radiation in the environmentuilis.unsyiah.ac.id/oer/files/original/cfa04b8af3fa1ce6f... · 2018. 1....

Radiation In The Environment

J. L. Hunt

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CONTENTS

Introduction

Chapter 1. Electromagnetic Radiation: The Classical Description.

Chapter 2. Electromagnetic Radiation: The Quantum Description.

Chapter 3. Radiometry and Photometry.

Chapter 4. Visible and Ultraviolet Radiation.

Chapter 5. Infrared and Radio Frequencies.

Chapter 6. Lasers and Hazards to the Eye

Chapter 7. The Atomic Nucleus and Radioactivity.

Chapter 8. The Interaction of Ionizing Radiation with Matter and its Biological Effects.

Chapter 9. Ionizing Radiation Detectors.

Chapter 10. Sound Chapter 10. Environmental Noise.

Appendix I. Numerical Constants.

Appendix II Symbols.

Appendix III Loudness Chart

Bibliography

Index

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Acknowledgment The author would like to acknowledge the important contribution to Chapters 1, 4, 5, 8 in the first edition of this textbook by Prof. W. G. Graham. This material has been very little altered in the later editions.

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INTRODUCTION

Radiation in Our World.

n the post-nuclear era the word ‘radiation’, in the popular lexicon, has taken on a specialized and negative meaning. Largely because of media misinterpretation the word now suggests to the non-scientist the malignant

effects associated with the misuse of nuclear energy. But ‘radiation’ is not all harmful, nor indeed is it all nuclear in origin. We live in an environment bathed in radiation of natural origin and essential to the maintenance of life on the planet. Sunlight and its secondary thermal emissions of heat radiation is the engine of life on the planet Earth. In addition to this ubiquitous low-energy radiation there are natural high-energy radiations in which life has developed and for which evolution and nature have provided effective defence mechanisms. High energy radiation damage to DNA molecules is repaired by appropriate enzymes, and our atmosphere protects us from most of the high-energy portion of the Sun’s radiation. To the overwhelming flux of natural radiations humankind has added a small amount of artificial ones for various specialized purposes. In general, the intensity of these is miniscule compared with the natural sources, but in certain cases they can be hazardous. Exposure to an unshielded nuclear reactor fuel element can harm or even kill you. Exposure to an intense laser beam can blind you (but so can the Sun). Uncontrolled exposure to intense microwaves can heat flesh and cook it. Human-made radiations are created with some purpose; usually the purpose is beneficent. X-rays are an invaluable tool in diagnosis and treatment in medicine and there is hardly anyone who would want to ban their use, but the potential for harm or misuse is always present. An example of such misuse was the widespread use of X-rays in the fitting of shoes in the 1940s and 50s. The accurate measurement of the exposure of both customers and salespersons led to the banning of this frivolous practice. It is essential, therefore, that we be able to understand radiation, be able to measure it in a reproducible way, and as a result of our understanding and measurements, be able to control it and protect ourselves and the public. Not all of the radiation of environmental concern is electromagnetic (e.g., light, X-rays) or consists of particles

I

Fitting shoes with X-rays. Photo provided by Oak Ridge Associated Universities

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(e.g., α, β-nuclear particles). Sound is also radiation and has an environmental impact, usually in the form of ‘noise’. Although there are certainly physical effects of intense sound, much of its impact depends on our psychological reaction, and so the subject of ‘psychoacoustics’ has been developed to measure human perception of sound radiation.

Energy in Radiation. The reason that radiation interacts with us and our environment is because it carries energy. The energy of radiation interacts with the environment through various atomic, molecular and nuclear mechanisms which can sometimes be usefully characterized by the amount of the energy involved in the process. The energy of a typical chemical bond is of the order of a few electron-volts (eV)1. Visible light and the near-ultraviolet (UV) also have energies of this magnitude, so it is not surprising that these radiations interact with the outer (or valence-bond-forming) electrons in atoms, but not at all with the inner, more tightly bound electrons. Thus visible and near UV radiation can influence or initiate chemical reactions as for instance in photosynthesis or in the complex process of skin tanning. With just a little more energy in the UV the chemical reactions can be violent enough to cause severe damage to sensitive biological molecules; e.g., sunburn. At the lower energies involved in the infrared (IR) only more subtle molecular processes, such as the denaturing (i.e. cooking) of proteins, are possible but this can be serious if it is caused by an intense IR laser beam ‘cooking’ the retinal cells in the eye. At even lower energies it becomes more and more difficult to couple electromagnetic radiation into biological systems as there are fewer and fewer available energy states in the molecules with which to interact. It is for this reason that most scientists are immediately sceptical about claims of the biological harm of radiation from 60 Hz power lines. At this frequency, the energy of the radiation is so low that any possible energy states are already activated by just the thermal energy of the biological system. What further can the EM radiation do? For higher energy EM waves, X- and gamma-rays for example, the energy range is from103 eV (keV) to 106 eV (MeV). These interactions can take place with the inner, more tightly bound, electrons in the atom detaching them (photoelectric effect) and producing a fast moving electric charge in the medium. There are also mechanisms by which high energy EM waves can interact with the outer weakly bound electrons (Compton Effect) and also produce a fast moving charge

in the medium. High energy particle radiation, such as α and β, also involves fast moving charges in the medium. A fast moving charge can detach electrons from the molecules of the medium creating chemically active species that can

1 The quantitative definition of the electron-volt will be given in Chapter 1.

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go on to produce extensive damage in living systems. Indeed almost all high-energy radiation damage in living cells is of this type. The interaction of radiation with the nuclei of atoms is negligible in the environmental context. The energy regime of nuclei is of the order of 106 eV (MeV) and there are certainly EM radiations of this energy. However, because of the small size of the nucleus, such interactions are extremely rare in the radiation fluxes encountered in the natural environment. To make such processes important the fluxes found in the cores of nuclear reactors are required.

Organization of the Text. Because so much of the natural radiation is electromagnetic, a review of electric and magnetic fields in Chapter 1 leads to a consideration of the properties of EM waves from a classical point of view in Chapter 1 and a quantum point of view in Chapter 2. Chapter 3 covers the measurement of visible light, a subject known as ‘photometry’. Here the human perception (seeing) of the physical phenomenon (EM waves) introduces the reader to one aspect of psychophysics and a plethora of new units and measures. Chapters 4 and 5 discuss the environmental effects of visible and UV radiation, and infrared and radio radiation respectively. Chapter 6 describes the construction of lasers and their classification by output power. This leads to an analysis of the hazard of laser radiation and the standards that have been established to minimize risk. Chapter 7 discusses the physics of nuclear radiations, and Chapter 8 their interactions with biological systems, and the associated hazards. Chapter 9 is a brief survey of some nuclear radiation detection and monitoring methods. Chapters 10 and 11 introduce sound radiation and the measurement and classification of a selection of examples of environmental noise. Of course, in an elementary survey, none of the topics are pursued to the detailed level that might be required of an environmental consultant or scientist in the field. Nevertheless, the physical principles are established and just with these, it is surprising how many realistic environmental situations can be understood and quantified. Throughout, there is an emphasis on quantitative methods with many numerical examples and problems.

CHAPTER 1: ELECTROMAGNETIC RADIATION: THE

CLASSICAL DESCRIPTION

1.1 Introduction.

his chapter is a review of the classical or wave nature of electromagnetic (EM) radiation; it is assumed that the reader is familiar with some of this material. More elementary and detailed treatments may be found in any introductory

university physics book. Since visible light and radio are familiar forms of EM radiation, these are used extensively as examples of respectively, short-wavelength and long-wavelength radiations.

It is well known that light transmits energy, can travel through the best laboratory vacuum, and exhibits diffraction and interference patterns under the proper conditions. This indicates that light (and other EM radiation) is some type of wave but not a mechanical wave (like sound) which requires matter for its transmission. Further, the familiar polarization effects with light (e.g., two ‘crossed’ polaroids; see Sec. 1.5) indicate that light is a transverse wave; there are no similar polarization phenomena with longitudinal waves such as sound.

In the 1860s, the Scots theorist, James Clerk Maxwell (1831-1879), after summarizing the known ‘laws’ of electromagnetism and adding significant new theory of his own, predicted the possibility of electromagnetic waves, i.e., waves consisting of oscillating electric and magnetic fields. Further, he predicted that the fields would oscillate in a direction perpendicular to the wave velocity, i.e., the waves would be transverse. Finally, from known electric and magnetic constants, he was able to calculate that the velocity of these waves in vacuum should be about 3×108 m/s. This value is identical to the speed of light in vacuum which had been measured fairly accurately by that time. These predictions immediately suggested that light is an electromagnetic wave. Maxwell suggested that such waves could be produced by any accelerating charge, for example, by an oscillating charge. Soon after, the German physicist-engineer, Heinrich Hertz (1857-1894), using a simple electrical spark device (a spark consists of a brief oscillating current), produced and detected the first waves we would now call radio waves; the precursor of modern radio and TV was born. Within a few decades,

the entire electromagnetic spectrum from radio to γ-radiation was discovered.

1.2 The Electric Field.

Since EM waves consist of electric and magnetic fields, the physical nature of these fields is reviewed here. By the term ‘field’, is simply meant any quantity which exists at every point in space, e.g., a ‘temperature field’. More specifically, electric and magnetic fields, or taken together, the electromagnetic field, is associated with the electromagnetic interaction or force, between charged particles.

A familiarity with the inherent property of electric charge possessed by particles such

T

HUNT: RADIATION IN THE ENVIRONMENT

1-2

as the electron (negative charge) and the proton (equal positive charge) is assumed. In the S.I. system, charge is measured in units of coulombs (C). One coulomb is the amount of charge on 6.242×1018 protons or electrons; conversely, the magnitude of the charge on these particles, the elementary charge (e), is the reciprocal: 1.602×10–19 C.

Atoms, molecules and macroscopic objects, having equal numbers of protons and electrons, have no net charge. However, they may become charged by losing or gaining electrons in various ways.

As implied above, the electromagnetic force is the force a particle or object experiences due to the net charge it possesses and its state of motion. This force has a double name (i.e., ‘electro’ and ‘magnetic’) because the force is, in general, composed of two parts which have different properties. The more familiar electric part depends only on the amount of charge the object has. The magnetic part depends not only on the amount of charge but also on its velocity relative to the observer’s reference frame; this is discussed in Sec. 1.3.

Many phenomena, including electromagnetic radiation, indicate that it is incorrect to think of one charge, say qA, somehow directly exerting a force on a second charge, say qB, some distance away as shown in Fig. 1-1(a). Rather, think of each charge creating in the space around itself, an electric and (if the charge is in motion) a magnetic field. The fields of qA have no effect on qA itself but exert electric and magnetic forces on other charges such as qB which enter qA’s field as shown in Fig. 1-1(b); similarly, qB’s field exerts forces on the charge qA.

An electric field is that condition at each point in space which exerts an electric force on a charged object at that point, i.e., a force which depends only on the charge on the object. Experiments show that at any point in space and instant in time, the electric force1 FE on a particle with a net charge of magnitude |q|, is proportional to |q|. (The letter q (or Q) is used as a general symbol for electric charge.) Equivalently, the electric force per unit charge, i.e., FE/|q| at each point in space and time is constant, and independent of the size of |q|. This ratio is used as a measure of the strength, E, of the electric field at each point, i.e., the Electric Field Strength is given by:

E = FE/|q| [1-1a]

This is usually written:

FE = E⋅|q| [1-1b] The S.I. units of E are obviously N/C.

1 This force is also called the ‘electrostatic force’ and the ‘Coulomb force’.

Fig. 1-1 Electric field of a point + charge.

1-ELECTROMAGNETIC RADIATION: THE CLASSICAL DESCRIPTION

1-3

The electric field has magnitude and direction, i.e., it is a vector quantity.2 At a point where there is an electric field, all positive charges will experience an electric force in the same direction and all negative charges will experience a force in the opposite direction. Of these two opposite directions, the direction of the force on positive charges is chosen arbitrarily, i.e., the direction of the electric field vector E is defined as the direction of the electric force on a positive charge. Figure 1-1 is drawn for the case of positive charges.

Example 1-1: At a certain point, a charge q = + 1.0 µC experiences an electric force FE of 2.0 N toward the south-west.

a) What is the electric field at this point? The magnitude of E is: E = FE/|q| = 2.0/1.0×10–6 C = 2.0×106 N/C; i.e., E = 2.0×106 N/C toward the south-west. b) If an electron (|q| = e = 1.60×10–19 C) is placed at the above spot, what electric force FE acts on it?

|FE| on the electron = |q|E = (1.60×10–19 C)(2.0×106 N/C) = 3.2×10–13 N

Since the electron is negative, the force on it is opposite to the direction of E, i.e.,

FE = 3.2×10–13 N toward the north-east. ________________

Recall that a ‘volt’, the unit of electrical potential (the potential energy per unit charge), is equivalent to a joule/coulomb (J/C) and a joule is a ‘newton-meter’. Verify that the units for electric field, i.e., N/C, are equivalent to volts/meter (V/m) (See Problem 1-2). Electric field strengths are frequently expressed in these alternate units; in Example 1-1 we could say that E = 2.0×106 V/m = 2.0×103 kV/m = 2.0 MV/m.

Since an electric field can exert a force on a charged particle, it can cause it to move, i.e., it can do work on the particle and give it energy. Thus, electric fields possess energy which must of course come from the source of the fields, which have not yet been discussed. Experiments and theory show that the electric field energy density uE, i.e., the field energy per unit volume (J/m3) at any point is proportional to the square of the field strength (E2) at that point. Thus, for fields in vacuum:

uE = ½ε0E2 [1-2]

where ½ε0 is a proportionality constant. (It is simply an historical convention that this

2 It is conventional to use bold-face symbols for vectors.


1-4

constant is written with the ½ factored out.) The constant ε0 is called the ‘permittivity of a vacuum’ and (as you should be able to show; see Problem 1-3) has units of C2/N·m2; it is an important electrical constant of nature; its value is:

ε0 = 8.85×10–12 C2/N⋅m2

Example 1-2: In Example 1-1, the field E = 2.0×106 N/C. What is the energy density at that point?

uE = ½ε0E2 = ½(8.85×10–12 C2/Nm2)(2.0×106 N/C)2 = 18 J/m3 _________________________ The previous discussion describes what an electric field does (it exerts a force and carries energy). How is it produced; what causes an electric field? An electric field is produced in two ways: (a) by charged particles. (b) by magnetic fields that are changing with time (electromagnetic induction);

magnetic fields are discussed in Sec. 1.3. For now, consider only method (a) above.

Each charged particle produces its own electric field which exerts a force on other charges. For example, the nucleus of an atom produces an electric field that exerts an attractive force on each electron in the atom; each electron produces a field which exerts an attractive force on the nucleus and a repulsive force on all the other electrons.

Consider a positive point charge Q (or a small spherical charge). Its field is spherically symmetric and at each point in space is directed away from the charge as shown by the vectors in Fig. 1-2(a).3 This direction follows from the well known fact that ‘like-charges repel’ (if there is no magnetic force present). Similarly, the field produced by a negative charge points toward the charge as shown in Fig. 1-2(b). As indicated by the vectors in Fig. 1-2(a) and (b), the field gets weaker as we move away from the charge;

more specifically, the field strength is given by:

EQ

r=

1

4 02πε

[1-3]

3 We assume that Q is at rest or if moving, its speed v is


1-5

where r is the distance from charge Q.4 Doubling the distance reduces the field to ¼ of its value. Vector diagrams such as Fig. 1-2(a) and (b) are useful for visualizing electric and magnetic fields. Usually these diagrams are changed to ‘field line’ diagrams such as Figs. 1-2(c) and (d). These lines are simply lines with the property that at each point the field vector (if drawn) would be in the direction of the line; the arrows give the field vector direction. Of course, the real fields are three-dimensional. In the line diagrams, the information about the vector length (i.e., magnitude of the field) at each point is lost. However, this information is to some extent retained by the convention of drawing the lines so that the number of lines per unit area (area perpendicular to the lines) is proportional to the field strength. Where the lines are close together such as near the charge, the field is strong.

Usually many charges contribute to the total electric field at a point. The total field is simply the vector sum of all the fields that would be produced by each contributing charge if it were there by itself. Figure 1-2(e) shows the field due to an electric dipole, i.e., two equal charges of opposite sign, separated by a small distance. Many asymmetric molecules such as the water molecule are electric dipoles with electric fields similar to Fig. 1-2(e).

Example 1-3:

4The quantity 1/(4πε0) is sometimes represented simply as ‘k’ and has the value 9.00×109 Nm2/C2. The symbol k is overworked in science notation; be careful to not confuse its various meanings

such as this one and the “spring constant” or the “wave vector” etc.

Fig. 1-2 (a) to (d) Electric field vectors and lines for positive and negative point charges. (e) The electric field of an electric dipole.


1-6

A charge q1 of +2e is situated 2 nm from a charge q2 of -2e (i.e. the two charges form an electric dipole). Find the electric field at a point P directly above the negative charge

which is elevated 30° above the q1 - q2 line as shown in the figure.

The distance from q1 to P is d1 = 2/cos30 = 2.309 nm The distance from q2 to P is d2 = 2 tan30 = 1.155 nm

The magnitude of the field E1 at P due to q1 = (9×109)2(1.6×10–19)/(2.309×10–9)2 = 5.4×108 V/m (up to the right) The magnitude of the field E2 at P due to q2 = (9×109)2(1.6×10–19)/(1.155×10–9)2 = 21.6×108 V/m (down) Component of the resultant field to the right = E1cos30 = 4.68×108 V/m Component of the resultant field down = E2 - E1sin30 = 18.9×108 V/m Resultant field E = (4.862 + 18.92)1/2×108 = 19.5×108 V/m

θ = tan–14.68/18.9 = 13.9° Check the qualitative result of this example with Fig. 1-2e _________________________

At the microscopic level, electric fields are everywhere, e.g., producing the force binding atomic electrons to the nucleus or binding atoms together to form molecules. At a more macroscopic level, E fields are important in controlling charge motion in devices as diverse as TV picture tubes, electrostatic air cleaners and xerographic printers.

At the human level, with the exception of our sensitivity to some types of EM radiation, we are usually not directly aware of E fields. However, we do live in large scale and sometimes fairly strong electric fields, usually of atmospheric origin. One such location is near (or within) large thunderstorm clouds. Due to strong vertical winds within and beneath these clouds, various water particles such as ice, hail, snow and liquid drops move both up and down. Due to collisions and other processes which are still not well understood, there is charge separation, i.e., these particles become charged. The overall result is that the top of the cloud becomes positively charged, the middle

Fig 1-3 Electric charges and fields associated with thunderstorm clouds, the ionosphere, and the Earth’s surface.


1-7

negative, and the bottom has both positive and negative regions. The ground beneath each bottom region has a charge of opposite sign as shown in Fig. 1-3. The result is an average electric field (E2) as large as 105 V/m between the ground and the cloud and within the cloud. If we are standing beneath the cloud, we are unaware of this field since we have no sense organs that detect these steady fields. Experiments show that there are detectable biophysical effects but the effects do not appear to be harmful. There is one major exception to the harmless nature of these fields and that is a lightning strike. Because of random turbulence within the cloud, the field at local points may increase to above 106 V/m. This is the dielectric breakdown strength for air containing rain-size water drops. The word ‘dielectric’ is another name for an electrical insulator, i.e., a material that normally has very few mobile charges; air is a fairly good insulator. As the E field increases, the air molecules are distorted and at about 106 V/m some electrons are pulled free of the molecules. The details are complex but for a brief time, this region of the air becomes a conductor and a giant spark, i.e., a lightning strike occurs between the cloud and ground, between different parts of the cloud, or between adjacent clouds.

The charge separation produced by thunderstorms has an effect even in regions that are far from any storms. In the atmosphere above about 80 km, many of the air molecules are ionized. This ionization is primarily caused by high energy solar radiation in the ultraviolet part of the spectrum; this region of the atmosphere is called the ‘ionosphere’. Charge is also fed into the ionosphere from below; positive charge travels upward from the tops of thunderstorm clouds. At any moment, there are always many thunderstorms occurring somewhere on Earth. The overall result is that thunderstorms remove positive charge from the ground, leaving it negatively charged and add positive charge to the ionosphere. Both the ground and the ionosphere are fairly good conductors and the charge quickly spreads uniformly over the Earth’s surface and ionosphere. The result is that even in fair weather, there is a vertical, downward-directed electric field E1 which at the surface has an average value of about 150 V/m (E1 in Fig. 1-3). We are living constantly in this field but are unaware of it.

Another place where there are fairly large electric (and also magnetic) fields is near high-voltage electric power transmission lines. Measurements show that on the ground directly beneath a 750 kV line, the electric field is about 5 to 10 kV/m. These are not steady fields, but oscillate at the electrical AC power frequency of 60 Hz in North America. The field depends on the height of the conductors and decreases rapidly as you move away from the line.

1.3 The Magnetic Field.

Originally, the term ‘magnetism’ referred only to natural and human-made magnets, the force between them or between them and the Earth (e.g., a compass needle). In 1820, it was discovered 5 that magnetism is associated with electricity; magnetic forces are, in fact, forces between moving (including spinning) charged particles or objects, in addition to the electric force. 5 In 1820, Hans Oersted (1777-1851) discovered that an electric current in a wire would deflect a compass needle.


1-8

As indicated earlier, these forces are considered to be due to magnetic fields. A magnetic field is defined as that condition in space which exerts on an object a force which depends on both the charge and the speed of the object relative to the observer’s reference frame. Thus, the Earth (somehow) sets up a magnetic field around itself; this field exerts a magnetic force on a compass needle or more correctly, on those electrons in the needle whose spin axes are aligned parallel to the needle’s long axis. The familiar compass needle can be used as a device to detect and to define the direction of the magnetic field at a point. If at any point a compass needle consistently aligns in one particular direction, a magnetic field exists at that point. The magnetic force on a compass needle is concentrated near the end regions which are called the ‘poles’ of the needle or magnet. On the Earth’s surface, one pole will consistently point approximately northward; that pole is called the N-pole and the opposite end the S-pole. Since the needle always rotates and aligns, in one direction, the forces on the poles must be in opposite directions as shown in Fig. 1-4(a). The direction of the magnetic field at any location is defined as the direction of the magnetic force on the N-pole of the needle or the direction the N-pole of the needle points after it comes to rest, as in Fig. 1-4(b). This choice is of course somewhat arbitrary; the opposite direction, i.e., the direction of the force on the S-pole could have been selected; the N-pole was the historical choice. As indicated in Fig. 1-4, the letter B is used as a symbol for the magnetic field strength; since it has direction, it is a vector quantity.6

It is neither easy nor fundamental to try to use a compass needle to compare or measure the strength of magnetic fields. Remember, the fundamental property of a magnetic field is that it exerts a force on a moving charge. Experiments on current-carrying conductors (and also, for example, on proton and electron beams) in magnetic fields give the results outlined in the following discussion.

Suppose a charge of magnitude |q| is moving with a velocity v in

a magnetic field B. Further, suppose θ is the smaller angle (i.e., θ ≤ 180°) between v and B as shown in Fig. 1-5. In general, the magnetic force FB on |q| has the properties: (i) FB is proportional to |q|

(ii) FB is proportional to v sinθ ≡ vζ, i.e., the component of v in the direction perpendicular to B.

6 In Chapter 6 it will be necessary to introduce another, related quantity H called the ‘magnetic

intensity’ which has different units.

Fig. 1-4 Direction of the magnetic field B relative to the magnetic force FB on the poles of a compass.

Fig. 1-5 A particle of charge |q| moving

with velocity v in a magnetic field B.


1-9

Therefore, FB % |q| v sinθ or FB % qvζ. Note that FB depends on the direction of v relative to B. If the charge is travelling

parallel (θ = 0) or anti-parallel (θ = 180°) to B, then FB is zero. For a given field |q|, and speed v, FB is a maximum when v is perpendicular to B.

Since FB is proportional to |q|vζ, we can write:

FB = (a proportionality constant) |q|vζ. This proportionality constant may be taken as our measure of the strength of the magnetic field; thus we can write:

FB = B |q| vζ [1-4]

To summarize the last few paragraphs, the direction of the magnetic field B is defined as the direction of the magnetic force on the N-pole of a compass needle. We define its

magnitude |B| by Eq. [1-4], i.e., B = FB/qvζ, the magnetic force per unit charge and per unit velocity component perpendicular to B. Since a magnetic field exerts a force on moving charges, it can therefore exert a force on current carrying conductors such as in an electric motor or spinning electrons (as in magnets), etc.

The S.I. units of B are derived from Eq. [1-4]: N·s·C–1m–1, or since one C/s is an ampere (A), N·A–1m–1; this unit is called the tesla (T).7 For example, the magnetic field of the Earth at Guelph Ontario has a value of about 57 µT (57×10–6T). Its direction is

about 75° below the horizontal plane (called the ‘dip angle’) and its horizontal component points about 7° west of north (called the ‘magnetic deviation’). Another unit used to express the strength of magnetic fields is the gauss (G). 8 One gauss is 10–4 T. Thus the Earth’s field at Guelph is about 0.57 G.

Magnetic fields of strong magnets are typically in the range of 2 to 10 T.

7 Named after the prominent Croatian-American inventor, Nikola Tesla (1856-1943). 8 Named after the German mathematician and physicist, Johann Karl Friedrich Gauss (1777-

1855).


1-10

The directional relationship between FB and B is not as simple as that between FE and E; FB does not point in the direction of B. Remember, B is the direction that the N-pole of a compass needle points, whereas FB is the force on a moving charged particle. Experiments show that FB is perpendicular to the plane formed by v and B. Figure 1-6 shows the specific case for a positive charge. To help you visualize this in three dimensions, x, y, z-axes have been added; v and B are in the x-y plane and FB lies along the z-axis as shown. Right-hand Rules are used to help remember the relative directions of v, B and FB. One such rule is illustrated in Fig. 1-6 and stated below.

"For a positive charge, using the fingers of your right hand, rotate the v vector into the B vector through the

smaller angle θ between them; your thumb points in the direction FB."

If the charge is negative, use the left hand instead.9

Example 1-4: A positively charged object is projected horizontally in a westward direction in the northern hemisphere

at a location where the magnetic deviation of the Earth’s field is zero. What is the direction of the magnetic force on the object?

Using the right-hand rule (rotate v into B through

the 90° angle indicated); F points down. ___________________________ Recall that electric fields store energy; in a similar way, magnetic fields store energy. Analogous to the electric field case, the energy density uB (J/m3) is proportional to B2 and this may be expressed as:

0

2

2

1

µB

uB = [1-5]

Compare Eq. [1-5] with Eq. [1-2]. (The fact that the proportionality constant µo is in the denominator in Eq. [1-5] is simply due to the historical way magnetic quantities were defined.) The constant µ0 is called the ‘permeability of free space’ and its S.I.

units are T⋅m/A (see Problem 1-8). The S.I. numerical value of µ0 is defined to be exactly:

µ0 = 4π×10–7 T⋅m/A

Example 1-5: The energy density of the Earth’s field in Guelph (B = 57 µT) is:

9 If you are familiar with vector algebra you will recognize that FB, v and B are related by: FB = qv×B, i.e. FB is in the direction of the vector cross product of v and B.

Fig. 1-6 The right hand rule. The direction of the magnetic force F on a charge +q moving

with a velocity v in a magnetic

field B.


1-11

uB

B = =× −

× − ⋅= × −

12

57 10 6

2 4 10 713 10 3

2

0µ π

( T)2

T m / A) J / m3

(.

________________ We now know what magnetic fields do: (a) they exert a magnetic force on charged particles in motion; (b) they store energy. Now we ask: What produces magnetic fields?

We are familiar with magnetic fields produced by magnets, the Earth, etc. At a more fundamental level, magnetic fields are produced in two ways: (a) by moving electric charge (e.g., electric currents). (b) by electric fields that vary with time. Note that no ‘magnetic charge’ (or magnetic monopole) has ever been observed, i.e., there are no radial magnetic fields that start (or terminate) on particles such as is the case for the electric fields of Figs. 1-2(c) and (d). Magnetic field lines form closed loops with no ‘start’ or ‘stop’ points, as illustrated in Fig. 1-7.

The magnetic field around a long straight current-carrying conductor (e.g., a metal wire) is an example of a field established by moving charges. Figure 1-7 illustrates the case for a vertical wire carrying a current I in the upward direction. The field lines are closed circular loops, concentric with the wire, in planes perpendicular to the wire, i.e. horizontal planes in this case, as illustrated by the plane P. The letters N, E, W, S represent the north, east, etc. directions. Figure 1-7 also illustrates another ‘Right Hand Rule’ to help you remember the relative direction of the field and current:

"Wrap the fingers of your right hand around the wire with your thumb pointing in the direction of the current, your fingers point in the direction of the magnetic field lines encircling the wire."

At each point, the magnetic field vector, B, is tangent to the line. For example, for a vertical upward current, as shown in Fig. 2-7, the field at a point to the west of the wire points toward the south, the field at a point to the south of the wire points east etc .

As you would expect, the magnitude of the field decreases with distance r as you move away from the wire; the magnitude of B produced by a current I is given by:

BI

r=

µπ0

2 [1-6]

Fig. 1-7 The magnetic field about a long, straight current-carrying conductor. The right hand rule for I and B is shown.


1-12

Example 1-6:

(a) What is the magnetic field 2.00 cm to the west of a long, straight vertical wire carrying a current I of 10.0 A upward?

BI

r= =

× ⋅= ×

−−µ

ππ

πµ0

74

2

4 10

00200100 10

(

( ..

T m / A)(10.0 A)

2 m) T = 100 T

This field would be about the same magnitude as the Earth’s field. Using the Right Hand Rule (Fig. 1-7), the direction of this field is horizontal, to the south.

(b) What is the magnetic force FB on a proton located at the point in part (a)?

The proton has a velocity v of 1.00×106 m/s upward, i.e., parallel to the current in the wire. Using Eq. [1-4]:

FB = qvζB = q(v sin θ)B = (1.60×10–19 C)(1.00×106 ⋅ sin 90° m/s)(1.00×10–4 T) = 1.60×10–17 N

The Right Hand Rule of Fig. 1-6 tells us that this force is toward the wire, i.e., horizontal, eastward.

________________

Figure 1-8(a) shows the important case of a loop of wire carrying a current I. At each point, the magnetic field vectors from each small segment of wire add to give the field shown.10 This field is particularly strong along the axis of the loop. For simplicity, the complete closed loops for each field line have not been drawn in the figure. Many magnetic devices (e.g. electromagnets) consist of loops of wire on a common axis (i.e., a coil), producing similar fields.

10 Imagine bending the wire in Fig. 2-7 into this loop and you will see how the field of the loop forms. You will also see that the field around the incoming and outgoing wires cancel out.

Fig. 1-8 Magnetic fields produced in various ways: (a) Current loop, (b) spinning spherical charge, (c) bar magnet.


1-13

A spinning charged particle (such as the spinning positively charged sphere of Fig. 1-8(b)) effectively forms a current loop with a magnetic field similar to the loop of Fig. 1-8(a). The field of an iron bar magnet is produced by the fields of many electrons in the iron with their spin-axes aligned and is shown in Fig. 1-8(c).

Figure 1-9 is a simplified approximation of the rather complex magnetic field of the Earth. It is produced by (poorly under-stood) electric currents in the molten core of the Earth. Near the surface, the field is modified by the magnetic properties of the local rock material, and at high altitude it is modified by the ‘solar wind’ of charged particles such as protons and electrons from the Sun. The Earth’s magnetic field has important environmental consequences: it protects us, living at the Earth’s surface, from the solar wind. These particles enter the Earth’s magnetic field and experience a magnetic force perpendicular to the plane formed by their velocity v and the field vector B. This results in their deflection into a spiral path along the field lines moving toward the north or south poles. As the magnetic field gets stronger near the poles, many of the particles are ‘reflected’ back toward the opposite pole. Thus, many of these particles are trapped at high altitudes in paths reflecting back and forth between the Polar Regions. The particles tend to be concentrated high above the equatorial and temperate regions of the Earth in belts called the Van Allen radiation belts. 11

Some of these high-energy charged particles from the Sun do manage to enter the upper atmosphere (above 100 km altitude) near the North and South Poles, where they collide with the relatively few oxygen and nitrogen molecules at those altitudes. The absorbed energy excites the electrons of the air molecules to higher energy states; as they fall back to the ground state, they emit the beautiful green and red colours we call the Aurora Borealis (Northern Lights) and Aurora Australis (Southern Lights) of the sub-polar regions.

So far we have described in some detail (i) electric fields produced by charged particles and (ii) magnetic fields produced by moving charges e.g. electric currents. However, as mentioned earlier, E fields can also be produced by changing B fields and similarly B fields can be produced by changing E fields. Fields produced in this manner are often called ‘induced’ E and B fields (electromagnetic induction). These processes have many

11 Named after American astrophysicist James A. Van Allen, b. 1914.

Fig. 1-9 The magnetic field of the Earth showing the Van Allen belts and the spiral paths of solar charged particles.


1-14

important applications (e.g. electrical generators and transformers) and are essential to the propagation of electromagnetic waves (see Sec. 1.4). We will not discuss this induction process further, however an important environmental application is referred to in the following paragraph.

In Sec. 1.2, when discussing electric fields, the relatively large electric fields encountered beneath high-voltage electric power transmission lines were mentioned. There are also relatively large magnetic fields beneath these lines. These are not steady fields but rather oscillating at the power frequency, e.g., 60 Hz. For several decades, there has been concern that these fields might cause health problems for those living very close to the lines or for electrical workers who spend a great deal of time working near the conductors. In addition to the fields produced directly from the charge and currents in the conductors, the oscillating magnetic fields could induce electric fields in people and other living organisms near the transmission lines. Similarly the oscillating electric fields can induce magnetic fields. Many laboratory experiments have been done investigating the biological effects of these low frequency fields. In addition, several studies have been done on electrical workers and on people living near such lines. The general consensus to date is that these fields are not strong enough to produce any medical effects. Further discussion of this subject is deferred to Chapter 6 5.


1-15

1.4 Electromagnetic (EM) Waves.

A. The General Nature of EM Waves. EM waves are produced by accelerating charge. A relatively simple and important example is a charge (or charges, e.g., an electric current) oscillating sinusoidally (simple harmonic motion) with a frequency f. The charge will radiate sinusoidal EM waves oscillating with the same frequency f. The radiation will carry energy away so, of course, the source must be continuously supplied with energy to maintain steady EM waves.

As an example, consider a simple vertical wire connected to an alternating current generator (G) at its centre as in Fig. 1-10. The generator pumps electrons up and down the wire i.e., it creates an oscillating current of frequency f; as a result the ends of the wire are alternately positively and negatively charged. At the instant shown, the upper end is positive and the current I is upward. By a suitable choice of electrical components in the generator, the frequency f can be adjusted over a wide range of values. For example, the frequency could be made f = 106 Hz or 1 megahertz (1 MHz); if so, this arrangement would be essentially a radio transmitting antenna.12 This arrangement is also called an ‘oscillating dipole’ since, at any instant, the ends of the wire have opposite charge. (Many details about efficient antenna design are ignored here. For example, for each frequency f, the length of the wire must be properly chosen so that standing waves of current are set up in the wire, i.e., the current resonates at frequency f.) The charges in the wire and the current create electric and magnetic fields around the wire. A complete analysis shows that the fields are extremely complex in the space 12 Radio station CFRB in Toronto broadcasts at ‘1010 on the dial’ which means 1010 kHz or 1.010 MHz.

Fig. 1-10 EM waves radiating from an oscillating electric dipole (radio transmitter).


1-16

near the antenna. The field lines shown in Fig. 1-10 are purely suggestive.13 Most of the oscillating fields near the antenna are not EM waves, i.e., they do not carry energy away. The fields close to the antenna are often called the near-fields or the reactive-fields. There are similar near-fields close to other radiating systems such as microwave antennas. These near-fields are mentioned because some people consider them to be a possible health hazard to technicians if they must work for extended periods close to the active antennas. This is similar to the concerns about the fields near high-voltage power transmission lines. Fortunately, the magnitude of these fields decreases rapidly as one moves away from the immediate vicinity of the antenna. This is discussed further in Chapter 5.

Of more interest are the EM waves or radiative fields that radiate energy away from the dipole. The fundamental reason that these (and other) EM waves can exist is that alternating E fields can induce B fields in their vicinity and also alternating B fields can induce E fields (refer to Sec. 1.3). Thus some of the alternating fields near the antenna propagate themselves outward, away from the antenna in almost all directions; these are the EM waves.

It is difficult to fully represent EM waves in a diagram. They are, of course, three-dimensional and the field lines form closed loops. In Fig. 1-10, an attempt is made to illustrate what the EM wave would look like if you could somehow see the fields at a given instant along a radial line (the ‘r-axis’ in Fig. 1-10) directed outward from the

centre of the dipole and at angle θ from the direction of the dipole, i.e., in this case, at angle θ from the vertical. The dipole-source controls the direction of the oscillating E-field of the waves for any given r-axis. The E-field oscillates in the plane defined by the dipole and the r-axis. In this case, since the dipole is vertical, E oscillates in a vertical plane and further, within this plane, it oscillates in a direction perpendicular to the r-axis, i.e., perpendicular to the direction of travel of the wave (see Fig. 1-10). At any instant, the magnitude of E varies sinusoidally along the r-axis as shown. The repeat-length, i.e., the wavelength

λ, is indicated. Associated with the E-field is an oscillating B-field. The B-field portion of the wave oscillates in a plane perpendicular to E, in this case, in the horizontal plane. Within this plane, B oscillates along a line perpendicular to the propagation direction (r-axis). Thus E and B are perpendicular to each other and to the direction of travel, the r-axis; i.e., the wave is a transverse wave.

In a travelling EM wave, there is a definite correlation at each point in space and time between the magnitude of E and B; their magnitudes are related by the simple equation:

E = cB [1-7]

13 Compare, however, with Fig. 1-2e for the electric field configuration of a dipole and Fig. 1-7 for the magnetic field configuration of a current-carrying wire.


1-17

Thus, as shown in Fig. 1-10, the fields have zero values at the same points; they have their positive maxima together and their negative maxima together i.e., the E and B parts of the wave are exactly in phase with each other. If, at a given instant, you looked along different radial lines, the adjacent points where the E and B-fields have their maximum values (amplitude) lie on spherical surfaces, concentric with the centre of the dipole. These are suggested by the curves in Fig. 1-10. These spherical surfaces are called wave fronts; this dipole radiation is called a spherical wave. For spherical waves, energy conservation (see below) requires that the wave amplitude decrease with increasing distance r from the dipole; in fact, the amplitude is proportional to 1/r. Finally, theory and observations show that for any given distance r, the E and B

amplitudes are not constant but are proportional to sinθ where, as shown, θ is the angle between the direction of the dipole (the vertical) and the line of observation (the r-axis). Thus a dipole does not radiate uniformly in all directions. (This of course is true for most sources of waves, EM, sound, etc.). The dipole does not radiate at all in

the direction of its ‘ends’ (i.e., θ = 0° or θ = 180°); the maximum amplitude is in the direction θ = 90°, i.e., in the horizontal plane in this case. In general, the magnitude of the E-field of the wave depends on distance r from the

dipole, time t, and angle θ. This wave can be modeled by the travelling wave equation:

E r tE

r

t

T

r

E r t kr

( , , )sin

sin

( , )sin( )

θθ π π

λ

θ ω

=

−

−

1

0

2 2

=

[1-8]

where: E1 = a constant with units of N⋅m/C which is proportional to the amplitude of the wave at r = 1 m and θ = 90°; this constant is a measure of the radiating strength of the dipole.

E0(r,θ) ≡ E1sinθ /r is the amplitude of the wave (N/C) at a given r and θ. Further: T = 1/f is the period of oscillation (s)

ω = 2πf = 2π/T is the angular frequency (radians/s) λ is the wavelength (m) k = 2π/λ is the ‘wavevector’ (radians/m)

Similarly, for the B-field:

B(r,θ ,t) = B0(r, θ)sin(ωt - kr) [1-9]

where, from Eq. [1-7], the amplitude B0(r, θ) = E0(r, θ)/c. Thus, at any point r, angle θ and time t, B = E/c. The pattern shown in Fig. 1-10 is for one instant; it is not static. The fields oscillate so that the sinusoidal pattern and associated energy propagates outward with the


1-18

speed common to all EM waves, that is, the speed of light, c = 2.998×108 m/s (to 4 significant figures).14 Further, as for any periodic wave, c, λ, T and f are related:

cT

fk

= = =λ

λω

[1-10]

Example 1-6: What is the wavelength of the EM waves broadcast by radio station CFRB (Toronto), where, as mentioned earlier (see footnote 12), f = 1.010×106 Hz?

λ = =×

×=

c

f

2998 10

1010 10297

8

6

.

.

m / s

Hz m

_________________________

Example 1-7: A measurement of the electric field 2.0 km horizontally from the CFRB transmitter shows that its amplitude is 100 mV/m.

a) What is the equation of the EM wave? b) What is the electric field amplitude 3.0 km away horizontally and 1.0 km above the ground?

a) Since f = 1.01×106 Hz, then ω = 2πf = 6.35×106 rad/s λ = 297 m (from Example 2-6), k = 2π/λ = 2.12×10–2 rad/m

E1 sinθ /r = (E1 × sin90)/2000 = 100×10–3 therefore E1 = 200 Nm/C

E(r,θ,t) = (200sinθ /r)sin(6.35×106t - 2.12×10–2r)

b) tan(90 - θ) = 1/3 therefore θ = 71.6° and r = (32 + 12)1/2 = 101/2 = 3.16 km

E0(3.16 km, 71.6°) = 200 sin71.6/3160 = 0.06 V/m ___________________________________

B. The EM Spectrum.

With a human-made device such as the radio transmitter described in the previous

section, frequencies can be generated over a range from f ≈ 0 Hz to about 100 GHz

14 The symbol ‘c’ and the given value are for waves in vacuum. If these waves are in air, the speed v will be slightly less than c; usually we can ignore the difference in speed between air

and vacuum. From EM theory, Maxwell showed that c is related to εo and µo, specifically: c

=1/√εo µ0___ If the values of εo and µo given earlier are substituted, the value for c given above will

be obtained.


1-19

(GHz = gigahertz = 109 Hz). However, there are many naturally occurring oscillating charge systems such as atoms, molecules, and nuclei that radiate EM waves. In fact, most of the EM radiation we are familiar with comes from these sources. Thus there is, in principle, no limit to the EM wave frequencies (and wavelengths) we may observe in nature. This range of frequencies is called the EM spectrum and is illustrated in Fig. 1-11. Note the logarithmic scale and the large range of frequencies involved.

Since c = λf, low frequency waves have long wavelengths and high frequency waves have short wavelengths as indicated. Also shown are the names given to the various regions. Obviously visible light, with wavelengths from about 400 nm to 700 nm (1 nm = 10–9 m) and frequencies around 1014 Hz, forms a very small region in the entire EM spectrum.

C. Energy, Power and Irradiance. As mentioned in Sec. 1.2 and 1.3, electric and magnetic fields possess energy with

energy densities given by: uE = ½ε0E2 and uB = B2/2µ0 (Eq. [1-2] and [1-5]). Hence, EM waves transmit energy which travels outward from the source at speed c (in air or vacuum). In an EM wave, the energy is associated with both the E-field and the B-field

portion. Since, in a travelling EM wave, E = cB and also since c = 1/√ε0µ0___

, it is easy to show (see Problem 1-15) that uE = uB for an EM wave. The custom is to express these

Fig. 1-11 The Electromagnetic spectrum.


1-20

energy densities in terms of the E-field, i.e., uE = uB = ½ε0E2. Therefore, since the total energy density uT = uE + uB, then

uT = ε0E2 [1-11]

for EM waves. Since, in general, E varies with position and time, uT also varies in a similar way. Often we are interested in the energy per unit time (power dP) falling on a surface (or passing through a surface) per unit area dA as shown in Fig. 1-12; this is called the irradiance I on the surface.15

I = dP/dA [1-12]

The S.I. units of irradiance are (W)/m2 or J/s⋅m2. Using Fig. 1-12, consider a short time interval dt. In this time interval the energy flows forward a distance cdt. Therefore the energy which falls on the area dA (or passes through it) is the energy on the incident side of dA

contained in the volume c⋅dt⋅dA. If dt is a short time, this volume is small (that is cdt


1-21

Bright sunlight gives an irradiance of about 1000 W/m2. Assuming it is a sinusoidal wave, what is the amplitude of the electric and magnetic fields in the wave?

From Eq. [1-14]:

EI

c0

012 8

2 2 1000

885 10 3 10868= =

× ×=

−ε( )

( . )( ) N / C

(All quantities in this calculation are in S.I. units; check that the calculation does give E0 in N/C.)

From Eq. [1-7], B = E/c, so the B-field amplitude is: B0 = E0/c = (868 N/C)/(3.00×108 m/s) = 2.89×10–6 T

(again, check the units).

________________ It was stated above that, because of energy conservation, the amplitude of the EM wave from a dipole radiator must decrease with distance r, from the source; more specifically Eo is proportional to 1/r. This is in fact true for any spherical wave of any type (EM, sound, etc.). This is related to the well known inverse square law.

Remember that the area of a sphere of radius r is A = 4πr2; thus if the size of a sphere is changed, the ratio of A/r2 (=

4π) remains constant; or equivalently A is proportional to r2. A similar relationship holds for the cone shown in Fig. 1-13. The cone has its apex at the centre (c) of the sphere and where it intersects the sphere, the segment of

spherical surface has an area ∆A which is a small portion of the total spherical area A. If the radius is changed ∆A changes, but for a given cone size, ∆A is proportional to r2 just as the total area A is proportional to r 2. That is, for a given cone:

∆A = kr2 [1-15] where k is a proportionality constant whose value depends on the apex angle of the

cone. If r is doubled, ∆A will increase by a factor of four, etc. The apex of a cone such as this is often called a solid angle or a ‘three-dimensional

angle’. Further, the constant ratio ∆A/r2 (the proportionality k in Eq. [1-15]) is used as a measure of the size of the solid angle and this dimensionless ratio is said to be measured in

steradians (sr).16 For example, if a cone subtended an area ∆A = 0.10 m2 at a distance r = 0.50 m from its apex, the solid angle of the cone would be 0.10 m2/(0.50 m)2 = 0.40

steradians or 0.40 sr. Often the symbol Ω or ∆Ω is used to represent a solid angle, and we could express Eq. [1-15] in the form (replacing k by ∆Ω): ∆Ω = ∆A/r2 or, 16 The entire concept is similar to the ‘radian’ used for two-dimensional angles.

Fig. 1-13 A cone of solid

angle ∆Ω and its associated spherical surface.


1-22

∆A = ∆Ω⋅r2 [1-16]

As a cone opens up, it eventually sweeps out the entire three-dimensional space

around its apex point and includes the whole sphere. For any given r, the ∆A becomes the total spherical area 4πr2 of the sphere surrounding the point at its centre. Thus, the total solid angle completely surrounding a point has a size of ∆A/r2 = 4πr2/r2 = 4π sr (i.e., 12.6 sr).17 If a source radiates in all directions, it radiates into 4π steradians. A ceiling lamp normally radiates downward into a hemisphere; it radiates into a solid

angle of 2π steradians. Return to the case of a ‘point source’ 18 radiating spherical waves. In general, it

radiates with a different amplitude in different directions (recall the sinθ factor for the dipole in Eq. [1-8]). We might be interested in how much power (∆P) it radiates in some particular direction (e.g., along the line shown in Fig. 1-14). If you think about it, it makes no physical sense to ask: "How much power radiates out along a line?" A line has no width. Rather we must consider a small cone or

solid angle (∆Ω) surrounding the line or direction of interest, with its apex at the source S as in Fig.

1-14. The source radiates some power ∆P into this small cone or solid angle. The ratio ∆P/∆Ω (watts/steradian) is called the radiated intensity (S) in the direction given by the centre line of the cone, i.e.,

SP

=∆∆Ω

[1-17]

(More correctly, the limit of this ratio is taken as ∆Ω→0). In general, S varies with direction from the source.

If there is no absorption of the radiated power, the power ∆P within the cone ∆Ω remains constant as does S. However, this constant power spreads out over larger

areas ∆A as it moves out to larger distances r. Therefore, the irradiance I decreases. Specifically, since I = ∆P/∆A and ∆A = r2 ∆Ω, then

IP

r

S

r= =

∆

∆Ω2 2 [1-18]

i.e., I ∝ 1/r2. This is the inverse square law valid for all spherical waves (point

17 This is equivalent to the total 2-dimensional angle around a point (i.e. 360 degrees) being equal to 2π radians.

18 No real source is a ‘point source’; however, any real source may be treated as a point if we are a distance r from the source which is greater than about 10 times the largest source

dimension. Thus, a 30 metre radio antenna would act as a point source if we are 300 m or more

away from it.

Fig. 1-14 The radiant intensity from a source (G) in a given direction.


1-23

sources) in any given direction.

For EM sinusoidal waves, I = ½ε0cE02 where E0 is the electric field amplitude. Combining this with Eq. [1-18] shows that E02 is proportional to 1/r2; therefore, E0 is proportional to 1/r as stated previously and shown explicitly in Eq. [2-8].

Example 1-9: A radio station radiates a total power of 50,000 W outward and upward from

its antenna, i.e., into the ‘above ground hemisphere’ or 2π steradians. The power radiated varies with direction. Your apartment is 10.0 km south-west of the transmitter and on the 20th floor of your building. In the direction of your apartment, the radiated intensity from the station is 5000 W/sr. (This would be a result of the way the transmitter antenna is designed.) (a) What is the average irradiance at 10.0 km from the station?

There are two equivalent ways to calculate the average irradiance (i.e., averaged over all directions). (i) use Eq. [1-18] with ∆Ω = 2π sr and ∆P = 50,000 W

IP

rav 3 2

5 250000 W

(10 10 m) (2 sr)7.96 10 W/ m= =

×= × −

∆

∆Ω2 π

(ii) consider the power ∆P radiated into a hemisphere; at a distance r, the

area of the hemisphere is 2πr2

IP P

rav

5 27.96 10 W/ m= = = × −∆∆

∆A 2 2π

as above. (b) What is (i) the irradiance, and (ii) the magnitude of the electric field amplitude at your apartment?

(i) The I in part (a) is the average value over all directions. The irradiance at

any given point varies with direction. In the direction of your apartment S = 5000 W/sr,

Therefore at your apartment

IS

r= =

×= × −

2

5000 W/ sr

(10 10 m)5.00 10 W/ m

3 25 2

i.e., somewhat less than the average value.

(ii) From Eq. [1-14], I = ½ε0cE02 for a sinusoidal wave.

EI

c0

0

2= =

×

× ×=

−

−ε2(5.0 10 )

(8.84 10 )(3.0 10 )0.19 N / C (or V / m)

5

12 8

Incidentally, this irradiance and field amplitude is more than adequate to give


1-24

you good radio reception. Modern radio receivers with built-in antennas can pick up signals as weak as a few hundred µV/m.

________________ D. Radiation Pressure and Momentum. In addition to energy, EM radiation also transports momentum which is now considered briefly. If an EM wave is incident normally on an absorbing surface as in Figure 1-15, a detailed analysis shows that the E and B fields of the wave exert a force on the charges in the atoms in the surface, a force perpendicular to the surface, i.e., a radiation pressure P.19 A complete analysis shows that the magnitude of the pressure is related to the irradiance I by (consult any optics text):

P = I/c [1-19]

For the irradiance values we normally encounter on Earth, this pressure is extremely small and we are unaware of it. For example, with bright sunlight, I = 1000 W/m2 and therefore P is about 3×10–6 Pa (pascals); recall that

1 atmosphere = 1×105 Pa.20 The radiation pressure due to light from the Sun is believed to be part of the cause of the tails of comets; the tail consists of particles of ice, rock and gas from the head of the comet. The tails exist only when the comet is near the Sun and the gas tail points directly away from the Sun.

We associate a force with a change in momentum. The pressure of the wind on a surface is caused by the momentum carried by the air molecules to the surface which then reflect from the surface. Similarly, the EM radiation carries linear momentum to the surface (even though the radiation does not have mass like the wind does). It can be shown from Eq. [1-19] and Newton’s Laws that, for any EM radiation, the energy E and the corresponding magnitude of the momentum p are related by:

p = E/c, or E = pc [1-20]

Example 1-10: Consider again the bright sunlight of irradiance I = 1000 W/m2. The corresponding momentum carried to each square metre of surface each second

is 1000 J/3.00×108 m/s = 3.3×10–6 kg m s–1, an amount even less than the momentum of a single raindrop striking the ground surface.

________________

19 Do not confuse this P with the same symbol for “power”. 20 If the radiation is perfectly reflected from the surface rather than absorbed, the pressure P is

doubled.

Fig. 1-15 Pressure P produced by EM radiation on an absorbing surface.


1-25

Under the proper conditions (circularly polarized light), a light beam will also exert a torque on an absorbing surface; this may be interpreted as the transport of angular momentum by the radiation. Thus, EM radiation transports energy and both linear and angular momentum.

1.5 Polarization. Because EM radiation is a transverse wave, it exhibits phenomena known as ‘polarization’. An example is the well known variation in the irradiance of light transmitted through two sheets of Polaroid when one of the sheets is rotated relative to the other. There are no such phenomena for longitudinal waves such as sound.

Refer to the description of the radio waves from the radio transmitter or dipole antenna discussed in Sec. 1.4. As described there and illustrated in Fig. 1-10, the electric field vector E along any r-axis, oscillates in the plane defined by the direction of the dipole and the r-axis and within that plane in the direction perpendicular to the r-axis. (In this discussion, we ignore the B-field which at all points is perpendicular to the E-field.) For example, if the dipole is vertical, the E-field radiated in any horizontal direction is also vertical. Therefore, such waves are said to be ‘plane-polarized’ or ‘linearly polarized’. Remember, the polarization direction is the direction of oscillation of the E-field and not the direction of travel of the radiation. Many radio transmitters are vertical so their radio signals are vertically polarized.21 Many TV signals are horizontally polarized.

Visible light from a lamp, the Sun, etc., is produced by billions of atoms, each radiating for a very short time (about 10–9 s). For purposes of this discussion, we may model each atom as a small dipole radiator of random orientation. Thus we may picture a sample of light from such a source as consisting of billions of short overlapping independent wave-trains, each individual wave-train being linearly polarized in some random direction perpendicular to the direction of travel of the light. With billions of wave-trains present at any point, essentially all possible polarization directions are present. The net result is that there is no particular polarization direction. Such light is said to be ‘unpolarized’; another name is ‘natural light’. Figure 1-16 illustrates two ways to represent unpolarized light in diagrams. Figure 1-

21 It is for this reason that you usually get the best reception, at least in open spaces, with the telescopic antenna of portable radios vertical.

Fig. 1-16 Two methods of representing unpolarized light.


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16(a) represents a beam of such light travelling in the x-direction, the several arrows in the y-z plane represent the many (in fact all possible) directions of oscillation of the E vectors. Figure 1-16(b) is another representation. All of the E vectors could be resolved into components along orthogonal axis (i.e., y- and z-axes) perpendicular to the direction of travel (the x-axis). For the present discussion, all directions perpendicular to the x-axis are equivalent, in this diagram, the y-axis has been chosen in the plane of the paper and the z-axis is perpendicular to it. The vertical lines (in the x-y plane) represent the one set of E components and the dots represent the orthogonal components, parallel to the z-axis or in the x-z plane. Because of the equivalence of all directions perpendicular to the direction of travel, the y-components and z-components are equal. Thus, one may consider unpolarized light as composed of two equal linearly polarized waves, polarized in orthogonal planes but with no fixed phase relationship (i.e., incoherent) relative to each other (or at least with any definite phase relationship changing at about 10–9 s intervals). Most of the EM radiation (visible, ultraviolet, infrared, etc.) produced by atomic and molecular sources in our environment is unpolarized, at least at the time of production. However, it is possible to convert at least some of the radiation into linearly polarized light. Three common methods are: (a) Selective absorption, (b) Reflection and (c) Molecular scattering

Selective Absorption. Selective absorption is the process whereby some structured materials (e.g., some crystals) absorb light by different amounts depending on the orientation of the electric field vector of the light relative to the structure of the material. One of the most common and important of these materials is ‘Polaroid’, the material used in some sunglasses. Using a special manufacturing process, long hydrocarbon molecules are aligned parallel to each other in a particular direction in the Polaroid sheet. When light is incident on the sheet, the electric field components parallel to the long molecules cause electrons to flow along the molecules, absorbing the radiant energy and converting it into thermal energy. Thus, the field components parallel to the hydrocarbon chains are absorbed and the fields perpendicular to them are not - i.e., they are transmitted through the sheet. This direction, perpendicular to the chains, is called the transmission axis of the sheet. For simplicity we will assume that 100% of the light with a polarization parallel to the transmission axis is transmitted and none of the perpendicular component is transmitted.

Now consider unpolarized light of irradiance I0(W/m2) incident on a sheet of Polaroid (sheet 1) with transmission axis (TA1) as oriented in the left half of Fig. 1-17. All the components of the E-fields of the incident light parallel to TA1 of sheet 1 are transmitted through the sheet and all of the components perpendicular to TA1 is absorbed. Thus, the light transmitted by sheet 1 is linearly polarized with its E-field parallel to TA1 and with irradiance I 1= ½ I 0 since in the original unpolarized light, one-half of the light energy is associated with each set of components. Thus a single sheet of Polaroid converts a beam of unpolarized light into a beam of plane polarized light of half the original intensity.


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If a second Polaroid sheet (2 in Fig. 1-17) is placed in the path of the light from sheet 1

with the transmission axis of sheet 2 (TA2) making an angle θ with TA1, then only the components of the E-field incident on sheet 2 which are parallel to TA2 will be transmitted through sheet 2. Therefore, the light emerging from sheet 2 is also linearly polarized parallel to TA2. Further, if the resultant amplitude of the light emerging from sheet 1 and incident on sheet 2 is E1, the resultant amplitude E2 emerging from sheet 2 is the component E2 =

E1cosθ. Since the irradiance is proportional to |E|2 for a sinusoidal wave, then I2 = I1 cos2θ = ½I0 cos2θ [1-21] a relationship known as ‘Malus’ Law’ after its discoverer.22 It applies to any two

polarizing elements whose axes make an angle θ relative to each other. The initial element (Polaroid sheet 1) that first produces the linear polarization is often called the polarizer and the second element is often called the analyzer since by rotating it one can determine the polarization direction of the light incident on it.

Obviously when θ = 90° or 270° (so-called “crossed Polaroids”), I2 is zero. Thus, if the direction of TA2 is known, the E1 direction may be determined.

Example 1-11: If in Fig. 1-17 a beam of sunlight of irradiance I0 = 500 W/m2 is incident on

Polaroid 1, and if θ = 60°, what is I2 emerging from sheet 2?

I I I2 12

02= = =

=cos cosθ θ½

500 W/ m

2cos 60 62.5 W/ m

22 2ο

This light is linearly polarized, parallel to TA2. ________________ Polarization by Reflection. The reflection of unpolarized light by the smooth surface of a transparent dielectric (non-conductor of electricity) such as water or a waxed floor can also produce linearly polarized light in the reflected beam.

First, let us review the ‘laws of reflection and refraction’ for such a smooth surface and for an isotropic material such as a liquid. The incident ray, reflected ray, refracted ray and surface normal are all in the same plane, as in Fig. 1-18a. Further:

(i) Angle of incidence (θ1) =

22 Etienne L. Malus (1775-1812), French Physicist. He thought that the two components of light had different “polarities” like magnetism and from this mistaken idea we get the word

“polarization”.

Fig. 1-17 Conversion of unpolarized to polarized light (Sheet 1) with subsequent analysis by Sheet 2.


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angle of reflection (θ1́) (ii) Snell’s Law: n1 sinθ1 = n2 sinθ2 [1-22] where n1 = c/v1 is the refractive index of material 1 and v1 is the speed of light in material 1; similarly for n2.

A complete analysis using the boundary conditions for the E and B-field components at the surface (or by experimental measurements) shows that the amount of the

incident light reflected varies with the angle of incidence θ1, being, for water, a minimum of about 2% at normal incidence (θ1 = 0°) and increasing to 100% at grazing incidence (θ1 = 90°). Further, as suggested earlier, the incident unpolarized light may be considered as the super-position of two randomly phased linearly polarized beams (as in Fig. 1-16b), polarized in orthogonal directions. In this case, the directions of physical significance are parallel to the plane of incidence (plane of the incident ray and normal i.e. the plane of the diagram in Fig 1-18) and perpendicular to the incident plane. These directions are given by the arrows and dots in Fig. 1-18a. The arrows represent E-field components parallel to the incident plane and the dots represent components per-pendicular to this plane. The incident beam consists of 50% of each component. An analysis of the reflection process shows that the reflected light contains more of the perpendicular component than the parallel component and the refracted beam is richer in parallel component. For example, in reflection of unpolarized light from a horizontal water surface, the plane of incidence is vertical. The reflected light is richer in the component perpendicular to the incident plane (a vertical plane), i.e., the component parallel to the water surface (or horizontal). The refracted beam is richer in the component in the plane of incidence. This is shown by the relative lengths of the arrows and size of the dots in the reflected and refracted rays in Fig. 1-18a.

Finally, it can be shown, both theoretically and experimentally, that at one particular

angle of incidence called the ‘polarizing angle’ θp, the reflected beam has no component parallel to the incident plane; it is 100% linearly polarized with its E-field perpendicular to the plane of incidence (or parallel to the reflecting surface) as shown in Fig. 1-18b.

At this special incident angle, the reflected and refracted rays are perpendicular to each other (see Fig. 1-18b) and hence Snell’s Law gives (see Problem 1-21):

Fig. 1-18 (a) Polarization changes in unpolarized light by reflection and refraction at a dielectric surface. (b) Light incident at the polarizing angle.


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tanθ pn

n= 2

1

[1-23]

This relationship is known as Brewster’s Law and θp is also called the Brewster angle.23 For example, for an air-water surface, n1 = 1.00 and n2 = 1.33 and θp is given by: tanθp = 1.33/1.00 and θp = 53°. Thus, if light is incident on a water surface at 53°, the reflected light is 100% polarized in a direction perpendicular to the plane of incidence (vertical) or parallel to the water surface. Note, the statement that the

reflected light (at θ1 = θp) is 100% linearly polarized in the component perpendicular to the plane of incidence does not mean that 100% of this component is reflected. It means only that none of the other component is reflected. Recall that the original beam consists of 50% of each component. At the polarizing angle, about 15% of the perpendicular component is reflected or about 7.5% of the total incident light. The remaining 92% appears in the refracted light which is richer in the component parallel to the incident plane.

As indicated earlier, Polaroid sunglasses will absorb about 50% of unpolarized light. In addition, these sunglasses have their transmission axis oriented to absorb an even larger portion of the reflected sunlight (glare) from horizontal surfaces. This ‘glare’ is rich in horizontally polarized light. Question: In what direction is the transmission axis of the sunglass lens? 24 If you have Polaroid sunglasses, try using them as an analyzer as in Fig. 1-17. Rotate the glasses while viewing reflected light through the lenses. Try to determine the degree of linear polarization and the direction of polarization of the light.

Polarization by Scattering. We are all familiar with the blue sky here on Earth. The blue colour is due to selective scattering of the shorter wavelengths (blue and also ultraviolet) of the Sun’s radiation, by the nitrogen and oxygen molecules in the atmosphere. If there were no air to scatter some of the Sun’s radiation, the sky would be black as it is on the Moon and on Earth at night.

23Named after Scottish physicist Sir David Brewster (1781-1868) who also invented the children’s

toy, the Kaleidoscope.

24Answer: The transmission axis should be vertical to absorb the horizontally polarized glare.

Fig. 1-19 Polarization of unpolarized light by molecular scattering. (a) Scattering model. (b) Application to atmospheric scattering of

sunlight.


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As a result of the molecular scattering process, the light scattered in directions perpendicular to the direction of travel of the incident light is linearly polarized. The following model, illustrated in Fig. 1-19(a), describes the process. The unpolarized incident beam is travelling in the positive x-direction. A single representative scattering molecule is shown. Axes y and z are two representative orthogonal directions, perpendicular to the incident rays. Relative to these axes, the incident unpolarized light may be considered to be comprised of two randomly-phased, linearly-polarized components, one in the x-y plane and one in the x-z plane. These oscillating electric fields of frequency f force the electrons in the molecule into oscillation at the same frequency. Relative to the y- and z-axes, these electron oscillations may be resolved into two dipole components, one in the y-direction (labelled 1) and one in the z-direction (labelled 2). Recall that a dipole does not radiate out of its ends (see Eq. [1-8] and Fig. 1-10); it radiates most strongly in directions perpendicular to itself. Therefore, in the y-direction, the scattered radiation comes entirely from the z-direction dipole and is linearly polarized in the z-direction (labelled 3). Similarly, the radiation travelling in the z-direction is linearly polarized in the y-direction (labelled 4). All directions perpendicular to the x-axis are equivalent, i.e., the y-axis can be in any direction perpendicular to the direction of the incident light rays. Therefore the light scattered in any direction perpendicular to the incident rays is linearly polarized in a direction perpendicular to the plane defined by the incident rays and the scattering direction. Note that in the scattering process, some of the energy from the incident radiation is absorbed by the molecule. However, since in general the incident light frequency is not one of the resonant frequencies of the molecule, there is no permanent absorption; the energy is immediately reradiated as scattered light. Equivalently, one could say that the incident photon does not have the correct energy for permanent absorption by the molecule.

Referring again to Fig. 1-19(a), consider scattering in directions other than perpendicular to the incident rays. For example, consider a direction in the x-y plane,

making an angle of θ = 60° to the x-axis. In this direction, scattered radiation comes from both the y- and z-direction oscillators. The ‘sinθ’ factor in Eq. [1-8] tells us that the z-component radiation has the larger amplitude (for the z-component θ = 90° and for the y-component oscillator θ = 30°). This superposition results in a mixture of unpolarized light (the y-component plus an equal amount of the z-component) and linearly polarized light (the remainder of the z-component). The light is said to be partially polarized. Similar considerations show that the light scattered in the forward and backward direction (i.e., along the x-axis) is unpolarized, consisting of equal y and

z components (θ = 90°). Figure 1-19(b) illustrates an application of the above concepts to blue light scattered by the atmosphere. Suppose it is late afternoon; the Sun is in the western sky as shown. An observer looks at light scattered from a region of the atmosphere near the zenith.25 This light has been scattered to the observer in a direction approximately perpendicular to the direct rays from the Sun. As indicated by the previous scattering

25 The zenith at any location is the point in the sky directly overhead.


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model, this light is linearly polarized in the direction perpendicular to the plane defined by the incident rays of the Sun and the direction from the scattering volume (the zenith) to the observer. The observer could check this prediction, for example, with Polaroid sunglasses, and would find that it is approximately correct, but that the polarization is not perfect. This is mainly due to multiple scattering of the sunlight.

As an experimental exercise you might try the following: Using the scattering model, try to predict the polarization of the blue light scattered by the atmosphere from various directions relative to the Sun’s position. When you have the opportunity, use Polaroid glasses or a Polaroid camera filter, to check your predictions. Also check the light reflected from clouds. Caution: do not look directly at the Sun; it is not safe to do so through Polaroid sunglasses; see Chapter 6. Polaroid filters are often used by photographers to darken the blue sky, enhancing the contrast between the sky and clouds or buildings. The polarization of skylight and its variation with direction relative to the Sun’s position is apparently used for navigation by some insects, for example, honey bees, that have eyes sensitive to the polarization state of the light.26

Other Polarization States. In addition to unpolarized and linearly polarized waves, other polarization states are possible. For example, the superposition of two identical plane polarized waves, polarized in planes perpendicular to each other and with a fixed phase difference of

90°, produces circularly polarized light. In circularly polarized light, the magnitude of the E-vector (and also the B-vector) does not change, instead, its direction changes. At any point, the E-vector rotates in the plane perpendicular to the direction of travel of the wave (the y-z plane in Fig. 1-20; the light is travelling in the positive x-direction). The tip of the E-vector at any x traces out a circle in the y-z plane. At any instant, the tips of the E-vectors at various positions x trace out a helical curve as shown in Fig. 1-20. In the most general case, if the original plane polarized waves are not of equal

amplitude, or if the phase difference is not 90°, the magnitude of E varies as it rotates, tracing out an ellipse, producing elliptically polarized light.

Circular or elliptical light is produced by various means such as reflection of unpolarized light from metallic surfaces. In the most general case, mixtures of unpolarized, and all types of polarized light are possible. See any text on physical optics for more details.

26The human eye is very insensitive to the polarization state of light. There is a very subtle

phenomenon called ‘Haidinger’s Brush’ which can be seen with difficulty. It is described in

several books on natural optical phenomena e.g. Light and Color in the Open Air by M.J.G.

Minnaert.

Fig. 1-20 Circularly polarized light.


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PROBLEMS Sec. 1.2 and 1.3: Electric and Magnetic Fields.

Note: An asterisk * denotes a problem for which additional data must be found elsewhere in

the text or estimated. 1-1. Define or explain the following concepts: (i) electric field, (ii) electric force, (iii) magnetic

field, (iv) magnetic force. State the S.I. units for each concept. 1-2. Verify that the units of electric field N/C are equivalent to V/m.

1-3. Verify that ε0 has units C2/N⋅m2. 1-4. A charge of +6.0 µC, at a certain point in space, experiences an electric force of 2.0 mN

in the +x-direction.

(a) What is the magnitude and direction of the electric field at this point? (b) What is the energy density of the electric field at this point? (c) If the +6.0 µC charge is replaced by a –2.0 µC charge, what is the electric force on

the

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