- Apply the law of reflection and describe images in plane mirrors
- Use Snell's law n = sin i / sin r and n = c/v with special angles and without a calculator
- Decide when total internal reflection happens, find the critical angle and explain optical fibres, prisms and dispersion
- Find the image of a thin lens with 1/u + 1/v = 1/f and describe it (real or virtual, size, orientation)
Put a pencil in a glass of water and it looks broken at the surface. A swimming pool looks shallower than it is, a diamond sparkles, and the internet reaches your phone through glass threads thinner than a hair. All of this comes from two simple rules: how light bounces off a surface and how it bends when it crosses into another material. Geometrical optics treats light as rays that travel in straight lines; the wave side of light (interference, diffraction) is in the lesson «Physical optics: interference and diffraction».
Reflection and plane mirrors
The incident ray, the reflected ray and the normal (the line perpendicular to the surface at the point of incidence) lie in one plane, and the angle of reflection equals the angle of incidence. Both angles are measured from the normal, not from the surface.
- iangle of incidence (from the normal)
- i′angle of reflection
Consequences: the angle between the incident and reflected rays is 2i; if the mirror turns by θ while the incident ray stays fixed, the reflected ray turns by 2θ. Rough surfaces scatter light in all directions (diffuse reflection, 漫反射), but every tiny piece still obeys i′ = i.
A plane mirror forms an image that is virtual (the rays only seem to come from behind the mirror, so it cannot be caught on a screen), upright, the same size as the object and as far behind the mirror as the object is in front of it. Image and object are symmetric about the mirror, which is why left and right appear swapped.
1) A ray makes an angle of 25° with the surface of a mirror. Find the angle of reflection and the angle between the incident and reflected rays.
2) The incident ray stays fixed while the mirror is turned by 10°. By how much does the reflected ray turn?
3) Leyla stands 1.5 m in front of a plane mirror and then walks 0.5 m towards it. How far is she from her image now?
4) Murad is 1.70 m tall. What is the shortest vertical mirror in which he can see his whole body?
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2) The reflected ray turns by 2 · 10° = 20°.
3) She is 1.0 m from the mirror and the image is 1.0 m behind it: 2.0 m.
4) Rays from his feet and from the top of his head reflect halfway up: the mirror must be at least 0.85 m tall (half his height), at any distance.
Refraction and Snell's law
When light crosses from one transparent material into another at an angle, it changes direction because its speed changes. Going into a medium where light is slower (optically denser, 光密介质), the ray bends towards the normal; going into a faster (optically less dense, 光疏介质) medium, it bends away from the normal. A ray along the normal (i = 0) goes straight through. Light paths are reversible (光路可逆): a ray sent backwards retraces the same path.
- nrefractive index of the medium (折射率), always > 1; air ≈ 1, water ≈ 4/3, glass ≈ 1.5
- iangle of incidence in air (vacuum), from the normal
- rangle of refraction in the medium
- cspeed of light in vacuum, 3 × 10⁸ m/s
- vspeed of light in the medium
Chinese textbooks define n for a ray coming from air (vacuum): n = sin i / sin r. Inside the medium light travels at c/n.
1) A ray in air hits glass at i = 45°, and the angle of refraction is 30°. Find n.
2) Find the speed of light in glass with n = 1.5.
3) Light travels at 2.25 × 10⁸ m/s in water. Find the refractive index of water.
4) A ray in air falls at 60° on a medium with n = √3. Find the angle of refraction.
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2) v = c/n = 3 × 10⁸ / 1.5 = 2 × 10⁸ m/s.
3) n = c/v = 3/2.25 = 4/3.
4) sin r = sin 60°/√3 = (√3/2)/√3 = 1/2 ⇒ r = 30°.
- n₁, n₂refractive indices of the first and second media
- θ₁, θ₂angles from the normal in each medium
- λ₀, λwavelength in vacuum and in the medium; the frequency does not change
- h, h′real and apparent depth when you look almost straight down from air
The speed ratio follows from n = c/v: v₁/v₂ = n₂/n₁. The colour of light is set by its frequency, which stays the same in every medium.
1) A ray passes from a liquid with n₁ = √2 into glass with n₂ = √3 at an angle of incidence of 60°. Find the angle of refraction.
2) Light with a wavelength of 600 nm in vacuum enters water (n = 4/3). Find its wavelength and frequency in the water.
3) A pool is 2.4 m deep. How deep does it look from above (n = 4/3)?
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2) λ = 600/(4/3) = 450 nm; the frequency stays the same: f = c/λ₀ = 3 × 10⁸ / (6 × 10⁻⁷) = 5 × 10¹⁴ Hz.
3) h′ = h/n = 2.4 · 3/4 = 1.8 m: rays from the bottom bend away from the normal at the surface, so the bottom seems raised.
Light passes obliquely from water (n = 4/3) into glass (n = 3/2). Which statement is correct?
A) It bends away from the normal and speeds up
B) It bends towards the normal, slows down and keeps its frequency
C) It bends towards the normal and its frequency increases
D) It can be totally reflected if the angle is large enough
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D is tempting, but total internal reflection is possible only when light goes from a denser into a less dense medium; C changes the frequency, and A reverses the direction.
Total internal reflection and optical fibres
Send light from glass or water into air and increase the angle of incidence. The refracted ray bends away from the normal and gets weaker, and at one angle, the critical angle C, it skims along the surface (r = 90°). For any larger angle no light leaves: all of it is reflected back. This is total internal reflection (全反射). Two conditions must both hold: light goes from the optically denser medium into the less dense one, and i ≥ C.
- Ccritical angle (临界角), measured from the normal inside the denser medium
- nrefractive index of the medium, with air on the other side
- n₁ > n₂the denser and the less dense medium
It follows from n sin C = 1 · sin 90°. The larger n, the smaller C, and the easier total internal reflection becomes.
1) Find the critical angle for glass with n = √2 and for a material with n = 2.
2) Water has n = 4/3. A ray inside the water hits the surface at 45°. Does it leave the water?
3) Diamond has n ≈ 2.4. Why does it sparkle so much more than glass?
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2) sin C = 3/4 = 0.75, while sin 45° = √2/2 ≈ 0.71 < 0.75. So 45° < C: the ray leaves the water, bending away from the normal. At 50° (sin ≈ 0.77) it would be totally reflected.
3) sin C ≈ 1/2.4 ≈ 0.42, so C ≈ 24°, much smaller than for glass (about 42°). Light entering a cut diamond is reflected inside many times and leaves only through a few faces, so those faces shine brightly.
An optical fibre (光导纤维) is a thin glass thread with a core of higher refractive index inside a cladding of lower index. Light entering one end meets the core–cladding boundary at angles larger than the critical angle, so it is totally reflected again and again and follows the fibre even around gentle bends, losing very little energy. Fibres carry internet and telephone signals and let doctors look inside the body with endoscopes. The same effect turns a 45°–90°–45° glass prism into a perfect mirror in periscopes and binoculars.
Light travels in a medium with refractive index √3 towards its boundary with air. For which angle of incidence is the light totally reflected? (sin 35° ≈ 0.57, sin 40° ≈ 0.64)
A) 20°
B) 30°
C) 34°
D) 40°
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C (34°) is tempting because it is close, but it is still below the critical angle; A and B are far below it.
Prisms and dispersion
A triangular glass prism bends a ray twice, both times towards its thicker part, so the ray leaving the prism is deflected towards the base. The refractive index of glass is slightly different for different colours: largest for violet, smallest for red. So white light entering a prism fans out into a spectrum from red (least deviated) to violet (most deviated). This is dispersion (色散); a rainbow is dispersion in raindrops.
| Quantity (in glass) | Red | Violet |
|---|---|---|
| Refractive index n | smallest | largest |
| Speed v = c/n | largest | smallest |
| Deviation in a prism | least | most |
| Critical angle C | largest | smallest (totally reflected first) |
| Wavelength in vacuum | longest (≈ 700 nm) | shortest (≈ 400 nm) |
| Frequency | lowest | highest |
1) Red and violet light enter the same glass. Which travels faster in it, and which has the smaller critical angle?
2) A ray enters a 45°–90°–45° glass prism (n = 1.5) perpendicular to one short face and meets the long face at 45°. What happens there?
3) Would the same prism still work as a mirror if it were made of a plastic with n = 1.3?
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2) sin C = 1/1.5 = 2/3 ≈ 0.67 < sin 45° ≈ 0.71, so 45° > C: total internal reflection; the ray leaves through the other short face, turned by 90°.
3) sin C = 1/1.3 ≈ 0.77 > 0.71, so C > 45°: no, part of the light would escape through the long face.
Thin lenses and images
A converging (convex, 凸透镜) lens is thicker in the middle and brings parallel rays together at its focus F; a diverging (concave, 凹透镜) lens is thinner in the middle and spreads them out, as if they came from a focus in front of it. The distance from the lens to F is the focal length f. Three rays are enough to draw any image: a ray through the optical centre goes straight on; a ray parallel to the axis passes through F; a ray through F comes out parallel to the axis.
- uobject distance (物距)
- vimage distance (像距): + for a real image behind the lens, − for a virtual image on the object's side
- ffocal length (焦距): + for a converging lens, − for a diverging lens
- mmagnification: image height ÷ object height (taken without sign)
The Chinese form, with real distances positive. Azerbaijani and Russian textbooks write the same law as 1/d + 1/f = 1/F (d to the object, f to the image, F the focal length), so watch the letters.
| Object position | Image | Use |
|---|---|---|
| u > 2f | between f and 2f; real, inverted, smaller | camera, the eye |
| u = 2f | at 2f; real, inverted, same size | the boundary between smaller and larger |
| f < u < 2f | beyond 2f; real, inverted, larger | projector |
| u = f | no image (the rays leave parallel) | a parallel beam of light |
| u < f | on the object's side; virtual, upright, larger | magnifying glass |
A converging lens has f = 10 cm. Find the image and describe it for an object at 1) u = 30 cm; 2) u = 15 cm; 3) u = 5 cm.
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2) 1/v = 1/10 − 1/15 = 1/30 ⇒ v = 30 cm; m = 30/15 = 2: real, inverted, twice as large (projector).
3) 1/v = 1/10 − 1/5 = −1/10 ⇒ v = −10 cm: the minus sign means a virtual image on the object's side; m = 10/5 = 2: upright and twice as large (magnifying glass).
An object stands between the focus F and the point 2F of a converging lens. What is the image like?
A) Real, inverted, larger, beyond 2F on the other side of the lens
B) Real, inverted, smaller, between F and 2F
C) Virtual, upright, larger, on the object's side
D) Real, upright, larger, beyond 2F
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B is the case u > 2f (camera); C is u < f (magnifying glass); D pairs «real» with «upright», but a real image of a converging lens is always inverted.
How CSCA asks about this
- Reflection angles: «30° to the mirror surface» → i = 60°; the angle between the rays is 2i; a mirror turned by θ turns the ray by 2θ.
- Plane-mirror images: distances doubled, the image's speed relative to you doubled, the same size whatever the distance.
- Snell's law with special angles: 30°, 45°, 60° and answers such as √2, √3, √6/2; also n = c/v, λ = λ₀/n and apparent depth.
- Total internal reflection: «does the ray leave?» — compare sin i with 1/n; fibres and prisms as applications.
- Dispersion: order the colours by n, speed, deviation or critical angle, often as «I, II, III» statements.
- Lenses: where the image is for u > 2f, f < u < 2f and u < f, or one quick use of 1/u + 1/v = 1/f.
| Term | 中文 | Pinyin |
|---|---|---|
| reflection | 反射 | fǎnshè |
| refraction | 折射 | zhéshè |
| normal | 法线 | fǎxiàn |
| angle of incidence | 入射角 | rùshèjiǎo |
| angle of refraction | 折射角 | zhéshèjiǎo |
| plane mirror | 平面镜 | píngmiànjìng |
| refractive index | 折射率 | zhéshèlǜ |
| optically denser / less dense medium | 光密介质 / 光疏介质 | guāngmì jièzhì / guāngshū jièzhì |
| total internal reflection | 全反射 | quánfǎnshè |
| critical angle | 临界角 | línjièjiǎo |
| optical fibre | 光导纤维 (光纤) | guāngdǎo xiānwéi (guāngxiān) |
| prism / dispersion | 棱镜 / 色散 | léngjìng / sèsàn |
| convex / concave lens | 凸透镜 / 凹透镜 | tū tòujìng / āo tòujìng |
| focal length | 焦距 | jiāojù |
| real / virtual image | 实像 / 虚像 | shíxiàng / xūxiàng |
Key points
- Reflection: i′ = i, with angles from the normal; a plane mirror gives a virtual, upright, same-size image as far behind the mirror as the object is in front.
- Refraction: n = sin i / sin r = c/v (n₁ sin θ₁ = n₂ sin θ₂); into a denser medium the ray bends towards the normal; the frequency never changes, λ = λ₀/n.
- Total internal reflection needs light leaving the denser medium with i ≥ C, where sin C = 1/n; it guides light in optical fibres and turns light in 45° prisms.
- Dispersion: n is largest for violet and smallest for red, so violet is deviated most, travels slowest and has the smallest critical angle.
- Thin lens: 1/u + 1/v = 1/f, m = v/u; u > 2f camera, f < u < 2f projector, u < f magnifying glass (virtual image, v < 0).
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