. The point where the reflected rays meet (for a real image), or appear to meet when produced backwards (for a virtual image), locates the top of the image.

1. A ray parallel to the principal axis, after reflection, passes through the principal focus F of a concave mirror, or appears to come from F of a convex mirror.
2. A ray passing through F (concave), or directed towards F (convex), becomes parallel to the principal axis after reflection.
3. A ray passing through the centre of curvature C (concave), or directed towards C (convex), is reflected straight back along its own path, because it strikes the mirror normally (perpendicular to it).
4. A ray striking the pole P is reflected making equal angles with the principal axis, since the principal axis is the normal at the pole.

For a concave mirror, the image changes a great deal as the object moves. Learn these six positions in order:

• Object at infinity → image at F; highly diminished, point-sized; real and inverted.
• Object beyond C → image between F and C; diminished; real and inverted.
• Object at C → image at C; same size as the object; real and inverted.
• Object between C and F → image beyond C; enlarged (magnified); real and inverted.
• Object at F → image at infinity; highly enlarged; real and inverted.
• Object between P and F → image behind the mirror; enlarged; virtual and erect.

So a concave mirror gives a virtual, erect image only when the object is between the pole and the focus. In every other position the image is real and inverted.

A convex mirror is much simpler. Whatever the object position, the image is always virtual, erect and diminished, and it always forms behind the mirror, between the pole P and the focus F. As the object moves from infinity towards the mirror, the image just shifts from near F outwards towards P, staying small and upright. Because it shrinks the scene, a convex mirror covers a much wider field of view.

Everyday uses of these mirrors:

• Concave mirrors are used as reflectors in torches, search-lights and vehicle head-lights, where the bulb is placed at the focus to throw a powerful parallel beam. They are used as shaving mirrors to see a larger, erect image of the face, by dentists to see enlarged images of teeth, and in solar furnaces and solar cookers/heaters to concentrate sunlight and produce heat.
• Convex mirrors are used as rear-view (wing) mirrors in vehicles, because they always give an erect, diminished image and show a wider area, letting the driver see more traffic behind. For the same wide-view reason they are also used as security mirrors in shops.

Before you use any mirror equation you must fix the signs. We follow the New Cartesian Sign Convention. The object is always placed on the left, so light travels from left to right, and every distance is measured from the pole (P) of the mirror along the principal axis.

• Distances measured in the direction of the incident light (to the right of P) are taken positive; distances measured against the incident light (to the left of P) are taken negative.
• So the object distance u is always negative, because the object is on the left.
• Heights measured upwards from the principal axis are positive; heights measured downwards are negative.
• Hence for a concave mirror the focal length f is negative, and for a convex mirror f is positive.

The mirror formula connects the three distances: 1/v + 1/u = 1/f, where u is the object distance, v the image distance and f the focal length. Always put in the values with their signs; the sign of your answer then tells you on which side the image forms.

The magnification (m) tells you how big the image is compared with the object, and whether it is upright or inverted: m = h'/h = −v/u, where h is the object height and h' is the image height. Read the sign of m like this:

• m is negative → image is real and inverted.
• m is positive → image is virtual and erect.
• Size of m: more than 1 → enlarged; equal to 1 → same size; less than 1 → diminished.

Worked example: an object 4 cm tall is placed 25 cm in front of a concave mirror of focal length 15 cm. Find the image distance, the magnification and describe the image.

1. Signed data: u = −25 cm, f = −15 cm, h = +4 cm (here F = 15 cm and C = 2f = 30 cm, so the object lies between F and C).
2. Mirror formula: 1/v = 1/f − 1/u = 1/(−15) − 1/(−25) = −1/15 + 1/25 = (−5 + 3)/75 = −2/75.
3. So v = −37.5 cm — the image is 37.5 cm in front of the mirror, on the same side as the object (beyond C).
4. Magnification: m = −v/u = −(−37.5)/(−25) = −1.5.
5. Image height: h' = m × h = −1.5 × 4 = −6 cm.

The magnification is negative and greater than 1 in size, so the image is real, inverted and enlarged (6 cm tall), formed 37.5 cm in front of the mirror. This agrees with the ray-diagram rule that an object placed between F and C of a concave mirror gives a real, inverted, magnified image beyond C.

Refraction is the bending of light when it passes obliquely (at an angle) from one transparent medium into another. It happens because the speed of light changes from one medium to the next. If a ray hits the surface along the normal (angle of incidence 0°), it does not bend at all — only its speed changes.

Which way does it bend?

• Going from a rarer to a denser medium (for example, air to glass), light slows down and bends towards the normal, so the angle of refraction r is smaller than the angle of incidence i.
• Going from a denser to a rarer medium (for example, glass to air), light speeds up and bends away from the normal, so r is larger than i.

There are two laws of refraction:

1. The incident ray, the refracted ray and the normal at the point of incidence all lie in the same plane.
2. Snell's law: for a given pair of media and a given colour of light, the ratio (sin i)/(sin r) is a constant. This constant is the refractive index of the second medium with respect to the first: (sin i)/(sin r) = n₂₁.

The refractive index tells us how much a medium slows light down and bends it. The absolute refractive index of a medium is n = c/v, where c is the speed of light in vacuum (about 3 × 10⁸ m s⁻¹) and v is the speed of light in that medium. Because light is fastest in vacuum, n for any medium is greater than 1 (for air, n ≈ 1.0003, taken as 1; for water n ≈ 1.33, for glass n ≈ 1.5, and diamond is highest at 2.42). The relative refractive index compares two media: for light going from medium 1 into medium 2, n₂₁ = v₁/v₂ = n₂/n₁.

Optically denser vs rarer: a medium with a higher refractive index is optically denser, and one with a lower refractive index is optically rarer. This is decided by the refractive index, not by mass density. For example, kerosene has a higher refractive index than water, so it is optically denser than water, even though its mass density is lower.

Refraction through a rectangular glass slab: when a ray enters the slab it bends towards the normal (air → glass), and when it leaves it bends away from the normal (glass → air) by an equal amount. So the emergent ray is parallel to the incident ray, but shifted sideways. This sideways shift is called the lateral displacement (lateral shift). The two bendings are equal and opposite, which is why the final direction is unchanged and the angle of incidence equals the angle of emergence.

Worked example: The refractive index of glass is 1.5. How fast does light travel in it? Using n = c/v, we get v = c/n = (3 × 10⁸ m s⁻¹) ÷ 1.5 = 2 × 10⁸ m s⁻¹. So light is slower in glass than in vacuum, exactly as expected for an optically denser medium.

A lens is a piece of transparent glass bound by two surfaces, at least one of which is curved. There are two kinds. A convex (converging) lens is thicker in the middle and thinner at the edges; it bends parallel rays inward so that they meet. A concave (diverging) lens is thinner in the middle and thicker at the edges; it spreads parallel rays outward.

Learn these terms first:

• Optical centre (O) — the central point of the lens. A ray passing through O goes straight, without any deviation.
• Principal axis — the straight line passing through the optical centre and the centres of curvature of the two surfaces.
• Principal focus (F) — for a convex lens, the point on the principal axis where rays coming parallel to the axis actually meet after refraction (a real focus). For a concave lens, the point from which such rays appear to diverge after refraction (a virtual focus).
• Focal length (f) — the distance between the optical centre O and the principal focus F.
• Every lens has two foci, F₁ and F₂, one on each side, because light can fall on it from either direction. Both lie at equal distances from O.

To locate an image, draw any two of these three standard rays and see where they meet (or appear to meet):

1. A ray travelling parallel to the principal axis, after refraction, passes through the principal focus F (in a concave lens it appears to come from F on the same side as the object).
2. A ray passing through the optical centre O passes straight on, without bending.
3. A ray passing through the principal focus F emerges parallel to the principal axis after refraction.

For a convex lens, the image changes as the object moves along the axis:

• Object at infinity → image at focus F; highly diminished (point-sized); real and inverted.
• Object beyond 2F → image between F and 2F on the other side; diminished; real and inverted.
• Object at 2F → image at 2F on the other side; same size as the object; real and inverted.
• Object between F and 2F → image beyond 2F on the other side; enlarged; real and inverted.
• Object at F → image at infinity; highly enlarged; real and inverted.
• Object between F and O → image on the same side as the object; enlarged; virtual and erect.

A concave lens is simpler. Wherever you place the object, the image is always virtual, erect and diminished, formed between the focus F and the optical centre O, on the same side as the object. As the object moves from infinity towards the lens, the image moves from F towards O, always staying small and upright.

Everyday uses:

• Convex lens — a magnifying glass (reading glass), the camera lens that forms a real image on the film or sensor, and spectacles to correct hypermetropia (long-sightedness).
• Concave lens — spectacles to correct myopia (short-sightedness).

First, fix the signs. We measure every distance from the optical centre of the lens, and we take the direction in which light travels as positive. So a distance measured in the direction of the incident light is positive, and one measured against it is negative. Because the object is always placed on the side the light comes from, the object distance u comes out negative. Heights measured upwards from the principal axis are positive; heights measured downwards are negative.

The lens formula connects the object distance u, the image distance v and the focal length f for a thin lens: 1/v − 1/u = 1/f. The same formula works for both types of lens, as long as you put in the correct signs. For a convex (converging) lens f is positive; for a concave (diverging) lens f is negative.

The magnification m tells you how tall the image is compared with the object: m = h'/h = v/u, where h' is the image height and h is the object height. Note carefully — for a lens there is no minus sign in v/u (this is different from a mirror, where m = −v/u).

Reading the sign and size of m:

• If m is positive, the image is virtual and erect (the same way up as the object).
• If m is negative, the image is real and inverted.
• If |m| > 1 the image is enlarged; if |m| < 1 it is diminished; if |m| = 1 it is the same size.

Worked example. An object is placed 15 cm in front of a convex lens of focal length 10 cm. Find the image distance, the magnification, and describe the image.

1. Write the signs: f = +10 cm (convex lens), u = −15 cm (object on the incoming-light side).
2. Rearrange the lens formula: 1/v = 1/f + 1/u = 1/10 + 1/(−15) = 1/10 − 1/15 = 1/30, so v = +30 cm.
3. Find the magnification: m = v/u = 30/(−15) = −2.
4. Describe the image: v is positive, so the image forms 30 cm from the lens on the far side (opposite the object), where a real image can be caught on a screen. m is negative, so the image is real and inverted, and |m| = 2, so it is twice the object's height (enlarged).

Now try to see this for yourself. Open the interactive lens ray-diagram simulator, drag the object along the principal axis, and watch the image flip from erect to inverted and change size as the object crosses the focus. The very numbers you just calculated will come alive in front of you.

The power of a lens is a measure of how strongly it bends the light rays passing through it — that is, the degree to which it converges or diverges the rays. A lens with a short focal length bends rays sharply, so it has a large power. A lens with a long focal length bends rays gently, so it has a small power.

Power is defined as the reciprocal of the focal length: P = 1/f, where the focal length f must be taken in metres. The SI unit of power is the dioptre, written as D. So 1 D = 1 m^-1. One dioptre is the power of a lens whose focal length is 1 metre.

• A convex (converging) lens has a positive focal length, so its power is positive.
• A concave (diverging) lens has a negative focal length, so its power is negative.
• The sign of the power (+ or −) follows directly from the sign of f in the New Cartesian sign convention.

When two or more thin lenses are placed in contact, the power of the combination is simply the sum of the individual powers: P = P₁ + P₂ + ... Opticians use this idea to work out the correct strength of the lenses in a pair of spectacles.

Worked example: Find the power of a convex lens of focal length 25 cm.

1. Convert the focal length to metres: f = 25 cm = 25/100 m = 0.25 m (positive, because the lens is convex).
2. Apply the formula: P = 1/f = 1/0.25 = +4 D. So the lens has a power of +4 D.

Now suppose this convex lens (power +4 D) is placed in contact with a concave lens of power −2 D. The net power is P = P₁ + P₂ = (+4 D) + (−2 D) = +2 D, so the combination still behaves as a converging lens.