The Physics of Energy, page 56
13.13 [T] Show that the ratio of the backwork (the work necessary to run the compressor) to the total work done by the turbine for the ideal Brayton cycle is . [Hint: use the 1st Law and the fact that both the compressor and turbine are adiabatic.] Check your result by comparing to the result stated in the text for the conditions shown in Figure 13.19.
13.14 [T] Calculate and plot the ratio of the efficiency of the Brayton cycle to the efficiency of the Otto cycle with the same compression ratio (with ).
13.15 [T] Verify the assertion made in §13.5.2 that the net efficiency of a CCGT is , where is the efficiency of the Brayton/Rankine cycle.
* * *
1 We have already made this type of idealization, e.g. for the combustion process, in our study of internal combustion engines in §11.
2 Although it does not pose a risk to the ozone layer, R-134a is a greenhouse gas that makes a significant contribution to climate change. In October 2016 over 170 countries reached an agreement to phase out the use of R-134a over the next several decades.
3 Although we have not plotted lines of constant enthalpy in the TS-plane, must lie below and to the right of because after throttling it has higher entropy and lower temperature.
4 Pumps are, of course, also used to move liquids up in a gravitational field.
5 A “ton of AC” is an archaic measure used in the US corresponding to the amount of heat required to melt a ton of ice in a day, 1 ton of AC kW.
6 For the ideal Rankine cycle, the temperature denotes the high temperature reached at point ; this is above the temperature at which the working fluid changes phase, unlike for the ideal VC cycle, where denoted the temperature of phase change. This notation matches the physical elements of the system; for the ideal VC cycle matches the operating temperature of the condenser, while corresponds to the nominal temperature of the boiler in the Rankine cycle. Note that only a fraction of the heat is added while the fluid is above the vaporization temperature .
7 Here, and throughout this chapter, we take the energy content of fuels to be the higher heating value (HHV) as defined in §9.4.3 and as is customary in the US. Values for the HHV for coal range from 15 GJ/t to 35 GJ/t (§33) depending on the type of coal. For definiteness, we use here a middle value of 25 GJ/t.
8 Although George Brayton’s original device, patented in 1872, employed reciprocating pistons.
9 We quantify this inefficiency in §36 (Systems).
Part IIEnergy Sources
CHAPTER 14
The Forces of Nature
Elementary particles and the forces that act between them are the fundamental ingredients in our most complete theoretical description of microscopic physics. Forces transform energy from one form to another when they act on matter. The study of energy thus inevitably leads to the question: What are the fundamental forces of nature, and what are the elementary particles on which these forces act? Gravity and electromagnetism are the most familiar two of the four forces that store and transform energy in our world. The other two, manifested by the strong interactions that hold nuclei together and the more subtle weak interactions, are usually the subject of more advanced physics courses.
All four forces contribute to the production and concentration of the forms of energy that we use on Earth. Gravity pulls together hydrogen atoms deep in the Sun. Strong nuclear interactions between these hydrogen atoms, facilitated by crucial weak interactions, result in the release of energy that is radiated from the Sun as electromagnetic waves, some of which hit Earth. Nuclear fission power plants rely on the interplay of strong nuclear forces, electromagnetism, and the weak interactions. Nuclear fuel is found in regions of high concentration in Earth’s crust that result from a combination of gravitational pressure and chemical electromagnetic interactions. Geothermal energy, as well, arises from the combined action of the four forces. Indeed, with the exception of tidal energy, which is essentially the result of gravity andelectromagnetic forces, all energy sources used by humanity rely on a happy conjunction among all four forces of nature.
The most familiar energy processes that we encounter on a day-to-day basis can be described in terms of electromagnetic andgravitational forces and interactions. The same is true of all the energy systems treated in this book so far. Much of the energy weuse, however, originates in the strong interactions, which are not encountered in everyday experience (unless, like Homer Simpson, you work at a nuclear reactor). The drama of the strong interactions involves some relatively familiar players such as the nuclei of atoms, but it also includes a menagerie of exotic particles such as quarks and gluons that, while they are just as real as electrons and atoms, are never seen outside of the atomic nucleus. In order to explain the origins of solar, nuclear, and geothermal energy, we must first describe some of the basic properties of the strong force. Along the way, we have the opportunity to put gravity and the electromagnetic force into a somewhat broader context.
Reader’s Guide
This chapter forms an “interlude” in which we survey the fundamental particles and forces that determine the structure of matterand its interactions with energy. Because they are less well known than gravity and electromagnetism, special attention is paid to the strong force that holds the nucleus together and to the weak force responsible for many nuclear decays. After discussing the four forces, we describe the properties of all the known elementary particles. We then present a simplified version of particle physics that is adequate for most of our purposes, in which only five particles, the proton, neutron, electron, photon, and antineutrino appear. The chapter closes with an introduction to β-decay and related processes that are important for understanding nuclear energy and related radiation.
Prerequisites: §2 (Mechanics), §3 (Electromagnetism), and §7 (Quantum mechanics), although much of this chapter is accessible with little technical background.
The strong and weak forces are important ingredients in the description of nuclei and nuclear energy in §17–§19, and ionizing radiation in §20. β-decay is key to understanding many nuclear decays. Gravity figures centrally in the large-scale structure of the universe (§21).
The fourth force, which is also only relevant in the microworld, is known as the weak interaction. Weak interactions play only a minor part in the actual generation of energy, but they act as a facilitator for many of the most important nuclear energy processes. Weak interactions play a role in fundamental physics a bit like fungi do in the environment. They are often overlooked; they do not contribute to building up great structures (such as nuclei and atoms); and, above all, they are the mediators of decay! Nevertheless we need to understand some of the features of the weak force because so many processes that figure significantly in energy physics would not occur without it.
Fortunately, the list of forces that affect our world stops at four: strong, electromagnetic, weak, and gravitational. There may be other forces locked away at very short distances, or that involve as-yet-undiscovered particles. Even if additional forces exist, however, we already know that they do not influence the dynamics of our familiar world in any significant way at the human scale (see Figure 14.1). We can be sure of this because we can account for essentially all the phenomena so far observed in nature in terms of the four known forces.1 This may sound like an exaggeration: after all there are many common phenomena not yet fully understood by science, from the turbulent flow of liquids to the origins of life. These are generally, however, phenomena characterized by complexity, and there is no evidence that they arise from additional, as yet unknown, fundamental interactions.
Figure 14.1 An illustration of the relative importance of the four forces over distances ranging from subnuclear ( m) to cosmological ( m).
If one focuses in on short distances and small systems, processes simplify, and everything that we can observe seems to be explained by a relatively simple set of fundamental equations. Our understanding is summarized in a meticulously tested and so far perfectly successful set of laws of physics, known as the Standard Model of particle physics. The word “model” connotes less certainty than “theory” or “law” – the terms typically used to designate well-established principles of physics. The Standard Model, however, which is formulated in terms of a theoretical framework known as relativistic quantum field theory, has described observed physical phenomena to a higher degree of precision than any other physical theory ever developed. Quantum field theory and the Standard Model are on a similar footing to Newton’s and Coulomb’s laws; they provide a set of simple and basic equations from which the behavior of matter can be deduced by mathematical analysis. The specific structure of the Standard Model dictates that matter is made of essentially pointlike particles with spin (quantum spin was discussed in §7). These particles interact with each other by emitting and absorbing photons and other massless and massive quanta that are cousins to the photon. The elementary spin-1/2 particles form families whose membership is limited by symmetry constraints and whose members have common properties. The details are beyond the scope of this book, but some aspects of the Standard Model are summarized in Tables 14.1 and 14.2 and in Figure 14.4.
The Standard Model of Particle Physics
The Standard Model of particle physics is a quantum field theory that, in concert with the classical theory of gravity, describes almost all observed phenomena to a high degree of accuracy in terms of four forces and 17 elementary particles.
Despite its successes, the Standard Model is neither a perfect nor a complete theory of nature. One glaring inadequacy is that it has about 20 parameters – numbers like the mass of the electron and the strength of the electric charge – whose origins we do not know. It also has a peculiar redundancy: each type of particle – the electron for example – comes in three versions that differ in mass, but little else. This redundancy, as well as the raison d'etre for the particular assortment of particles and fields that appear in nature, and other relationships in the Standard Model, are beyond our present understanding. There are some detailed aspects of the Standard Model, related to neutrinos and the Higgs particle, which are not fully settled experimentally. And, as mentioned above, it is possible that new forces or other novel phenomena are relevant at very high energies and short distance scales that have not yet been probed experimentally. Finally, and arguably at a more fundamental level, in our current understanding gravity is appended to the Standard Model only as a classical (rather than quantum) field, even though gravity also must exhibit quantum features at very short distance and length scales. Unifying gravity and quantum mechanics has challenged physicists for decades. An approach often called string theory provides a framework in which gravity and other forces can be described within a quantum-mechanical theory. Whether and how the Standard Model and the global structure of the universe in which we live is described within the framework of string theory is, however, still an open problem (§21).
All the limitations of the Standard Model that we have just listed are irrelevant, in any case, for the purposes of this book. All known energy sources can be understood with only a subset of the structure in the Standard Model, and it is hard to imagine any way in which physics beyond the Standard Model could have practical consequences for human energy use in the forseeable future. Thus, for all intents and purposes, the fundamental physics underlying any conceivable energy system that may be relevant in this century should be described within the Standard Model.
The aim of this chapter is to take a brief tour of the Standard Model, emphasizing those bits that are most important for the physics of energy and skirting as much as possible the complexities that usually relegate the subject to graduate-level physics courses. Although the concepts discussed here are relatively remote from everyday experience, they are nevertheless relevant to practical energy systems. In fact, the basic ideas presented here are essential for understanding the origins of nuclear and solar energy and the nature of radioactivity. In §14.1 we describe the four fundamental forces in more depth. Starting at the largest distance scales, where gravity is most important, we work our way down to sub-atomic distances, encountering electromagnetic, strong, and weak interactions in succession. Next (§14.2) we meet the players in the drama of subatomic physics, the elementary particles. Before exploring the whole zoo of particles, we introduce a stripped-down version of the Standard Model that includes almost everything needed to understand energy production and flow in the observable universe at the present time. This Standard Model “lite” comprises the proton, neutron, electron, photon, and one potentially less familiar addition, the (anti-)neutrino. We briefly survey the other particles in the Standard Model and then conclude the chapter (§14.3) with a close look at β-decay, a weak interaction of particular importance in energy physics.
Non-technical introductions to particle physics can be found in many books, for example [66]. Reference [67] provides a more technical introduction, at an advanced undergraduate level.
14.1Forces, Energies, and Distance Scales
To survey the hierarchy of strengths, distances, and energy scales that characterize the four forces, we start at astronomical distances and work our way into the interior of the atomic nucleus.
Gravitational and electromagnetic potential energies fall off like away from their sources. This is a special case of a more general form
(14.1)
where b is known as the range of the force and g is a measure of its strength. Equation (14.1) defines what is known as a Yukawa interaction, named after the Japanese physicist Hideki Yukawa who proposed it as a model for the strong nuclear force in the 1930s. The potential of gravity and electromagnetism emerges when the parameter b goes to . Hence, these are called infinite range forces. In the framework of quantum field theory, interactions between massive particles are the result of the exchange of particles called force carriers (Figure 14.2). For example, the photon is the force carrier for electromagnetism. In general, the exchange of a force carrier of mass m gives rise to a force of the form (14.1), where the parameter b is proportional to . Thus, the long-range nature of the electromagnetic force arises from the fact that the photon is massless. We now go through each of the forces in turn and discuss various aspects of each.
Figure 14.2 In quantum field theory, interactions between massive particles are mediated by the exchange of massless or massive force carriers. The figure depicts a Feynman diagram describing two electrons interacting through exchange of a photon, the force carrier for electromagnetic interactions.
14.1.1 Gravitational Interactions
Only gravitational forces are important at astronomical distances. Gravity alone persists at the scale of stars and galaxies not because gravity is strong – in fact gravity is by far the weakest of the four forces (see Table 14.1). The reason that gravity dominates over electromagnetism on astronomical scales is that gravity is universally attractive and its strength is proportional to the mass (or, more generally, the total energy) of an object. In contrast, electromagnetic forces depend on the signs of the electric charges; electromagnetic attraction and repulsion between charged particles are so strong that unless we work very hard to prevent it, matter always relaxes to a state in which it is electrically neutral. So matter at macroscopic scales is essentially electromagnetically uncharged, and the force of gravity can dominate at very large scales. Thus, Newton’s law of universal gravitation governs the motions of planets and stars. For very dense and very massive objects, Newton’s theory of gravity must be corrected or even replaced by Einstein’s General Theory of Relativity, which is described in §21.2
As soon as one leaves the astronomical domain, gravity subsides in importance and electromagnetic forces become equally or more important. Of course, gravity holds us here on Earth’s surface – again because a very large mass is involved, namely the mass of Earth. To support the claim that gravity is unimportant on human distance scales, consider the strength of the gravitational force between, for example, two one-gram masses, separated by one meter. According to Newton, the gravitational force between them is
(14.2)
which is far too weak to observe.
14.1.2 Electromagnetic Interactions
To gauge the strength of electromagnetic forces compared to gravity, we compute the electrostatic (Coulomb’s law) force between the same two one-gram lumps of matter described above if the electric charges were not neutralized. That is, we compute the electrostatic force between the positively charged protons in the two lumps. Since about half of the mass of one gram of material comes from protons (the other half comes from neutrons; the electrons are only about 1/2000th of the mass), one gram contains about protons, each with charge C, for a total charge of C. This generates a force of
(14.3)
Between comparable quantities of stuff, therefore, the electromagnetic force exceeds the gravitational force by ~36 orders of magnitude. Of course, the repulsive Coulomb forces among the protons are so strong that there is no way we could isolate and study a gram of material with charge 4.8 C. Electromagnetic forces subside to manageable scales because – to very high accuracy – matter is neutral.
Many of the large-scale processes on Earth’s surface result from the dynamic balance between electromagnetic and gravitational forces. For example, the electromagnetic repulsion between electrons in different atoms keeps matter from collapsing under huge gravitational pressures in Earth’s core, and – at a more human scale – keeps you from falling through your chair. Gravitational and electromagnetic forces maintain the dynamic balance that governs the circulation of the atmosphere and oceans; the Sun’s energy is absorbed and transmitted through electromagnetic forces while the overall vertical order is maintained by the pressure sustained by gravity.
13.14 [T] Calculate and plot the ratio of the efficiency of the Brayton cycle to the efficiency of the Otto cycle with the same compression ratio (with ).
13.15 [T] Verify the assertion made in §13.5.2 that the net efficiency of a CCGT is , where is the efficiency of the Brayton/Rankine cycle.
* * *
1 We have already made this type of idealization, e.g. for the combustion process, in our study of internal combustion engines in §11.
2 Although it does not pose a risk to the ozone layer, R-134a is a greenhouse gas that makes a significant contribution to climate change. In October 2016 over 170 countries reached an agreement to phase out the use of R-134a over the next several decades.
3 Although we have not plotted lines of constant enthalpy in the TS-plane, must lie below and to the right of because after throttling it has higher entropy and lower temperature.
4 Pumps are, of course, also used to move liquids up in a gravitational field.
5 A “ton of AC” is an archaic measure used in the US corresponding to the amount of heat required to melt a ton of ice in a day, 1 ton of AC kW.
6 For the ideal Rankine cycle, the temperature denotes the high temperature reached at point ; this is above the temperature at which the working fluid changes phase, unlike for the ideal VC cycle, where denoted the temperature of phase change. This notation matches the physical elements of the system; for the ideal VC cycle matches the operating temperature of the condenser, while corresponds to the nominal temperature of the boiler in the Rankine cycle. Note that only a fraction of the heat is added while the fluid is above the vaporization temperature .
7 Here, and throughout this chapter, we take the energy content of fuels to be the higher heating value (HHV) as defined in §9.4.3 and as is customary in the US. Values for the HHV for coal range from 15 GJ/t to 35 GJ/t (§33) depending on the type of coal. For definiteness, we use here a middle value of 25 GJ/t.
8 Although George Brayton’s original device, patented in 1872, employed reciprocating pistons.
9 We quantify this inefficiency in §36 (Systems).
Part IIEnergy Sources
CHAPTER 14
The Forces of Nature
Elementary particles and the forces that act between them are the fundamental ingredients in our most complete theoretical description of microscopic physics. Forces transform energy from one form to another when they act on matter. The study of energy thus inevitably leads to the question: What are the fundamental forces of nature, and what are the elementary particles on which these forces act? Gravity and electromagnetism are the most familiar two of the four forces that store and transform energy in our world. The other two, manifested by the strong interactions that hold nuclei together and the more subtle weak interactions, are usually the subject of more advanced physics courses.
All four forces contribute to the production and concentration of the forms of energy that we use on Earth. Gravity pulls together hydrogen atoms deep in the Sun. Strong nuclear interactions between these hydrogen atoms, facilitated by crucial weak interactions, result in the release of energy that is radiated from the Sun as electromagnetic waves, some of which hit Earth. Nuclear fission power plants rely on the interplay of strong nuclear forces, electromagnetism, and the weak interactions. Nuclear fuel is found in regions of high concentration in Earth’s crust that result from a combination of gravitational pressure and chemical electromagnetic interactions. Geothermal energy, as well, arises from the combined action of the four forces. Indeed, with the exception of tidal energy, which is essentially the result of gravity andelectromagnetic forces, all energy sources used by humanity rely on a happy conjunction among all four forces of nature.
The most familiar energy processes that we encounter on a day-to-day basis can be described in terms of electromagnetic andgravitational forces and interactions. The same is true of all the energy systems treated in this book so far. Much of the energy weuse, however, originates in the strong interactions, which are not encountered in everyday experience (unless, like Homer Simpson, you work at a nuclear reactor). The drama of the strong interactions involves some relatively familiar players such as the nuclei of atoms, but it also includes a menagerie of exotic particles such as quarks and gluons that, while they are just as real as electrons and atoms, are never seen outside of the atomic nucleus. In order to explain the origins of solar, nuclear, and geothermal energy, we must first describe some of the basic properties of the strong force. Along the way, we have the opportunity to put gravity and the electromagnetic force into a somewhat broader context.
Reader’s Guide
This chapter forms an “interlude” in which we survey the fundamental particles and forces that determine the structure of matterand its interactions with energy. Because they are less well known than gravity and electromagnetism, special attention is paid to the strong force that holds the nucleus together and to the weak force responsible for many nuclear decays. After discussing the four forces, we describe the properties of all the known elementary particles. We then present a simplified version of particle physics that is adequate for most of our purposes, in which only five particles, the proton, neutron, electron, photon, and antineutrino appear. The chapter closes with an introduction to β-decay and related processes that are important for understanding nuclear energy and related radiation.
Prerequisites: §2 (Mechanics), §3 (Electromagnetism), and §7 (Quantum mechanics), although much of this chapter is accessible with little technical background.
The strong and weak forces are important ingredients in the description of nuclei and nuclear energy in §17–§19, and ionizing radiation in §20. β-decay is key to understanding many nuclear decays. Gravity figures centrally in the large-scale structure of the universe (§21).
The fourth force, which is also only relevant in the microworld, is known as the weak interaction. Weak interactions play only a minor part in the actual generation of energy, but they act as a facilitator for many of the most important nuclear energy processes. Weak interactions play a role in fundamental physics a bit like fungi do in the environment. They are often overlooked; they do not contribute to building up great structures (such as nuclei and atoms); and, above all, they are the mediators of decay! Nevertheless we need to understand some of the features of the weak force because so many processes that figure significantly in energy physics would not occur without it.
Fortunately, the list of forces that affect our world stops at four: strong, electromagnetic, weak, and gravitational. There may be other forces locked away at very short distances, or that involve as-yet-undiscovered particles. Even if additional forces exist, however, we already know that they do not influence the dynamics of our familiar world in any significant way at the human scale (see Figure 14.1). We can be sure of this because we can account for essentially all the phenomena so far observed in nature in terms of the four known forces.1 This may sound like an exaggeration: after all there are many common phenomena not yet fully understood by science, from the turbulent flow of liquids to the origins of life. These are generally, however, phenomena characterized by complexity, and there is no evidence that they arise from additional, as yet unknown, fundamental interactions.
Figure 14.1 An illustration of the relative importance of the four forces over distances ranging from subnuclear ( m) to cosmological ( m).
If one focuses in on short distances and small systems, processes simplify, and everything that we can observe seems to be explained by a relatively simple set of fundamental equations. Our understanding is summarized in a meticulously tested and so far perfectly successful set of laws of physics, known as the Standard Model of particle physics. The word “model” connotes less certainty than “theory” or “law” – the terms typically used to designate well-established principles of physics. The Standard Model, however, which is formulated in terms of a theoretical framework known as relativistic quantum field theory, has described observed physical phenomena to a higher degree of precision than any other physical theory ever developed. Quantum field theory and the Standard Model are on a similar footing to Newton’s and Coulomb’s laws; they provide a set of simple and basic equations from which the behavior of matter can be deduced by mathematical analysis. The specific structure of the Standard Model dictates that matter is made of essentially pointlike particles with spin (quantum spin was discussed in §7). These particles interact with each other by emitting and absorbing photons and other massless and massive quanta that are cousins to the photon. The elementary spin-1/2 particles form families whose membership is limited by symmetry constraints and whose members have common properties. The details are beyond the scope of this book, but some aspects of the Standard Model are summarized in Tables 14.1 and 14.2 and in Figure 14.4.
The Standard Model of Particle Physics
The Standard Model of particle physics is a quantum field theory that, in concert with the classical theory of gravity, describes almost all observed phenomena to a high degree of accuracy in terms of four forces and 17 elementary particles.
Despite its successes, the Standard Model is neither a perfect nor a complete theory of nature. One glaring inadequacy is that it has about 20 parameters – numbers like the mass of the electron and the strength of the electric charge – whose origins we do not know. It also has a peculiar redundancy: each type of particle – the electron for example – comes in three versions that differ in mass, but little else. This redundancy, as well as the raison d'etre for the particular assortment of particles and fields that appear in nature, and other relationships in the Standard Model, are beyond our present understanding. There are some detailed aspects of the Standard Model, related to neutrinos and the Higgs particle, which are not fully settled experimentally. And, as mentioned above, it is possible that new forces or other novel phenomena are relevant at very high energies and short distance scales that have not yet been probed experimentally. Finally, and arguably at a more fundamental level, in our current understanding gravity is appended to the Standard Model only as a classical (rather than quantum) field, even though gravity also must exhibit quantum features at very short distance and length scales. Unifying gravity and quantum mechanics has challenged physicists for decades. An approach often called string theory provides a framework in which gravity and other forces can be described within a quantum-mechanical theory. Whether and how the Standard Model and the global structure of the universe in which we live is described within the framework of string theory is, however, still an open problem (§21).
All the limitations of the Standard Model that we have just listed are irrelevant, in any case, for the purposes of this book. All known energy sources can be understood with only a subset of the structure in the Standard Model, and it is hard to imagine any way in which physics beyond the Standard Model could have practical consequences for human energy use in the forseeable future. Thus, for all intents and purposes, the fundamental physics underlying any conceivable energy system that may be relevant in this century should be described within the Standard Model.
The aim of this chapter is to take a brief tour of the Standard Model, emphasizing those bits that are most important for the physics of energy and skirting as much as possible the complexities that usually relegate the subject to graduate-level physics courses. Although the concepts discussed here are relatively remote from everyday experience, they are nevertheless relevant to practical energy systems. In fact, the basic ideas presented here are essential for understanding the origins of nuclear and solar energy and the nature of radioactivity. In §14.1 we describe the four fundamental forces in more depth. Starting at the largest distance scales, where gravity is most important, we work our way down to sub-atomic distances, encountering electromagnetic, strong, and weak interactions in succession. Next (§14.2) we meet the players in the drama of subatomic physics, the elementary particles. Before exploring the whole zoo of particles, we introduce a stripped-down version of the Standard Model that includes almost everything needed to understand energy production and flow in the observable universe at the present time. This Standard Model “lite” comprises the proton, neutron, electron, photon, and one potentially less familiar addition, the (anti-)neutrino. We briefly survey the other particles in the Standard Model and then conclude the chapter (§14.3) with a close look at β-decay, a weak interaction of particular importance in energy physics.
Non-technical introductions to particle physics can be found in many books, for example [66]. Reference [67] provides a more technical introduction, at an advanced undergraduate level.
14.1Forces, Energies, and Distance Scales
To survey the hierarchy of strengths, distances, and energy scales that characterize the four forces, we start at astronomical distances and work our way into the interior of the atomic nucleus.
Gravitational and electromagnetic potential energies fall off like away from their sources. This is a special case of a more general form
(14.1)
where b is known as the range of the force and g is a measure of its strength. Equation (14.1) defines what is known as a Yukawa interaction, named after the Japanese physicist Hideki Yukawa who proposed it as a model for the strong nuclear force in the 1930s. The potential of gravity and electromagnetism emerges when the parameter b goes to . Hence, these are called infinite range forces. In the framework of quantum field theory, interactions between massive particles are the result of the exchange of particles called force carriers (Figure 14.2). For example, the photon is the force carrier for electromagnetism. In general, the exchange of a force carrier of mass m gives rise to a force of the form (14.1), where the parameter b is proportional to . Thus, the long-range nature of the electromagnetic force arises from the fact that the photon is massless. We now go through each of the forces in turn and discuss various aspects of each.
Figure 14.2 In quantum field theory, interactions between massive particles are mediated by the exchange of massless or massive force carriers. The figure depicts a Feynman diagram describing two electrons interacting through exchange of a photon, the force carrier for electromagnetic interactions.
14.1.1 Gravitational Interactions
Only gravitational forces are important at astronomical distances. Gravity alone persists at the scale of stars and galaxies not because gravity is strong – in fact gravity is by far the weakest of the four forces (see Table 14.1). The reason that gravity dominates over electromagnetism on astronomical scales is that gravity is universally attractive and its strength is proportional to the mass (or, more generally, the total energy) of an object. In contrast, electromagnetic forces depend on the signs of the electric charges; electromagnetic attraction and repulsion between charged particles are so strong that unless we work very hard to prevent it, matter always relaxes to a state in which it is electrically neutral. So matter at macroscopic scales is essentially electromagnetically uncharged, and the force of gravity can dominate at very large scales. Thus, Newton’s law of universal gravitation governs the motions of planets and stars. For very dense and very massive objects, Newton’s theory of gravity must be corrected or even replaced by Einstein’s General Theory of Relativity, which is described in §21.2
As soon as one leaves the astronomical domain, gravity subsides in importance and electromagnetic forces become equally or more important. Of course, gravity holds us here on Earth’s surface – again because a very large mass is involved, namely the mass of Earth. To support the claim that gravity is unimportant on human distance scales, consider the strength of the gravitational force between, for example, two one-gram masses, separated by one meter. According to Newton, the gravitational force between them is
(14.2)
which is far too weak to observe.
14.1.2 Electromagnetic Interactions
To gauge the strength of electromagnetic forces compared to gravity, we compute the electrostatic (Coulomb’s law) force between the same two one-gram lumps of matter described above if the electric charges were not neutralized. That is, we compute the electrostatic force between the positively charged protons in the two lumps. Since about half of the mass of one gram of material comes from protons (the other half comes from neutrons; the electrons are only about 1/2000th of the mass), one gram contains about protons, each with charge C, for a total charge of C. This generates a force of
(14.3)
Between comparable quantities of stuff, therefore, the electromagnetic force exceeds the gravitational force by ~36 orders of magnitude. Of course, the repulsive Coulomb forces among the protons are so strong that there is no way we could isolate and study a gram of material with charge 4.8 C. Electromagnetic forces subside to manageable scales because – to very high accuracy – matter is neutral.
Many of the large-scale processes on Earth’s surface result from the dynamic balance between electromagnetic and gravitational forces. For example, the electromagnetic repulsion between electrons in different atoms keeps matter from collapsing under huge gravitational pressures in Earth’s core, and – at a more human scale – keeps you from falling through your chair. Gravitational and electromagnetic forces maintain the dynamic balance that governs the circulation of the atmosphere and oceans; the Sun’s energy is absorbed and transmitted through electromagnetic forces while the overall vertical order is maintained by the pressure sustained by gravity.
