Rectilinear Propagation and the Law of Reflection
Rectilinear propagation means that light travels in straight lines in a uniform medium, which explains shadows and the pinhole camera, and the law of reflection states that the angle of incidence equals the angle of reflection, both measured from the normal, with the incident ray, reflected ray and normal in the same plane. In brief: light is represented by rays, straight lines that show its direction. A shadow forms when an opaque object blocks the rays. Smooth surfaces give specular (mirror-like) reflection and rough surfaces give diffuse reflection, which scatters light in all directions.
Geometric optics describes light using rays and ignores its wave nature, which is a good approximation when the objects are much larger than the wavelength of light. This guide describes the straight-line travel of light, shadows and eclipses, the law of reflection and types of reflection, compares specular and diffuse reflection in a table, works through shadow and pinhole camera calculations and corrects common mistakes.
How does light travel?
In a vacuum or a uniform transparent medium, light travels in straight lines at a constant speed: 3.00 × 108 m/s in a vacuum, slower in other media. A ray is a line showing the path of light, and a beam is a collection of rays (parallel, diverging or converging). Evidence for straight-line travel: sharp shadows, the beam of a torch in dusty air, and the inability to see around a corner. Objects are classified as luminous (they emit light, like the Sun or a lamp), transparent (transmit light, like clear glass), translucent (scatter transmitted light, like frosted glass) and opaque (block light). Light travels very fast: it takes about 1.3 s to go from the Moon to the Earth, and about 8.3 minutes from the Sun.
How do shadows and eclipses form?
A shadow is a region where light is blocked by an opaque object. A point source gives a sharp shadow. A larger source gives a dark central region, the umbra (no light from the source reaches it), surrounded by a partly lit penumbra (light from only part of the source reaches it). A solar eclipse occurs when the Moon moves between the Sun and the Earth, so that the Moon’s shadow falls on the Earth. A lunar eclipse happens when the Earth’s shadow falls on the Moon. A pinhole camera is a light-tight box with a small hole: rays from each point of an object travel through the hole in straight lines and form an inverted real image on the opposite wall. By similar triangles, the image height ÷ object height = image distance ÷ object distance.
What is the law of reflection?
When a ray strikes a surface, it can be reflected. The normal is the line perpendicular to the surface at the point of incidence. The two laws of reflection are:
- The angle of incidence i equals the angle of reflection r: i = r (both measured from the normal).
- The incident ray, the reflected ray and the normal all lie in the same plane.
On a smooth surface (a mirror, still water), parallel rays remain parallel after reflection: specular reflection, which produces images. On a rough surface (paper, a wall, cloth), the normals vary from point to point, so the reflected rays go in many directions: diffuse reflection. This is how most objects are seen from all directions, and why a wet road is more reflective at night, as water fills the roughness and increases specular reflection. The laws hold for each ray in both cases.
Specular and diffuse reflection compared
| Feature | Specular | Diffuse |
|---|---|---|
| Surface | Smooth (polished metal, still water) | Rough (paper, wall) |
| Reflected rays | Parallel rays stay parallel | Scattered in many directions |
| Image formed? | Yes | No |
| Law of reflection | Obeyed | Obeyed at each tiny part |
Worked example: shadows and pinhole images
- Shadow of a pole. A vertical pole 2.0 m tall casts a shadow 3.0 m long. At the same time, a tree casts a shadow 12 m long. By similar triangles (the Sun’s rays are parallel), the height of the tree = 2.0 × 12 ÷ 3.0 = 8.0 m.
- Pinhole camera. A candle flame 5.0 cm high is 40 cm from the pinhole of a camera whose back wall is 10 cm from the pinhole. Image height = 5.0 × 10 ÷ 40 = 1.25 cm, inverted.
- Law of reflection. A ray strikes a plane mirror so that it makes an angle of 30° with the surface. The angle of incidence from the normal is 90° − 30° = 60°, so the angle of reflection is 60°, and the angle between the incident and reflected rays is 120°.
- Turning a mirror. If the mirror is rotated by 10° while the incident ray stays fixed, the reflected ray turns by 20°. This principle is used in mirror galvanometers and laser scanners to amplify small rotations.
- Distance from echo of light. Laser pulses are reflected from a retroreflector left on the Moon, and the return takes 2.56 s. The distance = 3.00 × 108 × 2.56 ÷ 2 = 3.84 × 108 m.
How were these laws discovered?
Euclid’s Optica (about 300 BCE) described the straight-line travel of light and the equality of angles in reflection. Ibn al-Haytham (Alhazen), in the Book of Optics (about 1021), carried out experiments with the camera obscura and explained vision as light entering the eye. The law of reflection was stated by Hero of Alexandria in the first century CE, who proved by geometry that the path of reflected light is the shortest. Pierre de Fermat generalized the idea as the principle of least time in 1662.
Which mistakes are common?
Mistake 1: Measuring the angles from the mirror surface
Angles of incidence and reflection are measured from the normal.
Mistake 2: Saying that rough surfaces do not obey the law of reflection
Each ray obeys it, but the normals point in many directions.
Mistake 3: Thinking that light needs a medium to travel
Light travels through a vacuum.
Mistake 4: Believing that a larger hole in a pinhole camera gives a sharper image
A larger hole gives a brighter but blurrier image.