## Alice can see her whole reflection

Alice's image mirror mirror

Alice's image

### Figure 2.4 A full-length plane mirror.

mirror as any light rays from Alice reflected from points higher or lower have no chance of reaching her eyes. They would hit the mirror at the wrong angle, and go off into space, and will not be seen — at least not by Alice!

Alice has mounted the mirror in the best position to get a full-length reflection. It is exactly half her height and extends from a point halfway between the floor and her eye level, to a point midway between eye level and the top of her head.

What we see in a plane mirror is an illusion, reconstructed in our mind. There is of course nothing behind the mirror. The reflected rays appear to diverge from something. Such an image is called a virtual image.

### 2.2.2 Reversal from left to right

The image you see in a plane mirror is standing upright and facing in your direction. You see what appears to be a copy of yourself, "turned around", and looking back, face to face. On closer examination, you realise that what you see is not an exact copy, and is not quite how others see you.

The image in the mirror has 'turned around' in a peculiar way. Imagine that you went behind the mirror and turned around to 'face yourself. The act of turning consisted of swinging everything around a central axis, your right arm now appearing on the left side of the mirror, your left arm rotating to the other side. That is not what happened to the image. Every dot on the image has turned around independently. This results in a reversal of left and right, clockwise and anti-clockwise.

The young sportsman in the photograph does not quite see himself as others see him. He is holding the ball in the other hand. The number 13 on his shirt is written backwards. Even mi

The world in a mirror.

more noticeable is the image of the clock; it is going anticlockwise, and the numbers do not look right!

### 2.2.3 Reflection from a curved and uneven surface

So far we have only considered reflections at a plane surface. The same law of reflection applies if the mirror is curved, provided that we measure the angles relative to the normal at the point of reflection.

When a light ray meets a rough reflecting surface, the normal at the point of reflection can have practically any direction, resulting in diffuse reflection at all angles, as seen in Figure 2.5.

2.2.4 A spherical concave mirror

A spherical mirror consists of a reflecting surface which is curved, so that it forms part of the surface of an imaginary

The world in a mirror.

Figure 2.5 The incoming ray 'sees' only the normal at the precise point of reflection.

(large) hollow sphere. In the case of a concave mirror, the 'hollow' (concave) side is facing the incoming rays. Such a mirror has interesting properties which a plane mirror does not have, basically owing to the fact that the rays reflected from a concave surface tend to be brought together. For example, a parallel beam of light (provided it is not too far away from the axis) can be brought together at a point called the focus, and conversely a source of light placed at the focus will give rise to a reflected beam parallel to the axis. These properties can have many practical applications, some of which are illustrated in Figures 2.6 and 2.7.

A spherical mirror, however, will not bring all rays perfectly to the focus, and rays far from the optic axis will not cross that axis at quite the same point after reflection as those which came in close to the axis. To make a mirror with perfect focusing properties, the surface

sphere

Figure 2.6 A concave mirror will reflect parallel rays through a focal point.

sphere

Figure 2.6 A concave mirror will reflect parallel rays through a focal point.

sphere

Figure 2.7 The same mirror will give a beam of parallel reflected light if the source is placed at the focus.

sphere

Figure 2.7 The same mirror will give a beam of parallel reflected light if the source is placed at the focus.

has to be paraboloidal rather than spherical. The geometrical properties of a parabola exactly fulfil the criteria for perfect focusing. In Appendix 2.1 this property of a parabolic mirror is derived directly from Fermat's principle.

2.2.5 Applications of concave mirrors

Practical applications of concave mirrors are more common than one might expect. They can be divided into two classes:

### 1. Creation of a directed beam of light

A source of light (or any source of electromagnetic rays) at the focus of a concave mirror will give rise to a parallel beam. In the case of a lighthouse the mirror revolves around the source, producing a sweeping parallel beam of light.

2. Bringing light to a focus

Light shining over the entire surface area of a large concave mirror will be concentrated at the focus.

Lighthouse at Hook Head. ^se^a^ry and CaUe^.

Courtesy of the Commissioners of Irish Lights.

Lighthouse at Hook Head. ^se^a^ry and CaUe^.

Courtesy of the Commissioners of Irish Lights.

VLA. Courtesy of NRAO/AUI/NSF.

Reflecting telescope

An almost perfect example of a parallel light beam is light from a distant star. The fact that light from a dim source can be effectively gathered over the entire surface of a large mirror and focused at a point finds an obvious application in astronomical telescopes. The Hale telescope at Mount Palomar, for example, uses a parabolic mirror, with a diameter of 5.1 m (200 inches).

Mirrors are preferable to lenses in many optical applications. We will meet lenses in the next chapter; suffice it to say here that some wavelengths of the electromagnetic spectrum are absorbed by the glass of a lens — a problem which does not arise in the case of a mirror. In addition, high precision quality mirrors are easier and cheaper to manufacture than large lenses.

Radio waves from cosmic sources are brought to the focus of each mirror in a similar way to visible light.

The Very Large Array (VLA) of radio telescopes near Soccoro, New Mexico.

2.2.6 The 'death rays' of Archimedes

According to legend, Archimedes (287-212 BC) set fire to an invading fleet of Roman ships at the siege of Syracuse using an arrangement of mirrors and 'burning glasses'. This event is commemorated in a painting by Guiglio Parigi which may be seen in the Galleria degli Uffizi in Florence. That the artist was unfamiliar with the laws of physics is apparent, in that the painting shows divergent (rather than convergent) light rays — and these would be quite harmless!

While the arrangement depicted by Parigi may not be scientifically correct, the idea of using mirrors to focus sunlight at a distant point may not be as absurd as it at first appears. George Louis LeClerc, Comte de Buffon (1707-1788), set out in 1747 to produce fire using Archimedes' method. He assembled 168 mirrors in an arrangement designed to focus sunlight at a distance of about 50 m. It is said that he succeeded in igniting a plank of wood instantly — showing that the system could indeed be used as a formidable weapon at that range! He also claimed to have melted 3 kg of tin at 6 m using just 44 of his mirrors. Buffon's work, while it had not been subjected to rigorous scientific scrutiny, had enough credibility to be recognised, and his portrait appears on a postage stamp issued by the French post office.

In 2004, the Discovery Channel television programme MythBusters tried to ignite a fire in a set-up similar to that described in legend at the battle of Syracuse, but the attempt failed. The programme concluded that the legend could not be true, and that the myth of Archimedes' death rays had been 'busted'.

Burning mirrors. Courtesy of David Wallace and MIT Mechanical Engineering 2.009 class.

David Wallace and his 2.009 class at the Massachusetts Institute of Technology (MIT) were not convinced. In a feasibility project they assembled a total of 127 plane mirrors to form a giant concave mirror, which they used to focus sunlight on a wooden model of a ship at a distance of about 30 m. After about 10 minutes' exposure to the death rays the hull of the ship burst into flames. The MIT team admits that the conditions at the siege of Syracuse were not quite the same as in their reconstruction. The mirrors available to Archimedes would have been made of copper, not glass, and the ships would probably have been further away than 30 m. Nonetheless, they did show that the account of Archimedes and the burning ships is credible. The project has also confirmed the results of Buffon's experiments.

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