Reflection, Refraction and Lenses

Reading a ray diagram

A ray diagram is a two-dimensional model of light. Each straight line represents the direction in which light travels through a uniform medium, and an arrowhead shows which way it is travelling. Where a ray meets a surface or crosses into another material, its direction may change.

Accurate diagrams need a ruler, fine lines and clearly positioned arrowheads. At a boundary, draw a normal through the point where the ray arrives. The normal is a construction line at $90^\circ$ to the surface; angles are measured between rays and this normal.

Specular reflection at a mirror

A smooth mirror produces specular reflection. An incoming incident ray and the reflected ray obey the law of reflection:

$\text{angle of incidence}=\text{angle of reflection}$

Both angles are measured from the normal. To construct the reflected ray, mark where the incident ray meets the mirror, draw the normal there and draw the outgoing ray on the other side of the normal at the same angle. Conversely, if part of a diagram is missing, this equality lets you reconstruct it.

Parallel rays arriving at a plane mirror remain parallel after reflection, provided they strike the same flat surface. This is why a smooth surface gives a predictable reflected beam rather than scattering light in many directions.

Refraction at a plane boundary

Refraction occurs when a wave crosses into a different medium and its speed changes. For light passing obliquely from air into glass, water or Perspex, the ray slows and bends towards the normal. On passing back into air, it speeds up and bends away from the normal. A ray arriving along the normal changes speed but continues in the same direction, so no bending is shown.

The direction changes because one side of an oblique wavefront reaches the new medium first and changes speed first. The whole wavefront turns, so the ray—which is perpendicular to the wavefront—also turns.

When constructing a diagram through a parallel-sided transparent block, draw a normal at each face. The ray bends towards the normal as it enters the slower medium and away from the normal as it leaves. Because the two faces are parallel, the emerging ray is parallel to the original incident ray, although it is displaced sideways.

Speed, frequency and wavelength at a boundary

Wave speed, frequency and wavelength are related by

$v=f\lambda$

where $v$ is wave speed in metres per second, $f$ is frequency in hertz and $\lambda$ is wavelength in metres.

The source fixes the frequency. A boundary does not change the rate at which successive wavefronts arrive, so the frequency stays constant when a wave is transmitted into another medium. A change in speed must therefore cause a proportional change in wavelength:

• if speed decreases, wavelength decreases;
• if speed increases, wavelength increases;
• frequency remains unchanged in both cases.

This inter-relationship applies to sound transmitted between media as well as to light. For example, if sound enters a medium in which it travels faster, its frequency remains the same but its wavelength becomes longer. If the wave crosses the boundary at an angle, the speed change may also change its direction.

Dispersion by a prism

White light is a mixture of visible wavelengths, from violet at the shortest visible wavelength to red at the longest. In a glass prism, different wavelengths travel at different speeds and therefore refract by different amounts. Refraction at the two faces separates the colours, producing a spectrum. This separation is called dispersion.

To construct a qualitative prism diagram, draw a narrow white incident ray, show it refracting at both sloping faces, and draw the emerging colours spreading into separate rays. At the glass-to-air face, every colour must bend away from that face's normal. Violet is deviated more overall than red. The prism does not manufacture the colours: it separates wavelengths already present in white light.

The shared features of lens diagrams

A lens refracts light at its curved surfaces. Begin any lens diagram by drawing the principal axis through the optical centre, then mark the principal focus $F$ at the same distance on each side. That distance is the focal length.

In the thin-lens ray model, a ray through the optical centre continues undeviated. Both convex and concave lens diagrams use the principal axis, optical centre, foci and straight rays; their main difference is whether they bring parallel rays together or spread them apart.

Convex lenses: converging rays

A convex lens is thicker in the middle and is also called a converging lens. Three useful construction rays are:

1. a ray parallel to the principal axis refracts through the focus on the far side;
2. a ray through the optical centre continues straight;
3. a ray passing through the near focus emerges parallel to the principal axis.

Two rays are normally enough to locate an image. If the refracted rays actually meet, the image is real and can be projected onto a screen. An object more than $2F$ from the lens produces a real, inverted image. An object between $F$ and $2F$ produces a real, inverted and magnified image.

If the object is inside the focal length, the refracted rays spread out. Extend them backwards with dashed lines to find where they appear to meet. The resulting image is virtual, upright and magnified, as in a magnifying glass.

Concave lenses: diverging rays

A concave lens is thinner in the middle and is also called a diverging lens. A ray parallel to the principal axis emerges spreading away as though it had come from the focus on the incident side. Show this by extending the refracted ray backwards with a dashed line through that focus. A ray through the optical centre is drawn undeviated.

The actual rays leaving a concave lens do not meet. Their backward extensions locate a virtual image on the same side of the lens as the object; for a real object this image is upright and diminished. The focus of a concave lens is therefore a point from which rays appear to diverge, whereas the far focus of a convex lens is a point at which parallel rays actually converge.

Mirror reflection

• Draw a normal at $90^\circ$ to the mirror where the ray strikes.
• Measure both angles from the normal.
• Apply $i=r$.