# Light

## Electromagnetic waves

During the early stages of their study and development, electric and magnetic phenomena were thought to be unrelated. In 1865, however, James Clerk Maxwell
(1831–1879) provided a mathematical theory that showed a close relationship between all electric and magnetic phenomena. In addition to unifying the formerly separate fields of electricity and magnetism, his brilliant theory predicted that electric and magnetic fields can move through space as waves. The theory he developed is based on the following four pieces of information:

1. Electric field lines originate on positive charges and terminate on negative charges.
2. Magnetic field lines always form closed loops—they don’t begin or end anywhere.
3. A varying magnetic field induces an emf and hence an electric field. This is a
statement of Faraday’s law (ε = -ΔΦ/Δt).
4. Magnetic fields are generated by moving charges (or currents), as summarized
in Ampère’s law (B∼I).

The first statement is a consequence of the nature of the electrostatic force between charged particles, given by Coulomb’s law. It embodies the fact that free charges (electric monopoles) exist in nature.

The second statement—that magnetic fields form continuous loops—is exemplified by the magnetic field lines around a long, straight wire, which are closed circles, and the magnetic field lines of a bar magnet, which form closed loops. It says, in contrast to the first statement, that free magnetic charges (magnetic monopoles) don’t exist in nature.

The third statement is equivalent to Faraday’s law of induction, and the fourth is equivalent to Ampère’s law.

In one of the greatest theoretical developments of the 19th century, Maxwell used these four statements within a corresponding mathematical framework to prove that electric and magnetic fields play symmetric roles in nature. It was already known from experiments that a changing magnetic field produced an electric field according to Faraday’s law. Maxwell believed that nature was symmetric, and he therefore hypothesized that a changing electric field should produce a magnetic field. This hypothesis could not be proven experimentally at the time it was developed, because the magnetic fields generated by changing electric fields are generally very weak and therefore difficult to detect.

To justify his hypothesis, Maxwell searched for other phenomena that might be explained by it. He turned his attention to the motion of rapidly oscillating (accelerating)
charges, such as those in a conducting rod connected to an alternating voltage. Such charges are accelerated and, according to Maxwell’s predictions, generate changing electric and magnetic fields. The changing fields cause electromagnetic disturbances that travel through space as waves, similar to the spreading water waves created by a pebble thrown into a pool. The waves sent out by the oscillating charges are fluctuating electric and magnetic fields, so they are called electromagnetic waves. From Faraday’s law and from Maxwell’s own generalization of Ampère’s law, Maxwell calculated the speed of the waves to be equal to the speed of light, $c = 3 . 10^{8}$ m/s. He concluded that visible light and other electromagnetic waves consist of fluctuating electric and magnetic fields traveling through empty space, with each varying field inducing the other! This was truly one of the greatest discoveries of science, on a par with Newton’s discovery of the laws of motion. Like Newton’s laws, it had a profound influence on later scientific developments.

## HERTZ’S CONFIRMATION OF MAXWELL’S PREDICTIONS

The most dramatic prediction of Maxwell’s theory of electromagnetism, published in 1865, was the existence of electromagnetic waves moving at the speed of light, and the conclusion that light itself was just such a wave. In 1887, after Maxwell’s death, Heinrich Hertz (1857–1894) was the first to generate and detect electromagnetic waves in a laboratory setting. He used an apparatus similar to the one illustrated.

A high voltage power source caused sparks to oscillate across the gap as shown. Each visible spark is actually a series of many small sparks, jumping rapidly back and forth (oscillating) between the terminals. The size of the metal plates attached to the spheres from which the sparks are produced, controls the frequency of the sparks produced. A loop of wire held near the oscillating spark, had a spark jump across the air gap between the ends of the wire whenever a spark jumped across the high voltage (induction coil) spark gap. Hertz reasoned that as the spark jumps back and forth across the gap of the induction coil, it must set up rapidly changing electric and magnetic fields.  According to Maxwell’s theory, these changes propagate through space as electromagnetic waves.  Upon arrival at the loop of wire, the changing electric and magnetic fields produce a potential difference across the ends of the wire loop.  If the potential is large enough, a spark jumps across the gap. The wave produced has the same frequency as the sparks -this frequency was about 1.0 X 109 Hz.

Properties of Electromagnetic Waves

Hertz showed that:

• the speed of the waves were 3.00 X 108 m/s.
• these waves had many of the usual properties of light.
• EM waves were produced whenever electric charges are accelerated. If an electric charge is accelerated in a periodic fashion, the frequency of the electromagnetic waves equals the frequency of the oscillations of the charge.
• All electromagnetic waves travel in a vacuum at 3.00 X 108 m/s – the speed of light in vaccum
• Electromagnetic waves exhibit: interference, diffraction, polarization, refraction, etc.
• Electromagnetic waves consist of oscillating electric and magnetic fields in a constant phase relation perpendicular to each other and perpendicular to the direction of propagation as illustrated below. Thus EM waves are transverse waves in nature.
• Notice how the right hand rule applies to this diagram.   Point your fingers in the direction of motion of the wave.  Twist your arm so that you can curl your fingers into the direction of the electric field.  Your thumb points in the direction of the magnetic field.

• Hertz called the waves radio waves
• Harmonically moving electric charges emit EM waves called harmonic or monochromatic waves. The frequency of the monochromatic wave is the frequency of oscillation of the electric charge – the source of the wave. The relation between the wavelength λ, the frequency ν and the velocity c of the electromagnetic waves is the same as it is in the case of mechanical waves:  c = λ ν
• People like Marconi used these waves to build the first radios.

## Electromagnetic Wave Velocity in Empty Space

A spacecraft cruising through space may gain or lose speed, even if its engines are shut off, because gravity can accelerate it forward or backward. But an electromagnetic wave traveling through space never changes its speed. Not because gravity doesn’t act on light, for it does. Gravity can change the frequency of light or deflect light, but it can’t change the speed of light. What keeps light moving always at the same, unvarying speed in empty space? The answer has to do with electromagnetic induction and energy conservation.

Let’s assume the opposite – that the light could slow down (reductio ad absurdum). If light were to slow down, its changing electric field would generate a weaker magnetic field, which, in turn, would generate a weaker electric field, and so on, until the wave dies out. No energy would be transported from one place to another, So light cannot travel slower than it does. If light were to speed up, the changing electric field would generate a stronger magnetic field, which, in turn, would generate a stronger electric field, and so on, a crescendo of ever-increasing field strength and ever- increasing energy-clearly a no-no with respect to energy conservation. At only one speed does mutual induction continue indefinitely, carrying energy forward without loss or gain.

From his equations of electromagnetic induction, James Clerk Maxwell calculated the value of this critical speed and found it to be 300,000 kilometers per second, In his calculation, he used only the constants in his equations determined by simple laboratory experiments with electric and magnetic fields. He didn’t use the speed of light. He found the speed of light! Maxwell quickly realized that he had discovered the solution to one of the greatest mysteries of the universe-the nature of light. He discovered that light is simply electromagnetic radiation within a particular frequency range, 4.3 x 10^ 14 to 7 x 10^ 14 vibrations per second. Such waves activate the “electrical antennae” in the retina of the eye. The lower-frequency waves appear red, and the higher-frequency waves appear violet. Maxwell realized, at the same time, that electromagnetic radiation of any frequency propagates at the same speed as light.

### The Electromagnetic Spectrum

In a vacuum, all electromagnetic waves move at the same speed and differ from one another in their frequency. The classification of electromagnetic waves according to frequency is the electromagnetic spectrum. Electromagnetic waves have been detected with a frequency as low as 0.01 hertz (Hz). Electromagnetic waves with frequencies of several thousand hertz (kHz) are classified as very low frequency radio waves. One million hertz (MHz) lies in the middle of the AM radio band. The very high frequency (VHF) television band of waves starts at about 50 MHz, and FM radio waves are between 88 and 108 MHz. Then come ultrahigh frequencies (UHF), followed by microwaves, beyond which are infrared waves, often called “heat waves.” Further still is visible light, which makes up less than 1 millionth of 1 % of the measured electromagnetic spectrum. The lowest frequency of light visible to our eyes appears red. The highest frequencies of visible light, which are nearly twice the frequency of red light, appear violet. Still higher frequencies are ultraviolet. These higher-frequency waves cause sunburns. Higher frequencies beyond ultraviolet extend into the X-ray and gamma-ray regions. There are no sharp boundaries between the regions, which actually overlap each other. The spectrum is separated into these arbitrary regions for classification.

Recall that the frequency of a wave is the same as the frequency of the vibrating source. The same is also true for EM waves: the frequency of an electromagnetic wave as it vibrates through space is identical to the frequency of the oscillating electric charge generating it. Different frequencies correspond to different wavelengths-waves of low frequency have long wavelengths and waves of high frequencies have short wavelengths. For example, since the speed of the wave is 300,000 kilometers per second, an electric charge oscillating once per second (1 hertz) will produce a wave with a wavelength of 300,000 kilometers. This is because only one wavelength is generated in 1 second. If the frequency of oscillation were 10 hertz, then 10 wavelengths would be formed in 1 second, and the corresponding wavelength would be 30,000 kilometers. A frequency of 10,000 hertz would produce a wavelength of 30 kilometers. So the higher the frequency of the vibrating charge, the shorter the wavelength of radiant energy.

We tend to think of space as empty, but only because we cannot see the montages of electromagnetic waves that permeate every part of our surroundings. We see some of these waves, of course, as light. These waves constitute only a microportion of the electromagnetic spectrum. We are unconscious of radio waves, which engulf us every moment, Free electrons in every piece of metal on the Earth’s surface continually dance to the rhythms of these waves. They jiggle in unison with the electrons being driven up and down along radio- and television- transmitting antennae. A radio or television receiver is simply a device that sorts and amplifies these tiny currents. There is radiation everywhere. Our first impression of the universe is one of matter and void, but actually the universe is a dense sea of radiation in which occasional concentrates are suspended.

## Light – introduction

Until the beginning of the 19th century, light was modeled as a stream of articles emitted by a source that stimulated the sense of sight on entering the eye. The chief architect of the particle theory of light was Newton. With this theory, he provided simple explanations of some known experimental facts concerning the nature of light—namely, the laws of reflection and refraction.

Most scientists accepted Newton’s particle theory of light. During Newton’s lifetime,
however, another theory was proposed. In 1678, the Dutch physicist and astronomer
Christian Huygens (1629–1695) showed that a wave theory of light could also
explain the laws of reflection and refraction.

The wave theory didn’t receive immediate acceptance, for several reasons. First, all the waves known at the time (sound, water, and so on) traveled through some sort of medium, but light from the Sun could travel to Earth through empty space. Further, it was argued that if light were some form of wave, it would bend around obstacles; hence, we should be able to see around corners. It is now known that light does indeed bend around the edges of objects. This phenomenon, known as diffraction, is difficult to observe because light waves have such short wavelengths. Even though experimental evidence for the diffraction of light was discovered by Francesco Grimaldi (1618–1663) around 1660, for more than a century most scientists rejected the wave theory and adhered to Newton’s particle theory, probably due to Newton’s great reputation as a scientist.

The first clear demonstration of the wave nature of light was provided in 1801 by Thomas Young (1773–1829), who showed that under appropriate conditions, light exhibits interference behavior. Light waves emitted by a single source and traveling
along two different paths can arrive at some point and combine and cancel each other by destructive interference. Such behavior couldn’t be explained at that time by a particle model, because scientists couldn’t imagine how two or more particles could come together and cancel each other.

The most important development in the theory of light was the work of Maxwell, who predicted in 1865 that light was a form of high-frequency electromagnetic wave. His theory also predicted that these waves should have a speed of 300 000 km/s, in agreement with the measured value. Although the classical theory of electricity and magnetism explained most known properties of light, some subsequent experiments couldn’t be explained by the assumption that light was a wave. The most striking of these was the photoelectric effect discovered by Hertz. Hertz found that clean metal surfaces emit charges when exposed to ultraviolet light.

In 1905, Einstein published a paper that formulated the theory of light quanta (“particles”) and explained the photoelectric effect. He reached the conclusion
that light was composed of corpuscles, or discontinuous quanta of energy. These corpuscles or quanta are now called photons to emphasize their particlelike nature. According to Einstein’s theory, the energy of a photon is proportional to the frequency of the electromagnetic wave associated with it, or:

E = h f

where $h = 6,626.10^{-34}$  J.s is Planck’s constant. This theory retains some features of both the wave and particle theories of light. As we will discuss later, the photoelectric effect is the result of energy transfer from a single photon to an electron in the metal. This means the electron interacts with one photon of light as if the electron had been struck by a particle. Yet the photon has wavelike characteristics, as implied by the fact that a frequency is used in its definition.

In view of these developments, light must be regarded as having a dual nature : In some experiments light acts as a wave and in others it acts as a particle. Classical electromagnetic wave theory provides adequate explanations of light propagation and of the effects of interference, whereas the photoelectric effect and other
experiments involving the interaction of light with matter are best explainedby assuming that light is a particle.

So in the final analysis, is light a wave or a particle? The answer is neither and both: light has a number of physical properties, some associated with waves and others with particles.

Light originates from the accelerated motion of electrons. It is an electromagnetic phenomenon and only a tiny part of a larger whole – a wide range of electromagnetic waves called the electromagnetic spectrum.

If you shake the end of a stick back and forth in still water, you will produce waves on the surface of the water, Similarly, if you shake an electrically charged rod to and fro in empty space, you will produce electromagnetic waves in space. This is because the moving charge is actually an electric current. What surrounds an electric current? The answer is, a magnetic field. What surrounds a changing electric current? The answer is, a changing magnetic field. Recall that a changing magnetic field generates an electric field- electromagnetic induction. If the magnetic field is oscillating, the electric field that it generates will be oscillating, too. And what does an oscillating electric field do? It induces an oscillating magnetic field. The vibrating electric and magnetic fields regenerate each other to make up an electromagnetic wave, which emanates (moves outward) from the vibrating charge. There is only one speed, it turns out, for which the electric and magnetic fields remain in perfect balance, reinforcing each other as they carry energy through space. But why is this  so?

### Transparent Materials

Light is an energy-carrying electromagnetic wave that emanates from vibrating electrons in atoms. When light is transmitted through matter, some of the electrons in the matter are forced into vibration, In this way, vibrations in the emitter are transmitted to vibrations in the receiver. This is similar to the way sound is transmitted.

Thus the way a receiving material responds when light is incident upon it depends on the frequency of the light and on the natural frequency of the electrons in the material. Visible light vibrates at a very high frequency, some 100 trillion times per second (10^14 hertz). If a charged object is to respond to these ultrafast vibrations, it must have very, very little inertia. Because the mass of electrons is so tiny, they can vibrate at this rate.

Such materials as glass and water allow light to pass through in straight lines. We say they are transparent to light. To understand how light travels through a transparent material, visualize the electrons in the atoms of transparent materials as if they were connected to the nucleus by springs.  When a light wave is incident upon them, the electrons are set into vibration.

Materials that are springy (elastic) respond more to vibrations at some frequencies than at others. Bells ring at a particular frequency, tuning forks vibrate at a particular frequency, and so do the electrons of atoms and molecules. The natural vibration frequencies of an electron depend on how strongly it is attached to its atom or molecule, Different atoms and molecules have different “spring strengths.” Electrons in the atoms of glass have a natural vibration frequency in the ultraviolet range. Therefore, when ultraviolet waves shine on glass, resonance occurs and the vibration of electrons builds up to large amplitudes, just as pushing someone at the resonant frequency on a swing builds to a large amplitude. The energy any glass atom receives is either reemitted or passed on to neighboring atoms by collisions, Resonating atoms in the glass can hold onto the energy of the ultraviolet light for quite a long time (about 100 millionths of a second). During this time, the atom makes about 1 million vibrations, and it collides with neighboring atoms and gives up its energy as heat. Thus, glass is not transparent to ultraviolet light.

At lower wave frequencies, such as those of visible light, electrons in the glass atoms are forced into vibration, but at lower amplitudes. The atoms hold the energy for a shorter time, with less chance of collision with neighboring atoms, and with less energy transformed to heat. The energy of vibrating electrons is reemitted as light. Glass is transparent to all the frequencies of visible light. The frequency of the reemitted light that is passed from atom to atom is identical to the frequency of the light that produced the vibration in the first place. However, there is a slight time delay between absorption and reemission.

It is this time delay that results in a lower average speed of light through a transparent material. Light travels at different average speeds through different materials, We say average speeds because the speed of light in a vacuum, whether in interstellar space or in the space between molecules in a piece of glass, is a constant 300,000 kilometers per second. We call this speed of light c.  The speed of light in the atmosphere is slightly less than in a vacuum, but it is usually rounded off as c. In water, light travels at 75% of its speed in a vacuum, or 0.75 c. In glass, light travels at about 0.67 c, depending on the type of glass. In a diamond, light travels at less than half its speed in a vacuum, only 0.41 c. When light emerges from these materials into the air, it travels at its original speed, c.

Infrared waves, with frequencies lower than those of visible light, vibrate not only the electrons, but entire atoms or molecules in the structure of the glass, This vibration increases the internal energy and temperature of the structure, which is why infrared waves are often called heat waves. So we see that glass is transparent to visible light, but not to ultraviolet and infrared light.

Different materials have different molecular structures and therefore absorb or reflect light from various spectral ranges differently.

### Opaque Materials

Most things around us are opaque-they absorb light without reemIttmg it. Books, desks, chairs, and people are opaque. Vibrations given by light to their atoms and molecules are turned into random kinetic energy-into internal energy. They become slightly warmer.

Metals are opaque, Because the outer electrons of atoms in metals are not bound to any particular atom, they are free to wander with very little restraint throughout the material (which is why metal conducts electricity and heat so well). When light shines on metal and sets these free electrons into vibration, their energy does not “spring” from atom to atom in the material but, instead, is reflected. That’s why metals are shiny.

Earth’s atmosphere is transparent to some ultraviolet light, to all visible light, and to some infrared light, but it is opaque to high-frequency ultraviolet light. The small amount of ultraviolet that does get through is responsible for sunburns. If it all got through, we would be fried to a crisp. Clouds are semitransparent to ultraviolet, which is why you can get a sunburn on a cloudy day. Dark skin absorbs ultraviolet before it can penetrate too far, whereas it travels deeper in fair skin. With mild and gradual exposure, fair skin develops a tan and increases protection against ultraviolet. Ultraviolet light is also damaging to the eyes-and to tarred roofs. Now you know why tarred roofs are covered with gravel.

Have you noticed that things look darker when they are wet than they do when they are dry? Light incident on a dry surface bounces directly to your eye, while light incident on a wet surface bounces around inside the transparent wet region before it reaches your eye. What happens with each bounce? Absorption! So more absorption of light occurs in a wet surface, and the surface looks darker.

Insightslonger-wavelength ultraviolet, called UV-A, is close to visible light and isn’t harmful. Short-wavelength ultraviolet, called UV-C, would be harmful if it reached us, but is almost completely stopped by the atmosphere’s ozone layer. It is the intermediate ultraviolet, UV-B, that can cause eye damage, sunburn, and skin cancer.

InsightsMetals are shiny because light that shines on them forces free electrons into
vibration, and these vibrating electrons then emit their “own” light waves as reflection.

## 1    P r o p a g a t i o n   of    l i g h t

In vacuum all electromagnetic waves have the same speed:

c=299792458 108 m/s

Electromagnetic wave with exact value of its frequency is called monochromatic wave. The wave length λ, the frequency f and the speed c of monochromatic wave are related through formula:

c = λf

Light has both wave and particle properties:

Particle-like: emission, absorption…

Wave-like: propagation, interference…

Visible spectrum

The visible spectrum is the portion of the electromagnetic spectrum that is visible to the human eye. Electromagnetic radiation in this range of wavelengths is called visible light or simply light. A typical human eye will respond to wavelengths from about 390 to 700 nm. In terms of frequency, this corresponds to a band in the vicinity of 430–770 THz.

Perception of color begins with specialized retinal cells containing pigments with different spectral sensitivities, known as cone cells. In humans, there are three types of cones sensitive to three different spectra, resulting in trichromatic color vision. The three types of cone photoreceptors could be classified as short-preferring (blue), middle-preferring (green), and long-preferring (red), according to their response to the wavelengths of light striking the retina. The relative strengths of the signals detected by the three types of cones are interpreted by the brain as a visible color. For instance, yellow light uses different proportions of red and green, but little blue, so any hue depends on a mix of all three cones, for example, a strong blue, medium green, and low red.

Many species can see light with frequencies outside the human “visible spectrum”. Bees and many other insects can detect ultraviolet light, which helps them find nectar in flowers. Plant species that depend on insect pollination may owe reproductive success to their appearance in ultraviolet light rather than how colorful they appear to humans. Birds, too, can see into the ultraviolet (300–400 nm).

 Color Wavelength Frequency Photon energy violet 380–450 nm 668–789 THz 2.75–3.26 eV blue 450–495 nm 606–668 THz 2.50–2.75 eV green 495–570 nm 526–606 THz 2.17–2.50 eV yellow 570–590 nm 508–526 THz 2.10–2.17 eV orange 590–620 nm 484–508 THz 2.00–2.10 eV red 620–750 nm 400–484 THz 1.65–2.00 eV

Colors that can be produced by visible light of a narrow band of wavelengths (monochromatic light) are called pure spectral colors. The various color ranges indicated in the diagram above are an approximation: The spectrum is continuous, with no clear boundaries between one color and the next.

Refractive index

In optics the refractive index or index of refraction n of an optical medium is a dimensionless number that describes how light, or any other radiation, propagates through that medium. It is defined as

n = c / v

where c is the speed of light in vacuum and v is the speed of light in the medium.

The refractive index determines how much light is bent, or refracted, when entering a material. In air the speed is only slightly less than the speed in vacuum. The index of refraction is never less than 1, and values for various materials are given in the table:

The frequency of light does not depend on the medium but only on the source of light:

λ0=c/f , λ=v/f    →   λ=λ0/n

therefore the wave length depend on the index of refraction of the medium.

The ray model of light

A great deal of evidence suggests that light travels in straight lines under a wide
variety of circumstances. For example, a source of light like the Sun casts distinct
shadows, and the light from a laser pointer appears to be a straight line. In fact, we
infer the positions of objects in our environment by assuming that light moves
from the object to our eyes in straight-line paths. Our orientation to the physical
world is based on this assumption.

This reasonable assumption is the basis of the ray model of light. This model
assumes that light travels in straight-line paths called light rays. Actually, a ray is
an idealization; it is meant to represent an extremely narrow beam of light. When
we see an object, according to the ray model, light reaches our eyes from each
point on the object. Although light rays leave each point in many different directions,
normally only a small bundle of these rays can enter an observer’s eye. If the person’s head moves to one side, a different bundle of rays will enter the eye from each point.

We saw that light can be considered as an electromagnetic wave. Although the ray model of light does not deal with this aspect of light , the ray model has been very successful in describing many aspects of light such as reflection, refraction, and the formation of images by mirrors and lenses. Because these explanations involve straight-line rays at various angles, this subject is referred to as geometric optics.

Types of light rays

While there are numerous names for types of light rays, the most common ones are incident rays, reflected rays, and refracted rays. Incident rays are the rays that approach and hit a particular surface — they are said to be ‘incident’ on the surface.

Reflected rays are what you get if the surface is in some way reflective, such as in the case of a mirror. The ray that bounces off the surface at an angle is known as the reflected ray.

Refracted rays are when the light goes through the surface, bending due to the change of material (or medium). For example if a ray of light is travels from air into water, or into a block of glass the light ray will bend as a result. This effect is called refraction, and the ray that bends as it moves through the material is called a refracted ray.

## 2. Reflection and refraction of light

When light strikes the surface of an object, some of the light is reflected. The rest can be absorbed by the object (and transformed to thermal energy) or, if the object is transparent like glass or water, part can be transmitted through. For a very smooth shiny object such as a silvered mirror, over 95% of the light may be reflected.

When a narrow beam of light strikes a flat surface, we define the angle of incidence, θto be the angle an incident ray makes with the normal (perpendicular) to the surface, and the angle of reflection, θr, to be the angle the reflected ray makes with the normal. It is found that the incident and reflected rays lie in the same plane with the normal to the surface, and that

the angle of reflection equals the angle of incidenceθi = θr

This is the law of reflection. It was known to the ancient Greeks, and you can confirm it yourself by shining a narrow flashlight beam or a laser pointer at a mirror in a darkened room.

When light is incident upon a rough surface, even microscopically rough such as this page, it is reflected in many directions. This is called diffuse reflection. The law of reflection still holds, however, at each small section of the surface. Because of diffuse reflection in all directions, an ordinary object can be seen at many different angles by the light reflected from it. When you move your head to the side, different reflected rays reach your eye from each point on the object (such as this page).

Let us compare diffuse reflection to reflection from a mirror, which is known as specular reflection. (“Speculum” is Latin for mirror.) When a narrow beam of light shines on a mirror, the light will not reach your eye unless your eye is positioned at just the right place where the law of reflection is satisfied. This is what gives rise to the special image- forming properties of mirrors.

Refraction

When light passes from one transparent medium into another with a different index of refraction, part of the incident light is reflected at the boundary. The remainder passes into the new medium. If a ray of light is incident at an angle to the surface (other than perpendicular), the ray changes direction as it enters the new medium. This change in direction, or bending, is called refraction.

Figure below shows a ray passing from air into water. Angle θ1 is the angle the
incident ray makes with the normal (perpendicular) to the surface and is called the
angle of incidence. Angle θ2 is the angle of refraction, the angle the refracted ray makes with the normal to the surface. Notice that the ray bends toward the normal
when entering the water. This is always the case when the ray enters a medium where
the speed of light is less. If light travels from one medium into a second where its speed is greater, the ray bends away from the normal.

Refraction is responsible for a number of common optical illusions. For example, a person standing in waist-deep water appears to have shortened legs. The rays leaving the person’s foot are bent at the surface. The observer’s brain assumes the rays to have traveled a straight-line path (dashed red line, fig. bellow), and so the feet appear to be higher than they really are.

Similarly, when you put a straw in water, it appears to be bent.

### Snell’s Law

The angle of refraction depends on the speed of light in the two media and on the
incident angle. An analytical relation between  θ1 and θ2 was arrived at experimentally about 1621 by Willebrord Snell (1591-1626). It is known as Snell’s law and is written:

$n_1$ sin θ1 = $n_2$ sin θ2

θ1 is the angle of incidence and θ2 is the angle of refraction;  $n_1$ and $n_2$ are the
respective indices of refraction in the materials. The incident and refracted rays lie in the same plane, which also includes the perpendicular to the surface. Snell’s law is the law of refraction.

It is clear from Snell’s law that if  $n_2$ > $n_1$, then  θ1 θ2That is, if light enters
a medium where n is greater (and its speed is less), then the ray is bent toward the normal. And if  $n_2$ < $n_1$, then  θ2 θ1, so the ray bends away from the normal.

### Total Internal Reflection

When light passes from one material into a second material where the index of refraction is less (say, from water into air), the light bends away from the normal, as for rays I and J in Fig. 2-7. At a particular incident angle, the angle of
refraction will be 90°, and the refracted ray would skim the surface (ray K) in this case. The incident angle at which this occurs is called the critical angle, θc.

From Snell’s law,  θc is given by

sin θc = $\frac{n_2}{n_1}$ sin 90º = $\frac{n_2}{n_1}$

For any incident angle less than ()e, there will be a refracted ray, although part of the light will also be reflected at the boundary. However, for incident angles greater than θc, Snell’s law would tell us that sin θ2 is greater than 1.00. Yet the sine of an angle can never be greater than 1.00. In this case there is no refracted ray at all, and all of the light is reflected, as for ray L in Fig. 2-7. This effect is called total internal reflection. Total internal reflection can occur only when light strikes a boundary where the medium beyond has a lower index of refraction.

CONCEPTUAL EXAMPLE:  View up from under water.

Describe what a person would see who looked up at the world from beneath the perfectly smooth surface of a lake or swimming pool.

RESPONSE:  For an air-water interface, the critical angle is given by

sin θc = 1/1.33= 0.75

Therefore,  θc = 49°. Thus the person would see the outside world compressed into a circle whose edge makes a 49° angle with the vertical. Beyond this angle, the person would see reflections from the sides and bottom of the lake or pool.

Fiber Optics

Total internal reflection is the principle behind fiber optics. Glass and plastic fibers as thin as a few micrometers in diameter are common. A bundle of such tiny fibers is called a light pipe or cable, and light t can be transmitted along it with almost no loss because of total internal reflection. Figure 2-9 shows how light traveling down a thin fiber makes only glancing collisions with the walls so that total internal reflection occurs. Even if the light pipe is bent into a complicated shape, the critical angle still won’t be exceeded, so light is transmitted practically undiminished to the other end. Very small losses do occur, mainly by reflection at the ends and absorption within the fiber.

Important applications of fiber-optic cables are in communications and medicine. They are used in place of wire to carry telephone calls, video signals, and computer data. The signal is a modulated light beam (a light beam whose intensity can be varied) and data is transmitted at a much higher rate and with less loss and less interference than an electrical signal in a copper wire. Fibers have been developed that can support over one hundred separate wavelengths, each modulated to carry up to 10 gigabits ($10^{10}$ bits) of information per second. That amounts to a terabit ( $10^{12}$bits) per second for the full one hundred wavelengths.

The sophisticated use of fiber optics to transmit a clear picture is particularly useful in medicine, Fig. 2-10. For example, a patient’s lungs can be examined by inserting a light pipe known as a bronchoscope through the mouth and down the bronchial tube. Light is sent down an outer set of fibers to illuminate the lungs. The reflected light returns up a central core set of fibers. Light directly in front of each fiber travels up that fiber. At the opposite end, a viewer sees a series of bright and dark spots, much like a TV screen-that is, a picture of what lies at the opposite
end. Lenses are used at each end. The image may be viewed directly or on a monitor screen or film. The fibers must be optically insulated from one another, usually by a thin coating of material with index of refraction less than that of
the fiber. The more fibers there are, and the smaller they are, the more detailed the picture. Such instruments, including bronchoscopes, colonoscopes (for viewing the colon), and endoscopes (stomach or other organs), are extremely useful for examining hard-to-reach places.

### 3. Dispersion of Light

If we make careful measurements, we find that the index of refraction in anything but vacuum depends on the wavelength of light. The dependence of the index of refraction on wavelength is called dispersion. Fig. 3-1 is a graphical representation of this variation in the index of refraction with wavelength. Because n is a function of wavelength, Snell’s law indicates that the angle of refraction made when light enters a material depends on the wavelength of the light. As seen in the figure, the index of refraction for a material usually decreases with increasing wavelength. This means that violet light (λ≅ 400 nm) refracts more than red light (λ≅ 650 nm) when passing from air into a material.

To understand the effects of dispersion on light, consider what happens when light strikes a prism, as in Figure 3-2. A ray of light of a single wavelength that is incident on the prism from the left emerges bent away from its original direction of travel by an angle δ, called the angle of deviation. Now suppose a beam of white light (a combination of all visible wavelengths) is incident on a prism. Because of dispersion, the different colors refract through different angles of deviation, and the rays that emerge from the second face of the prism spread out in a series of
colors known as a visible spectrum, as shown in F-g 3-2. These colors, in order of decreasing wavelength, are red, orange, yellow, green, blue, and violet. Violet light deviates the most, red light the least, and the remaining colors in the visible spectrum fall between these extremes.

Prisms are often used in an instrument known as a prism spectrometer. This instrument is commonly used to study the wavelengths emitted by a light source, such as a sodium vapor lamp. Light from the source is sent through a narrow, adjustable slit and lens to produce a parallel, or collimated, beam. The light then passes through the prism and is dispersed into a spectrum. The refracted light is observed through a telescope. The experimenter sees different colored images of the slit through the eyepiece of the telescope. The telescope can be moved or the prism can be rotated in order to view the various wavelengths, which have different angles of deviation.

All hot, low-pressure gases emit their own characteristic spectra. One use of a prism spectrometer is to identify gases. For example, sodium emits only two wavelengths in the visible spectrum: two closely spaced yellow lines. (The bright linelike images of the slit seen in a spectroscope are called spectral lines.) A gas emitting these, and only these, colors can thus be identified as sodium. Likewise, mercury vapor has its own characteristic spectrum, consisting of four prominent wavelengths—orange, green, blue, and violet lines—along with some wavelengths of lower intensity. The particular wavelengths emitted by a gas serve as “fingerprints” of that gas. Spectral analysis, which is the measurement of the wavelengths emitted or absorbed by a substance, is a powerful general tool in many scientific
areas. As examples, chemists and biologists use infrared spectroscopy to identify molecules, astronomers use visible-light spectroscopy to identify elements on distant stars, and geologists use spectral analysis to identify minerals.

THE RAINBOW

The dispersion of light into a spectrum is demonstrated most vividly in nature through the formation of a rainbow, often seen by an observer positioned between the Sun and a rain shower. To understand how a rainbow is formed, consider Figure 3-3. A ray of light passing overhead strikes a drop of water in the atmosphere and is refracted and reflected as follows: it is first refracted at the front surface of the drop, with the violet light deviating the most and the red light the least.

At the back surface of the drop, the light is reflected and returns to the front surface, where it again undergoes refraction as it moves from water into air. The rays leave the drop so that the angle between the incident white light and the returning violet ray is 40° and the angle between the white light and the returning red ray is 42°. This small angular difference between the returning rays causes us to see the bow as explained in the next paragraph.

Now consider an observer viewing a rainbow, as in Figure 3-4. If a raindrop high in the sky is being observed, the red light returning from the drop can reach the observer because it is deviated the most, but the violet light passes over the observer because it is deviated the least. Hence, the observer sees this drop as being red. Similarly, a drop lower in the sky would direct violet light toward the observer and appear to be violet. (The red light from this drop would strike the ground and not be seen.) The remaining colors of the spectrum would reach the observer from raindrops lying between these two extreme positions. Figure 3-4 shows a beautiful rainbow and a secondary rainbow with its colors reversed.

Diamond color

Diamonds achieve their brilliance from a combination of dispersion and total internal reflection. Because diamonds have a very high index of refraction of about 2.4, the critical angle for total internal reflection is only 25°. The light
dispersed into a spectrum inside the diamond therefore strikes many of the internal surfaces of the diamond before it strikes one at less than 25° and emerges. After many such reflections, the light has traveled far enough that the colors have become sufficiently separated to be seen individually and brilliantly by the eye after leaving the diamond.

## Summary of terms:

Electromagnetic wave:  An energy-carrying wave emitted by a vibrating charge (often electrons) that is composed of oscillating electric and magnetic fields that regenerate one another.

Electromagnetic spectrum:  The range of electromagnetic waves extending in frequency from radio waves to gamma rays.

Transparent:  The term applied to materials through which light can pass in straight lines.

Opaque: The term applied to materials that absorb light without reemission and thus through which light cannot pass.

Shadow:  A shaded region that appears where light rays are blocked by an object.

Umbra:  The darker part of a shadow where all the light is blocked.

Penumbra:  A partial shadow that appears where some but not all of the light is blocked.

Ray: A thin beam of light

Reflection:  The return of light rays from a surface.

Refraction:  The bending of an oblique ray of light when it passes from one transparent medium to another.

Law of reflection: The angle of reflection equals the angle of incidence.

Diffuse reflection: Reflection in irregular directions from an irregular surface.

Critical angle:  The minimum angle of incidence inside a medium at which a light ray is totally reflected.

Total internal reflection:  The total reflection of light traveling within a denser medium when it strikes the boundary with a less dense medium at an angle greater than the critical angle.

Review Questions

1. What evidence can you cite to support the claim that the frequency of light does not change upon reflection?

Answer:  The color of an image is identical to the color of the object forming the image. When you look at yourself in a mirror, the color of your eyes doesn’t change. The fact that the color is the same is evidence that the frequency of light doesn’t  change upon reflection.

2. If the speed of light were the same in air of various temperatures and densities, would there still be slightly longer daytimes, twinkling stars at night, mirages, and slightly squashed Suns at sunset?

3. If the speed of light were the same in all media, would refraction still occur when light passes from one medium to another?

4. If light traveled at the same speed in raindrops as it does in air, would we still have rainbows?

5. The unvarying speed of electromagnetic waves in space is a remarkable consequence of what central principle in physics?

Answer:  The underlying principle that makes light and all other electromagnetic waves travel at one fixed speed is the conservation of energy.

6. Is it correct to say that a radio wave can be considered a low-frequency light wave? Can a radio wave also be considered to be a sound wave?

Answer:  Both a radio wave and a light wave are electromagnetic waves, which originate in the vibrations of electrons. Radio waves have lower frequencies than light waves, so a radio wave may be considered to be a low-frequency light wave (and a light wave, similarly, can be considered to be a high-frequency radio wave). But a sound wave is a mechanical vibration of matter and is not electromagnetic. A sound wave is fundamentally different from an electromagnetic wave. So a radio wave is definitely not a sound wave.

7. Why is glass transparent to visible light but opaque to ultraviolet and infrared?

Answer:  Because the natural vibration frequency for electrons in glass is the same as the frequency of ultraviolet light, resonance occurs when ultraviolet waves shine on glass. The absorbed energy is passed on to other atoms as heat, not reemitted as light, making the glass opaque at ultraviolet frequencies. In the range of visible light, the forced vibrations of electrons in the glass are at smaller amplitudes- vibrations are more subtle, reemission of light (rather than the generation of heat) occurs, and the glass is transparent. lower-frequency infrared light causes whole molecules, rather than electrons, to resonate; again, heat IS generated and the glass is opaque to infrared light.

8.What does a changing magnetic field induce?   Answer: An electric field

9. What does a changing electric field induce?     Answer: A Magnetic field

10. What produces an electromagnetic wave?      Answer: Accelerating charges

11. Which have the longest wavelengths-light waves, X-rays, or radio waves?

12. Which has the shorter wavelengths, ultraviolet or infrared? Which has the higher frequencies?

Answer: Ultraviolet has shorter wavelengths than infrared. Correspondingly, ultraviolet also has the higher frequencies.

13. What is it, exactly, that waves in a light wave?

Answer: What waves in a light wave are the electric and magnetic fields. Their oscillation frequency is the frequency of the wave.

14. Which have the longest wavelengths – light waves, X-rays, or radio waves?

15. Knowing that interplanetary space consists of a vacuum, what is your evidence that electromagnetic waves can travel through a vacuum?

Answer: We can see the Sun and stars.

16. What is the principal difference between a gamma ray and an infrared ray?

Answer: Gamma rays have shorter wavelength.

17. What is the speed of X-rays in a vacuum?

Answer: The same as the speed of light c=299792458 m/s or approximately 300 000 km/s

18. Which travels faster through a vacuum – an infrared ray or a gamma ray?

Answer: Both travel at the same speed c=299792458 m/s (approximately 300 000 km/s)

19. Is it true that any radio wave travels appreciably faster than any sound wave?

20.  Suppose a light wave and a sound wave have the same frequency. Which has the longer wavelength?

21. Suppose a light wave and a sound wave have the same frequency. Which has the longer wavelength?

Hint: Use c=λν

22. Which requires a physical medium in which to travel-light, sound, or both?

23. Do radio waves travel at the speed of sound, or at the speed of light, or somewhere in between?

24. How is the fact that an electromagnetic wave in space never slows down consistent with the conservation of energy?

25. How is the fact that an electromagnetic wave in space never speeds up consistent with the conservation of energy?

26. Knowing that interplanetary space consists of a vacuum, what is your evidence that electromagnetic waves can travel through a vacuum?

27. What is the principal difference between a gamma ray and an infrared ray?

28. What is the speed of X-rays in a vacuum?

29. Which travels faster through a vacuum-an infrared ray or a gamma ray?

30.When you look at a distant galaxy through a telescope, how is it that you’re looking backward in time?

Problems

1. The Sun is 1,50 . 10¹¹ meters from the Earth. How long does it take for the Sun’s light to reach the Earth?
2. How long does it take for a pulse of laser light to reach the Moon and to bounce back to the Earth?
3. The wavelength of yellow sodium light in air is 589 nm. What is its frequency?
4. A light-year is the distance that light travels in vacuum in one Julian year (365.25 days). Calculate that distance.