5.1 Introduction. 5.2 History

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$Lecture 5 5 Refraction of Light Reading Assignment: Read Kipnis Chapter 4 Refraction of Light, Section I, II. I have also summarized much of this material below. 5.1 Introduction In previous experiments we learned that when light falls on certain materials some of it is reflected back.$

1 Lecture 5 5 Refraction of Light Reading Assignment: Read Kipnis Chapter 4 Refraction of Light, Section I, II. I have also summarized much of this material below. 5.1 Introduction In previous experiments we learned that when light falls on certain materials some of it is reflected back. In some materials the light also goes through the material or body such as glass, plastic, or water. For example, I can see my face when light is reflected from the surface of water such as a swimming pool, which means that water reflects light. At the same time, I can see the bottom of the swimming pool, which means light is transmitted through the water and then reflected from the bottom of the pool. Light travels in straight lines until it encounters another material where it is partially reflected and partially transmitted. We learned that the angle of incidence is equal to the angle of reflection and the angles do not depend on the nature of the material. In refraction we will learn that the angle of the ray when transmitted through the material changes and depends on the speed of light in the two materials. Many phenomena encountered in our daily lives can be simply explained on the basis of refraction and reflection. Some of these are : why do fish look larger in the water, what causes the bending of a spoon in a water cup, why does light travel indefinitely in an optical cables, and of course how are rainbows, and mirages formed, etc. We begin our discussion of refraction from a historical perspective. 5.2 History When Napoleon's army occupied Egypt in soldiers frequently saw pools of water and lakes in a desert. but when the thirsty men approached them, the lakes disappeared. Napoleon had several prominent scientists with his expedition.

2 Fig. 5.1 The mathematician Gaspard Monge ( ) explained the phenomenon by an abnormal refraction of light in the atmosphere. When the ground is very hot, the temperature of air near the ground is high and the optical density is low. As the altitude increases, the temperature of the air decreases and its optical density increases. For this reason, light coming from the object AB is refracted as shown in Fig The eye of the soldier riding the camel imagines light coming from the direction A' as if reflected by water at the ground level. To understand this further we will consider a more simplified situation and see what happens to light when traveling from air into such bodies as water, glass and plastic. You may want to revisit this discussion after you understand the Law of Refraction Aristotle (Dates) The phenomena of refraction has been known since Aristotle, who for instance, mentioned that a straight stick partly immersed in water appears to be broken as shown below in Fig Can you think of a better way to start your studies on refraction than with a problem that challenged Aristotle? You will study this phenomena in detail in this home lab Ptolemy (Dates) Long interested in atmospheric refraction, Ptolemy conducted a series of measurements of the angles of refraction when light passed from air to water, from air to glass, and from water to glass. In Fig. 5.3 below imagine light traveling from air to water form point A to point E. Observe as the ray strikes the boundary between air and water the ray is bent and then continues in straight line again. The angle i is called the angle of incidence and the angle r is called the angle of refraction. Note how the angles are defined between the light ray and the dashed line at C, which is a perpendicular to the surface at C.

$Fig. 5.3 Ptolemy expected the ratio of the angle of incidence (i) to the angle of refraction (r) to be constant for the given two media.$

3 Fig. 5.3 Ptolemy expected the ratio of the angle of incidence (i) to the angle of refraction (r) to be constant for the given two media. However, the only thing he was able to deduce from his measurements was that when the angle of incidence increased, the ratio of this angle to the angle of refraction also increased. Subsequently, scientists adopted this ratio to be constant for angles below 30, which is a good approximation Willebrod Snellius ( ) The true law of refraction was discovered experimentally by Willebrod Snellius. He did not publish it, but it became known to people after his death. Snell's law states that CE/CF = const, the ratio of the length of specific lines CE/CF (instead of trigonometric functions) is constant for the given two media and the ratio does not depend on the angle of incidence Rene Descartes ( ) After Snellius death Descartes published the law in 1637 in a different form. Some scientists even claimed that Descartes was familiar with Snellius' work but no evidence was ever presented. Rene Descartes was born at La Haye in France in a family of minor nobility. He received a modem secondary education at the Jesuit school of La Fleche in Anjou. where he studied languages, mathematics, physics, and philosophy. In 1616, he graduated from the University of Poitiers with a degree in law. After that he enrolled as a volunteer in the army. Soon, however, he became more interested in mathematics and science than in pursuing a military career. On November 10, 1619, after a day of concentrated thinking he decided that he had to discover true knowledge by himself and to

achieve this he had to question everything known in philosophy and science. That night Descartes had three consecutive dreams which reinforced him in his decision.

4 achieve this he had to question everything known in philosophy and science. That night Descartes had three consecutive dreams which reinforced him in his decision. That made such an impression on him that he vowed a pilgrimage to Our Lady of Loretto. However, he did not start his program immediately. First, he spent nine years studying, primarily mathematics, as the result of which he discovered analytical geometry. To execute his program he went to Netherlands where he could enjoy much more freedom of thought than in France and where he lived almost until his death. Descartes' greatest contributions were in philosophy, mathematics, mechanics, optics, and physiology. His approach to the problem of refraction exhibits such typical features of his method as a tendency to mathematize physical phenomena and to proceed from the general concepts to particular ones. Although Descartes' formulation of his law looks similar to Snell's law, there is a profound difference between them. Descartes gave the first physical explanation of refraction. He stated that when light when it changes its velocity in going from one medium to another it changes direction. He incorrectly presumed that when light crosses the boundary of two media only the normal component of its velocity changes but its tangential component remains constant. But he came up with right formula. Fig. 5.4

$V 1t = V 1 sini V 2t = V 2 sin r V 1t = V 2t V 1 sini = V 2 sin r sini sinr = V 2 V 1 Thus, the ratio of the sine of the incidence angle to the sine of the refracted angle equals the inverse ratio of$

5 V 1t = V 1 sini V 2t = V 2 sin r V 1t = V 2t V 1 sini = V 2 sin r sini sinr = V 2 V 1 Thus, the ratio of the sine of the incidence angle to the sine of the refracted angle equals the inverse ratio of velocities of light in the two media Pierre Fermat ( ) Descartes' derivation of the law of refraction raised a number of objections. The French mathematician Pierre Fermat ( ) found the preservation of the tangential velocity in different media utterly unconvincing. He adopted Hero's law that light always takes the shortest path and tried to apply it to refraction. In 1662, he arrived at the hypothesis that light always selects such a path between any two given points which takes the least time. One will understand how to pass from this hypothesis to the law of refraction by considering the following analogy by Fermat. Imagine a messenger who proceeds from point A in Fig. 5.5 in a meadow, where his velocity is V 1 to point B on a sandy beach, where his velocity is V 2. At which point must he cross the boundary of the two media to accomplish his mission in the shortest time? For instance, if AD=DB, it is obvious that the straight line AB is not the solution, for it is preferable to extend the route in grass and reduce it in the sand. Fermat proved that the necessary solution is provided by such a point C at which Snell s Law is true. 1. A Fig.5.5. Illustration of Fermat's principle. It is interesting that one of Fermat's objections to Descartes was that resolving the velocity of light into two components is meaningless if the velocity is infinite, as Descartes

6 stated. It is possible, however, that Descartes' term "infinite" meant simply extremely high. It was not until 1676 that the Danish astronomer Olaf Roemer ( ) proved that the velocity of light is finite and measured it. Cartesians objected to Fermat saying that light has no free will to choose one way or another. But soon his principle of the least time was accepted as a useful mathematical (not physical!) concept. Isaac Newton rescued Descartes' derivation by explaining the conservation of the tangential velocity. According to him, light changes its velocity due to an attraction from a refracting body. Since this force is perpendicular to the surface of the body, it does not affect the tangential component of the velocity Christiaan Huygens ( ) Christiaan Huygens offered another explanation, based on his wave theory. Huygens was born in The Hague in a family of politicians and diplomats. While studying law at Leyden University he became very interested in mathematics. The mathematician Marin Mersenne ( ), who corresponded with the 17-year old Christiaan, predicted that he will become a new Archimedes. Huygens began with mathematics and mechanics and then switched to optics. In 1652, he discovered how to reduce spherical aberration of spherical lenses and decided to build an improved telescope. He learned how to grind high quality lenses and in three years he built his 12-foot telescope. With this telescope, one of the best in Europe, he discovered a satellite of Saturn. Soon he built a 24-foot telescope which allowed him to discover the Saturn's ring. From astronomy his interests returned to mechanics: he made important contributions to pendulum clocks and spring watches, discovered the laws of centripetal force and of the center of vibrations. By 1666, Huygens became so famous that he was invited to Paris to lead the newly founded Academy of Sciences. It was there that he presented in 1678 the wave theory of light (he published it in 1690), based on what became known as Huygens' Principle. According to this principle, every point of a wave front can be treated as a source of secondary waves, moving only forward, and the common envelope of all secondary waves represents a new wave front. For instance, let AB be the initial wave front (Fig. 4.16) which travels with the velocity v. If we want to know its position after time t, we draw secondary waves of the radius R=vt with centers on AB: their envelope A'B' will be the new wave front. Fig Huygens' principlebear in mind that the direction of propagation of light in each medium is perpendicular to the wave front.). Huygens' result looks similar to that of Descartes, however, his ratio of velocities is inverted. This means that to Huygens (and Fermat) light travels slower in a denser medium while, according to Descartes and

7 Newton, light travels faster in a denser medium. For the next century and a half this distinction became an important issue in the debate between the emission (or corpuscular) theory of light and the wave theory. 5.3 Waves or particles of light? From the 17th century on, among the hypotheses on the nature of light the chief rivals were the corpuscular (or emission) hypothesis and the wave hypothesis. According to the emission hypothesis, light was a stream of very small particles emitted by a luminous body and traveling to the eye. Newton was the best-known supporter of this hypothesis in the 17th century. In the wave hypothesis, what was transferred from a luminous body to the eye was a motion rather than matter. This motion spreads like sound or waves on the surface of water. It was Descartes who originated this hypothesis, although he applied it to pressure rather than motion; Hooke and Huygens developed the concept further. The light waves were supposed to travel in the ether, an extremely elastic and rare medium which penetrated all bodies as well as vacuum. There was a certain parity between the two hypotheses in the second half of the 17th century because both explained equally well reflection and refraction of light. In the 18th century, however, the emission hypothesis dominated the scene, and only a few scientists (Leonard Euler included) continued to support the concept of light waves. Here are some of the arguments used in the debate between the two hypotheses. The emissionists stated that, no matter how small the resistance of the ether to the motion of the planets, after thousands of years the change in their orbits could become noticeable. The undulationists countered this by assuming that the ether had an extremely low density, too low to have any impact on planetary motions. In their turn, they pointed out that since the particles of light travel with such a high velocity they must communicate a considerable momentum to illuminated bodies. Experiments, however, did not show any trace of the pressure of light. The emissionists answered that if the mass of light particles is extremely small, their momentum should be insignificant. Euler indicated that since light particles took away some of the sun's mass, its brightness must decrease with time. The emissionists replied that if the mass of these particles were extremely small the diminution of the sun's mass would be unnoticeable even after a thousand years. One of the major objections to the emission hypothesis was that if two beams of light particles cross, their collision would cause scattering of light and produce distorted images. 'The emissionists pointed out that the distance between particles in a ray of light may be so large that particles of one beam could pass through the interstices between the particles of another beam without any collisions. Since most of the arguments on both sides were then unverifiable by experiment, the controversy about the nature of light had a rather abstract character. The emission hypothesis enjoyed more popularity since it explained some optical phenomena unexplained by its rival, including stellar aberration, dispersion, colors of thin films, diffraction, and others. The explanations were qualitative, of course, but that was the accepted standard for the eighteenth century.

8 References: 1. N. Kipnis, Rediscovering Optics, Bena Press, 1993, ISBN X Chapter I, p.

9.4 Light: Wave or Particle?

9.4 Light: Wave or Particle? Huygens principle every point on a wave front can be considered as a point source of tiny secondary wavelets that spread out in front of the wave at the same speed as the wave itself rectilinear propagation