Light

Reflection and the law of reflection

When light hits a surface, some of it bounces back. This is reflection. To describe reflection properly you always measure angles from the normal, which is the line drawn perpendicular to the surface at the point where the ray hits it. The angle of incidence, i, is the angle between the incident ray and the normal, and the angle of reflection, r, is the angle between the reflected ray and the normal.

The law of reflection states that the angle of incidence is equal to the angle of reflection: i = r. The incident ray, the reflected ray and the normal all lie in the same plane. A smooth surface such as a mirror gives regular reflection, so a clear image is seen, while a rough surface scatters the light in all directions, which is why you cannot see your face in a painted wall.

You can show the law with a ray box and a plane mirror. A narrow ray is aimed at the mirror and the two angles are measured from the normal with a protractor. Move the ray to a new angle of incidence and the angle of reflection changes by exactly the same amount, so the two angles are always equal.

There is one useful calculation. If the mirror is turned through an angle, the reflected ray turns through twice that angle, because the normal turns with the mirror. Rotating a mirror by 10 degrees therefore swings the reflected ray by 20 degrees.

• Angle of incidence = angle of reflection (i = r).
• Angles are always measured from the normal, not from the surface.
• The incident ray, reflected ray and normal all lie in the same plane.
• Turning the mirror through an angle turns the reflected ray through twice that angle.

Images formed by a plane mirror

A plane mirror forms an image with four characteristics you should be able to state: the image is the same size as the object, it is as far behind the mirror as the object is in front, it is upright, and it is virtual. It is also laterally inverted, which is why the word AMBULANCE is written back to front on the front of the vehicle.

The image is virtual because the reflected rays diverge. They only appear to come from a point behind the mirror, and no light actually passes through that point, so a screen placed behind the mirror shows nothing at all. This is the opposite of a real image, where the rays really do meet and the image can be projected onto a screen.

You can construct the image by drawing two rays from a point on the object. Reflect each ray at the mirror using i = r, then extend the reflected rays backwards until they cross. That crossing point is the position of the image of that point. Repeating this for the top and the bottom of the object gives the whole image, and it always comes out the same size and the same distance behind the mirror as the object is in front.

Because the image is as far behind the mirror as the object is in front, you can calculate distances directly. If you stand 1.5 m from a plane mirror, your image is 1.5 m behind it, so the distance between you and your image is 3.0 m. If you walk towards the mirror at 0.5 m/s, your image also moves towards you at 0.5 m/s, so the gap between you closes at 1.0 m/s.

Refraction at a boundary

Refraction is the change in direction of light when it crosses the boundary between two transparent materials. It happens because the speed of light is different in different materials: light travels fastest in a vacuum and in air, and more slowly in water and glass. When light enters a denser medium, one in which it travels more slowly, it slows down and bends towards the normal. When it leaves the denser medium it speeds up again and bends away from the normal.

The angle of refraction, r, is the angle between the refracted ray and the normal. If light enters along the normal the angle of incidence is zero, and although the light slows down it does not change direction, so the angle of refraction is also zero.

You can show refraction with a ray box and a glass block. A ray is directed at the block at a known angle of incidence, and as it enters the glass it bends towards the normal. As it leaves the far side it bends away from the normal, so the emergent ray is parallel to the incident ray but displaced sideways. Repeat the experiment with blocks of different shapes, such as a rectangular block, a semicircular block and a triangular prism, and the same rules hold at every boundary.

Light passing through a prism is refracted twice, once at each surface, and this time the two refractions do not cancel because the faces of a prism are not parallel. The ray is turned through an angle, and this is the basis of the dispersion effect in the last block.

Refractive index, critical angle and total internal reflection

The refractive index, n, of a material compares the speed of light in a vacuum with its speed in the material: n = speed of light in a vacuum / speed of light in the material. Because it is a ratio of two speeds, n has no unit. Air has a refractive index of about 1.00, water 1.33 and glass about 1.50.

Refractive index can also be found from the angles. For light going from air into a material, n = sin i / sin r, where i is the angle of incidence in air and r is the angle of refraction in the material. The larger the refractive index, the more strongly the light is bent.

Here is the equation working on real numbers. Light passes from air into water at an angle of incidence of 50 degrees, and the refractive index of water is 1.33. Rearranging n = sin i / sin r gives sin r = sin 50 / 1.33 = 0.766 / 1.33 = 0.576, so r = 35 degrees. The ray has bent towards the normal, exactly as expected when it enters a denser medium.

When light tries to leave a denser medium such as glass, it bends away from the normal. As the angle of incidence is increased, the angle of refraction reaches 90 degrees and the refracted ray runs along the surface. The angle of incidence at which this first happens is the critical angle, c. For any angle of incidence greater than c the light cannot escape at all, and it is all reflected back inside the material. This is total internal reflection.

The critical angle and the refractive index are linked by n = 1 / sin c, or sin c = 1 / n. For glass of refractive index 1.50, sin c = 1 / 1.50 = 0.667, so c = 41.8 degrees. For water, sin c = 1 / 1.33 = 0.752, so c = 48.8 degrees. A larger refractive index always gives a smaller critical angle.

Total internal reflection has two conditions that must both be true: the light must be travelling from a denser to a less dense medium, and the angle of incidence must be greater than the critical angle. Everyday examples include the sparkle of a cut diamond, the reflection seen at the surface of a glass of water viewed from the side, and optical fibres.

An optical fibre is a thin strand of glass with a core of higher refractive index surrounded by cladding of lower refractive index. Light entering the core at a shallow angle strikes the core-cladding boundary at an angle greater than the critical angle, so it is totally internally reflected and stays inside the core. This happens many thousands of times along the fibre, so light can carry a signal for many kilometres with very little loss. Optical fibres are used in telecommunications because they carry far more information than copper cables and are not affected by electrical interference.

Thin lenses: converging and diverging

A lens is a piece of transparent material shaped so that it refracts light. A converging (convex) lens is thicker in the middle. It takes a beam of light parallel to the principal axis and brings it together at a single point called the principal focus, or focal point. The principal axis is the line passing through the centre of the lens at right angles to it, and the focal length, f, is the distance from the centre of the lens to the principal focus.

A diverging (concave) lens is thinner in the middle. It spreads a parallel beam out, so the rays appear to come from a point on the principal axis behind the lens. That point is its principal focus, and the distance to it is again the focal length. A shorter focal length means a more strongly curved lens that bends the light more, and so a more powerful lens.

You can find the focal length of a converging lens by focusing the image of a distant object, such as a window or a tree, onto a screen. When the image is sharpest, the distance from the lens to the screen is the focal length, because light from a very distant object arrives almost parallel and is brought to a focus at the principal focus.

Ray diagrams: real and virtual images

To draw a ray diagram for a converging lens, use two or three rays from the top of the object. The three standard rays are: a ray parallel to the principal axis, which passes through the principal focus after the lens; a ray through the centre of the lens, which carries straight on without bending; and a ray through the principal focus, which emerges parallel to the principal axis. Where two of these rays cross after the lens is where the image of the top of the object is formed.

If the object is further from the lens than the focal length, the rays really do cross after the lens and a real image is formed. A real image is inverted and it can be projected onto a screen. Its size depends on the object distance: an object beyond twice the focal length gives a diminished image, as in a camera, while an object between one and two focal lengths from the lens gives an enlarged image, as in a projector.

If the object is closer to the lens than the focal length, the rays diverge after passing through the lens. They never actually meet, but when they are extended backwards they appear to come from a point on the same side of the lens as the object. This is a virtual image. It is upright and enlarged, and it cannot be projected onto a screen because no light actually passes through it. This is exactly how a magnifying glass works.

Lenses are also used to correct vision. In short-sightedness the eye focuses the light in front of the retina, so a diverging lens is used to spread the rays slightly before they enter the eye. In long-sightedness the eye cannot focus the light enough and the image would form behind the retina, so a converging lens is used to bring the rays together sooner.

Dispersion: splitting white light into a spectrum

White light is a mixture of all the colours of the visible spectrum. When a narrow beam of white light passes through a triangular glass prism, it is refracted at both surfaces and spread out into a band of colours called a spectrum. This spreading is called dispersion.

The reason is that the refractive index of glass is slightly different for each colour. Violet light has the highest frequency and is slowed down the most by the glass, so it has the largest refractive index and is refracted through the largest angle. Red light has the lowest frequency, is slowed down the least, and is refracted the least. The colours therefore leave the prism at slightly different angles and separate out.

The traditional seven colours, in order of decreasing frequency and therefore of increasing wavelength, are red, orange, yellow, green, blue, indigo, violet. Red has the longest wavelength and the lowest frequency, and violet has the shortest wavelength and the highest frequency. The order is remembered by the mnemonic Richard Of York Gave Battle In Vain.

Light of a single frequency is described as monochromatic. A monochromatic beam is not split by a prism, it is only refracted. A rainbow is a natural spectrum, produced when sunlight is refracted and reflected inside millions of raindrops.