Light: Mirrors, Lenses and the Human Eye
2. Spherical mirrors: terms, sign convention and the mirror formula
A concave mirror curves inward (reflecting surface on the inside of the sphere) and converges parallel rays; a convex mirror bulges outward and diverges them. The pole P is the centre of the mirror, the centre of curvature C is the centre of the sphere it was cut from, the radius of curvature R = PC, and the principal focus F is where rays parallel to the axis meet (concave, real) or appear to come from (convex, virtual). The focal length f = PF and f = R/2.
New Cartesian sign convention: measure all distances from the pole; the object is always placed to the left, and distances in the direction of the incident light (to the right) are positive, the opposite direction negative. Heights above the axis are positive, below negative. So a real object has u negative; a concave mirror has f negative and a convex mirror f positive. The mirror formula is 1/v + 1/u = 1/f and the magnification is m = h′/h = −v/u. A negative m means a real, inverted image; positive means virtual and erect.
Worked example 1. A concave mirror has f = −10 cm and the object is at u = −15 cm. Then 1/v = 1/f − 1/u = −1/10 + 1/15 = −1/30, so v = −30 cm: the image is 30 cm in front of the mirror, and m = −v/u = −(−30)/(−15) = −2, meaning real, inverted and twice as large. Worked example 2. A convex mirror has f = +20 cm and u = −30 cm. Then 1/v = 1/20 + 1/30 = 1/12, so v = +12 cm (12 cm behind the mirror) and m = −12/(−30) = +0.4: virtual, erect, diminished.
3. Images by mirrors and their uses
| Object position (concave mirror) | Image position | Nature and size |
|---|---|---|
| At infinity | At F | Real, inverted, highly diminished |
| Beyond C | Between F and C | Real, inverted, diminished |
| At C | At C | Real, inverted, same size |
| Between C and F | Beyond C | Real, inverted, enlarged |
| At F | At infinity | Real, inverted, highly enlarged |
| Between P and F | Behind the mirror | Virtual, erect, enlarged |
- Concave mirror (converges light; enlarged virtual image at close range): shaving and make-up mirrors, dentist's and doctor's head mirrors, reflectors of torches and vehicle headlights (the lamp sits at F to give a parallel beam), solar cookers and furnaces.
- Convex mirror (always erect and diminished, so a wide field of view): rear-view mirrors of vehicles, security mirrors in shops and on road bends.
4. Refraction, refractive index and lenses
Refraction is the bending of light when it passes obliquely from one transparent medium to another, because its speed changes. Entering a denser (optically denser) medium it bends towards the normal; entering a rarer medium it bends away. Snell's law: sin i / sin r is a constant for a given pair of media. The absolute refractive index is n = c/v, where c = 3 × 10⁸ m/s is the speed of light in vacuum; it has no unit and is never below 1. Typical values: water about 1.33, crown glass about 1.5, diamond about 2.42. In glass of n = 1.5 light travels at 3 × 10⁸/1.5 = 2 × 10⁸ m/s. A rectangular glass slab shifts the ray sideways but the emergent ray is parallel to the incident ray.
A convex (converging) lens is thicker at the centre; a concave (diverging) lens is thinner at the centre. The lens uses the same sign convention with distances from the optical centre: the lens formula is 1/v − 1/u = 1/f, magnification m = h′/h = v/u (note the sign differs from the mirror's −v/u), and power P = 1/f with f in metres; the unit is the dioptre (D). Convex lens: f and P positive. Concave lens: f and P negative. Powers of lenses placed in contact add algebraically: P = P₁ + P₂.
Worked example 3. A convex lens f = +10 cm, object at u = −30 cm: 1/v = 1/f + 1/u = 1/10 − 1/30 = 1/15, so v = +15 cm (real image on the far side) and m = v/u = −0.5: inverted and half-size. Worked example 4. A concave lens f = −20 cm, u = −20 cm: 1/v = −1/20 − 1/20 = −1/10, v = −10 cm and m = +0.5: virtual, erect, half-size. Power: f = +25 cm gives P = +4 D; f = −50 cm gives P = −2 D; a +3 D lens in contact with a −1 D lens gives +2 D, so f = 50 cm.
5. The human eye and its defects
Light enters through the transparent cornea, which does most of the bending; behind it the iris (the coloured muscle) controls the size of the pupil, the opening that regulates how much light enters. The eye lens fine-tunes focus and throws a real, inverted image on the retina, where light-sensitive rod and cone cells send signals along the optic nerve to the brain. The ciliary muscles change the lens curvature: this is accommodation. For a normal eye the near point is about 25 cm and the far point is infinity.
| Defect | What the person sees | Cause | Correction |
|---|---|---|---|
| Myopia (short-sightedness) | Near clear, distant blurred | Eyeball too long or lens too curved; image forms in front of retina | Concave (diverging) lens, negative power |
| Hypermetropia (long-sightedness) | Distant clear, near blurred | Eyeball too short or lens too flat; image falls behind retina | Convex (converging) lens, positive power |
| Presbyopia | Near point recedes with age | Ciliary muscles weaken, lens loses flexibility | Bifocal lens (concave above, convex below) |
Worked example 5 (myopia). A person's far point is 80 cm. The lens must form the image of a very distant object (u = −∞) at 80 cm, so f = −80 cm = −0.8 m and P = 1/f = −1.25 D, a concave lens. Worked example 6 (hypermetropia). A person's near point is 50 cm but wants to read at 25 cm. The lens must make an object at u = −25 cm appear at v = −50 cm: 1/f = 1/v − 1/u = −1/50 + 1/25 = 1/50, so f = +50 cm and P = +2 D, a convex lens. Cataract is a different problem: the crystalline lens becomes cloudy with age, and it is treated by surgery, not spectacles.
6. Dispersion, atmospheric refraction and scattering
Dispersion: a glass prism splits white light into its seven colours, VIBGYOR (violet, indigo, blue, green, yellow, orange, red), because each colour has a different speed in glass and so bends by a different amount. Violet bends the most and red the least. Newton recombined the colours with a second inverted prism. A rainbow forms when sunlight is dispersed and internally reflected inside raindrops; it is seen with the Sun behind the observer, against a region of rain.
Atmospheric refraction: the air layers have different densities, so starlight bends gradually and its apparent position wavers; this is why stars twinkle. Planets are much closer and act as an extended source, so the fluctuations average out and they do not twinkle. The Sun is seen about 2 minutes before actual sunrise and about 2 minutes after actual sunset for the same reason.
Scattering: the Tyndall effect is the scattering of light by particles in a colloid or in air (a beam seen through a dusty room or forest mist). Scattering by air molecules is much stronger for shorter wavelengths, so scattered blue light fills the sky and it looks blue; at sunrise and sunset the sunlight passes through a thick air path, blue is scattered away and the Sun looks reddish. Red light is scattered least by fog and smoke, so it is used for danger signals, including railway signals, as it can be seen from far away. An astronaut sees a dark sky because there is no air to scatter light.
Key takeaways
- Angles of incidence and reflection are measured from the normal; a plane mirror image is virtual, erect, same size and laterally inverted, as far behind as the object is in front.
- Mirror: 1/v + 1/u = 1/f, m = −v/u, f = R/2; lens: 1/v − 1/u = 1/f, m = v/u, P = 1/f in dioptres (f in metres). Concave mirror and concave lens have negative f; convex mirror and convex lens have positive f.
- Real images are inverted and virtual images erect; a concave mirror or convex lens gives a virtual enlarged image only when the object is inside F; convex mirror and concave lens always give virtual, erect, diminished images.
- Refractive index n = c/v (c = 3 × 10⁸ m/s); light bends towards the normal in a denser medium. Lens powers in contact add algebraically.
- Myopia is corrected by a concave lens (negative power), hypermetropia by a convex lens (positive power), presbyopia by bifocals; lens power is P = 1/f in dioptres; for myopia f equals the far point distance (taken negative), and for hypermetropia use 1/f = 1/v − 1/u.
- Violet bends most and red least in a prism; stars twinkle from atmospheric refraction; the sky is blue, and the setting Sun red, because of scattering; red is used for danger signals as it scatters least.
Practice questions (16)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
Which mirror is fitted as a rear-view mirror in vehicles, and for what reason?
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Answer: A — Convex mirror, because it always gives an erect, diminished image and a wide field of view
A convex mirror gives an erect, diminished virtual image for any real object, and the diminished image lets a large area fit in a small mirror. A concave mirror gives a magnified image only at close range and an inverted real image for far objects, so it is not suitable; a plane mirror gives a virtual, not real, image and a narrow field.A concave mirror has a focal length of 10 cm. An object is placed 15 cm from its pole. Where is the image formed?
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Answer: B — 30 cm in front of the mirror
With f = −10 cm and u = −15 cm, 1/v = 1/f − 1/u = −1/10 + 1/15 = −1/30, so v = −30 cm: the image is real, 30 cm in front. The value 6 cm comes from adding 1/10 and 1/15 (wrong sign for f), and a positive v would mean a virtual image behind the mirror, which cannot happen for an object beyond the focus.Two plane mirrors are inclined to each other at an angle of 60°, and a small object is placed between them. How many images are formed?
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Answer: C — 5
The number of images is 360°/θ − 1 when 360°/θ is an even integer, so 360/60 − 1 = 5. The answer 6 forgets to subtract one (that count includes the object itself), and 3 is the result for mirrors at 90°.The speed of light in vacuum is 3 × 10⁸ m/s. What is its speed in a glass of absolute refractive index 1.5?
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Answer: D — 2 × 10⁸ m/s
Since n = c/v, v = c/n = 3 × 10⁸/1.5 = 2 × 10⁸ m/s. The value 4.5 × 10⁸ comes from multiplying instead of dividing, which would make light faster than c in a material and is impossible.A convex lens of focal length 10 cm forms an image of an object placed 30 cm from the lens. What is the image like?
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Answer: A — Real, inverted, at 15 cm on the other side of the lens
1/v = 1/f + 1/u = 1/10 − 1/30 = 1/15, so v = +15 cm, a real image on the far side with m = v/u = −0.5 (inverted, half-size). The 7.5 cm option arises from adding the two reciprocals, and a 30 cm image distance would need the object at 15 cm (u = −15 cm), not 30 cm; an object at 2f = 20 cm gives an image at 20 cm.What is the power of a convex lens of focal length 25 cm?
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Answer: B — +4 D
P = 1/f with f in metres: f = 0.25 m so P = +4 D, positive for a convex lens. Using f = 25 without converting to metres gives 1/25 = 0.04, which is wrong, and −4 D would belong to a concave lens of the same focal length.Consider these statements. P: A convex mirror always forms a virtual, erect, diminished image of a real object. Q: A concave lens can form a real image of a real object. R: A concave mirror forms a virtual, enlarged image when the object lies between its pole and focus. S: The image formed by a plane mirror is real. Which statements are correct?
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Answer: C — P and R only
P and R are correct. Q is false because a concave lens always diverges the light, giving a virtual, erect, diminished image. S is false because the rays only appear to come from behind a plane mirror, so its image is virtual; this is the most tempting error.Which of the following pairings of a device and its mirror is NOT correct?
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Answer: D — Reflector behind a vehicle headlight: convex mirror
A headlight reflector is a concave mirror with the lamp at its focus, so that the diverging light becomes a parallel beam; a convex mirror would only spread the beam. The other three pairings are standard: dentists and solar cookers use concave mirrors, and road-bend security mirrors use convex mirrors.A ray of light strikes a plane mirror so that it makes an angle of 30° with the surface of the mirror. What is the angle between the incident ray and the reflected ray?
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Answer: A — 120°
The angle of incidence is measured from the normal, so it is 90° − 30° = 60°. The reflected ray also makes 60° with the normal, so the angle between the two rays is 120°. The answer 60° is the tempting one, as it is the angle of incidence itself, and 30° wrongly treats the surface angle as the angle of incidence.A convex mirror has a focal length of +20 cm and a real object is placed 30 cm in front of it. What are the image distance and the magnification?
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Answer: B — 12 cm behind the mirror, m = +0.4
1/v = 1/f − 1/u = 1/20 + 1/30 = 5/60, so v = +12 cm (behind the mirror), and m = −v/u = −12/(−30) = +0.4: virtual, erect, diminished. The 60 cm option comes from subtracting the reciprocals instead of adding, which would wrongly give an enlarged image; a convex mirror never enlarges.A person can see nearby objects clearly but cannot see distant objects clearly. Which defect is this and which lens corrects it?
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Answer: C — Myopia, concave lens
Clear near vision with blurred distance vision is myopia, where the image forms in front of the retina; a concave (diverging) lens moves the image back onto the retina. Hypermetropia is the reverse defect and needs a convex lens; presbyopia is an age-related loss of accommodation.The far point of a short-sighted person is 80 cm. What is the power of the lens that lets this person see distant objects clearly?
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Answer: D — −1.25 D
The lens must form the image of an infinitely distant object at the far point, so f = −80 cm = −0.8 m and P = 1/f = −1.25 D. The value −0.8 forgets to take the reciprocal, and +1.25 D has the right size but a convex lens, which would worsen short sight.A long-sighted person has a near point of 50 cm. What lens power lets this person read a book held at the normal near point of 25 cm?
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Answer: A — +2 D
The lens must make an object at u = −25 cm appear at v = −50 cm: 1/f = 1/v − 1/u = −1/50 + 1/25 = 1/50, so f = +50 cm and P = +2 D. The answer +4 D is the tempting one that ignores the 50 cm near point and uses only 25 cm, while −2 D has the wrong sign for hypermetropia.Which sequence correctly traces the path of light entering a normal human eye to the point where it is converted into signals for the brain?
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Answer: B — Cornea, pupil, lens, retina, optic nerve
Light passes through the transparent cornea, then the pupil (whose size the iris controls), then the eye lens, and is focused on the retina, whose rod and cone cells send signals along the optic nerve. The pupil sits behind the cornea and in front of the lens, which is the step most often swapped.Why is red light used for danger signals, including railway signals?
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Answer: C — Red light is scattered the least by fog, dust and smoke, so it can be seen from far away
Scattering is strongest for short wavelengths and weakest for red, the longest visible wavelength, so a red signal passes through fog and smoke with the least loss. Red bends the least, not the most, in a prism, and its frequency is the lowest of the visible colours, so the first two reasons are wrong.The twinkling of stars is caused mainly by which of the following?
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Answer: D — Atmospheric refraction through air layers of changing density
Starlight passes through air layers of different and fluctuating densities, so it is refracted by slightly different amounts from moment to moment and the star's apparent position and brightness waver. Dispersion splits colours and does not make a star flicker, and the stars' own light does not change on a one-second scale.