This note covers how light reflects in plane mirrors, how it bends when it enters a transparent material (refraction), when it is totally internally reflected, how lenses form images, and how white light splits into colours. These ideas are used in periscopes, optical fibres, cameras, glasses and magnifying glasses.
Key ideas
Reflection in a plane mirror. Light obeys the law of reflection: the angle of incidence equals the angle of reflection, both measured from the normal. The image in a plane (flat) mirror is:
- virtual (the light only appears to come from it, and it cannot be caught on a screen),
- the same size as the object,
- as far behind the mirror as the object is in front,
- laterally inverted (left and right are swapped).
Refraction of light. Light travels more slowly in glass, water or perspex than in air, so it bends at the boundary.
- Air into glass (slower): the ray bends towards the normal.
- Glass into air (faster): the ray bends away from the normal.
- A ray entering along the normal (at 90° to the surface) does not change direction.
For a ray passing from air into a material, the refractive index n is:
n = sin i ÷ sin r
where i is the angle in air and r is the angle in the material, both measured from the normal. A larger n means more bending. Typical values are about 1.33 for water and 1.5 for glass.
Through a rectangular glass block, the ray bends towards the normal on entry and away from the normal on exit. The emerging ray is parallel to the incident ray but shifted sideways.
Total internal reflection. When light travels from a more optically dense material (such as glass) into a less dense one (such as air), the angle in the air is larger than the angle in the glass. As the angle in the glass increases, the refracted ray bends closer to the surface. At the critical angle c, the refracted ray travels along the boundary (an angle of 90°). For any larger angle of incidence, no light leaves and all of it reflects back inside. This is total internal reflection. It requires both:
- the light to be travelling in the denser material towards the boundary, and
- the angle of incidence to be greater than the critical angle.
(Extended) The critical angle is linked to the refractive index by sin c = 1 ÷ n.
Uses. Optical fibres carry light along a thin glass strand by repeated total internal reflection. They are used for telephone and internet signals and for endoscopes. Prisms in periscopes and binoculars turn light through 90° or 180° with total internal reflection.
Thin converging lenses. A converging (convex) lens refracts parallel rays so they meet at the principal focus. The distance from the centre of the lens to the principal focus is the focal length, f. To draw the image of a point on an object, use two rays:
- a ray parallel to the principal axis, which refracts through the principal focus on the far side,
- a ray through the centre of the lens, which carries straight on.
Where the rays meet is where that point of the image forms. The image depends on the object's distance u from the lens.
- Object beyond 2f: image is real, inverted, smaller, between f and 2f (camera).
- Object between f and 2f: image is real, inverted, larger, beyond 2f (projector).
- Object closer than f: image is virtual, upright, larger, on the same side of the lens as the object (magnifying glass).
A real image is formed by rays that actually meet, so it can be shown on a screen. A virtual image cannot. A diverging (concave) lens spreads parallel rays out as though from a point on the incoming side. It always gives a virtual, upright, smaller image.
Dispersion. White light is a mixture of colours. In a glass prism each colour has a different speed and bends by a different amount, so a spectrum is produced. Violet bends the most and red bends the least. Each colour has a different frequency. Monochromatic light has a single frequency and so a single colour.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Normal | Line at 90° to a surface. All angles of reflection and refraction are measured from it. |
| n = sin i ÷ sin r | Refractive index for light entering a material from air. No units. |
| Critical angle c | Angle of incidence inside a dense material at which the refracted ray travels along the boundary. |
| sin c = 1 ÷ n (Extended) | Gives the critical angle from the refractive index (or the other way round). |
| Principal focus | Point where rays parallel to the axis meet after a converging lens. |
| Focal length f | Distance from the centre of the lens to the principal focus. |
| Real / virtual image | Real: formed where rays meet, can be on a screen. Virtual: rays only appear to come from it. |
| Dispersion | Splitting of white light into colours because each colour refracts by a different amount. |
Worked examples
Example 1 A ray of light in air hits a glass block of refractive index 1.50. The angle of incidence is 40°. Find the angle of refraction.
n = sin i ÷ sin r, so sin r = sin i ÷ n.
sin r = sin 40° ÷ 1.50 = 0.6428 ÷ 1.50 = 0.4285.
r = sin−1(0.4285) = 25.4°.
The ray bends towards the normal, as expected for air into glass.
Answer: 25.4°
Example 2 In an experiment, light passes from air into a clear plastic with an angle of incidence of 45° and an angle of refraction of 28°. (a) Find the refractive index of the plastic. (b) (Extended) Find the critical angle for light travelling from the plastic into air.
(a) n = sin 45° ÷ sin 28° = 0.7071 ÷ 0.4695 = 1.506, so n = 1.51.
(b) sin c = 1 ÷ n = 1 ÷ 1.506 = 0.664.
c = sin−1(0.664) = 41.6°.
Answer: (a) 1.51; (b) 41.6°
Example 3 (Extended) An optical fibre has a refractive index of 1.60. A ray inside the fibre hits the wall at an angle of incidence of 50°. Does the ray escape from the fibre into the air? Justify your answer.
Critical angle: sin c = 1 ÷ 1.60 = 0.625, so c = sin−1(0.625) = 38.7°.
The ray is inside the denser material and its angle of incidence (50°) is greater than c (38.7°).
So total internal reflection occurs and no light escapes.
Answer: No. 50° is greater than the critical angle of 38.7°, so the ray is totally internally reflected.
Common errors
- Common error: measuring angles of incidence or refraction from the surface instead of from the normal.
- Common error: using sin i ÷ sin r the wrong way round. For light entering a material from air, i is the angle in the air and r the angle in the material, so n is greater than 1.
- Common error: saying total internal reflection can occur when light goes from air into glass. It only happens inside the denser material.
- Common error: drawing a refracted ray bending the wrong way. In glass or water the ray is always closer to the normal than it is in air.
- Common error: drawing a real image with dashed lines. Dashed lines are used for virtual images and construction lines behind the lens or mirror, and solid lines for real rays.
Exam tip: Check that your calculator is in degree mode before you start any sin or inverse sin calculation. Then check your answer against the rule: air to glass should give an angle smaller than the one in air.
Check yourself
- An object is 0.80 m in front of a plane mirror. How far is the object from its image?
Answer
The image is 0.80 m behind the mirror, so the distance from object to image is 0.80 + 0.80 = 1.60 m. - (Extended) Water has a refractive index of 1.33. Find the critical angle for light going from water into air.
Answer
sin c = 1 ÷ 1.33 = 0.752, so c = 48.8°. - A ray in air meets a glass surface (n = 1.50) at an angle of incidence of 30°. Find the angle of refraction.
Answer
sin r = sin 30° ÷ 1.50 = 0.5 ÷ 1.50 = 0.3333, so r = 19.5°. - Describe the image formed by a magnifying glass.
Answer
The object is closer to the lens than the principal focus. The image is virtual, upright and larger than the object, and it is on the same side of the lens as the object.