Reflection and Refraction of Light
AdvancedLight reflects and refracts at a boundary: the angle of reflection equals the angle of incidence, and refraction follows Snell's law n₁ sin θ₁ = n₂ sin θ₂, with refractive index n = c/v measuring how much light slows in a medium. Going from a denser to a less dense medium at an angle beyond the critical angle, light is totally internally reflected — the principle behind optical fibres.
- Law of reflection: angle of incidence equals angle of reflection, all angles measured from the normal
- Refraction arises from the change of light speed: into a denser medium bend towards the normal, into a less dense one away from it
- n = c/v; Snell's law n₁ sin θ₁ = n₂ sin θ₂ is the core of refraction calculations
文體小知識：科學知識說明文——以光線模型講解反射定律與折射定律，引入折射率與全反射，並用光纖通訊說明原理怎樣變成科技，計算以斯涅爾定律為核心。
§Text Passage
1Geometrical optics describes light with the ray model: a straight line with an arrow shows the path light travels. Light striking a plane mirror or smooth surface reflects with the angle of reflection equal to the angle of incidence, both measured from the normal — the line perpendicular to the surface — with the incident ray, normal and reflected ray in one plane. For example, a mirror image is symmetric with the object because every ray retraces its path by this law.
2When light passes from one medium into another its speed changes, and the direction of travel bends — this is refraction. Entering a denser medium (slower light) the ray bends towards the normal; entering a less dense medium (faster light) it bends away. For example, a chopstick standing slanted in a glass of water looks bent upwards at the surface because light leaving the water bends away from the normal; a fish seen from the bank appears shallower than it really is for the same reason.
3The refractive index measures how much slower light is in a medium: it equals the speed of light in vacuum divided by the speed in the medium. Vacuum has index 1, air about 1.00, water 1.33, glass about 1.5 and diamond as high as 2.4. The larger the refractive index, the slower light travels and the more strongly rays bend towards the normal; comparing the two indices tells you the direction of bending — the first step of every refraction calculation.
4The heart of refraction calculations is Snell's law: the refractive index of the first medium times the sine of the incidence angle equals that of the second medium times the sine of the refraction angle, both measured from the normal. When light travels from a denser to a less dense medium the refraction angle exceeds the incidence angle; as the incidence angle grows to the critical angle the refracted ray skims along the surface at 90°, and beyond it all the light is reflected back — total internal reflection.
5Total internal reflection needs two conditions: light going from a denser to a less dense medium, and an incidence angle larger than the critical angle. Optical fibre is the flagship application: signal light advances by repeated total internal reflection inside the core with almost no loss, letting submarine cables carry internet data across oceans. Water's critical angle is about 48.8°, so a diver looking up sees a bright cone overhead and a mirror-like surface outside it — a natural demonstration of total internal reflection.
1Summary 重點摘要
Light reflects and refracts at a boundary: the angle of reflection equals the angle of incidence, and refraction follows Snell's law n₁ sin θ₁ = n₂ sin θ₂, with refractive index n = c/v measuring how much light slows in a medium. Going from a denser to a less dense medium at an angle beyond the critical angle, light is totally internally reflected — the principle behind optical fibres.
2Vocabulary cards Vocabulary
The angle of reflection equals the angle of incidence, both measured from the normal, with the rays coplanar.
The line perpendicular to the surface at the point of incidence; all angles are measured from it.
The bending of light when it enters another medium and its speed changes.
n = c/v, measuring how much slower light is in a medium than in vacuum; water 1.33, glass about 1.5.
n₁ sin θ₁ = n₂ sin θ₂, relating the angles of incidence and refraction.
The angle between the incident ray and the normal.
The angle between the refracted ray and the normal; smaller than the incidence angle when entering a denser medium.
When light in a denser medium strikes the boundary beyond the critical angle, all of it reflects back.
The incidence angle at which the refracted ray travels along the surface; sin C = 1/n.
A thin strand that guides light along its core by total internal reflection; used in communications and endoscopes.
3Step by step Steps
- Step 1: The ray model and the law of reflectionGeometrical optics draws light as rays: straight lines with arrows. When light strikes a boundary, the incident ray, the normal and the reflected ray lie in one plane, and the angle of reflection equals the angle of incidence, both measured from the normal. For example, your mirror image is symmetric with you, left and right swapped, because every ray obeys this law; at night, a lit room is reflected in the window glass for the same reason. In ray diagrams the normal is drawn as a dashed line perpendicular to the surface.
- Step 2: Refraction and its causeCrossing a boundary changes the speed of light and so bends the direction of travel: into a denser medium (slower light) the ray bends towards the normal; into a less dense medium (faster light) it bends away; at normal incidence (0°) the direction is unchanged. For example, a coin at the bottom of a cup looks shallower than it is because light leaving the water bends away from the normal and the eye traces the rays back along straight lines; pouring water into a cup can reveal a coin that was hidden by the rim — a classic classroom demonstration.
- Step 3: Refractive index n = c/vThe refractive index measures how much slower light is in a medium than in vacuum: n = c/v, where c = 3 × 10⁸ m s⁻¹. Vacuum has n = 1, air about 1.00, water 1.33, glass about 1.5 and diamond as high as 2.4. For example, light in water travels at v = c ÷ n = 3 × 10⁸ ÷ 1.33 ≈ 2.26 × 10⁸ m s⁻¹, and in glass at about 2 × 10⁸ m s⁻¹. The larger the refractive index, the slower the light and the stronger the bending towards the normal; diamond's small critical angle traps light in many internal reflections, which is why it sparkles.
- Step 4: Using Snell's lawn₁ sin θ₁ = n₂ sin θ₂ is the master key for refraction: θ is measured from the normal, not from the surface — if the question gives the angle to the surface, subtract it from 90° first. For example, light in air (n = 1.00) striking water (n = 1.33) at 40° incidence: 1.00 × sin40° = 1.33 × sinθ₂, so sinθ₂ = 0.643 ÷ 1.33 = 0.483 and θ₂ = sin⁻¹(0.483) ≈ 29°. The refraction angle 29° is smaller than the 40° incidence angle, the ray bending towards the normal — exactly what 'into a denser medium, towards the normal' predicts.
- Step 5: The two conditions for total internal reflectionTotal internal reflection requires both conditions: first, light must travel from a denser medium towards a less dense one — water to air, glass to air; second, the incidence angle must exceed the critical angle C. Then no light is refracted at the surface: all of it reflects back, obeying the law of reflection. For example, looking up from the bottom of a clear pool, the surface acts like a mirror reflecting the underwater lights, because rays reaching it from below at more than about 48.8° are totally reflected. Conversely, light going from air into water can never be totally reflected, whatever the angle.
- Step 6: The critical angle sin C = 1/nFind the critical angle from sin C = 1/n, where n is the refractive index of the denser medium with respect to the less dense one. For water: sin C = 1 ÷ 1.33 = 0.752, C = sin⁻¹(0.752) ≈ 48.8°. For glass to air: sin C = 1 ÷ 1.5 = 0.667, C ≈ 41.8°. For diamond: sin C = 1 ÷ 2.4 ≈ 0.417, C ≈ 24.6°. The larger the refractive index, the smaller the critical angle and the easier total internal reflection becomes — which is why diamond out-sparkles glass: rays are locked in by many more reflections before escaping through the crown.
- Step 7: Optical fibres and applicationsAn optical fibre is a hair-thin strand of glass or plastic whose core has a higher refractive index than the cladding. Light strikes the wall at more than the critical angle and advances along the core by thousands of total internal reflections with very little loss. Submarine fibre cables carry intercontinental internet traffic, and medical endoscopes pipe light and images into the body. For example, a fibre cable as thick as a finger holds dozens of fibres, each shifting huge volumes of data per second; signals stay crisp after a thousand reflections, letting data between Hong Kong and the world travel in milliseconds.
4Diagrams Diagrams
5Worked example Example
Example: a ray of light in air strikes a water surface at an incidence angle of 40°; the refractive index of water is 1.33. (a) Find the angle of refraction. (sin 40° = 0.643) (b) Find the critical angle for light going from water to air. (c) Name the principle that optical fibres rely on and state the conditions required.
- (a) Snell's law: n₁ sin θ₁ = n₂ sin θ₂, i.e. 1.00 × sin40° = 1.33 × sinθ₂
- (a) sinθ₂ = 0.643 ÷ 1.33 = 0.483
- (a) θ₂ = sin⁻¹(0.483) ≈ 28.9° ≈ 29°
- (b) sin C = 1 ÷ n = 1 ÷ 1.33 = 0.752, so C = sin⁻¹(0.752) ≈ 48.8°
- (c) Total internal reflection: light must travel from a denser medium (water or the glass core) to a less dense one (the cladding) at an angle beyond the critical angle
- Answer: (a) The angle of refraction is about 29°, the ray bending towards the normal in water. (b) The critical angle is about 48.8°. (c) Optical fibres rely on total internal reflection: light must go from a denser to a less dense medium at an incidence angle beyond the critical angle, so it advances along the core by repeated reflections. Always measure angles from the normal, and check that the refraction angle makes sense for the direction of travel.
6Common mistakes Mistakes
7Quick check Quick check
Q1State the law of reflection.
Q2Why does a coin in water look shallower than it really is?
Q3Calculate: light in air strikes glass (n = 1.5) at 30° incidence. What is the refraction angle? (sin 30° = 0.5)
Q4State the two conditions for total internal reflection.
Q5How do optical fibres carry light signals over long distances?
8Recap Recap
- Law of reflection: angle of incidence equals angle of reflection, all angles measured from the normal
- Refraction arises from the change of light speed: into a denser medium bend towards the normal, into a less dense one away from it
- n = c/v; Snell's law n₁ sin θ₁ = n₂ sin θ₂ is the core of refraction calculations
- Total internal reflection needs two conditions: denser to less dense medium, and incidence angle beyond the critical angle, sin C = 1/n
- Optical fibres carry signals by total internal reflection — submarine cables and endoscopes are its applications