The Physics of Energy, page 16
While nonlinearities play some role in many wave systems, two important exceptions are electromagnetic waves and quantum mechanical systems, which are governed by linear wave equations for all purposes relevant for this book.
Discussion/Investigation Questions
4.1 Explain why the strings that play lower frequency notes on a guitar or violin are generally thicker and less tightly strung than the strings playing higher notes.
4.2 Suppose an electromagnetic plane wave propagates in a direction perpendicular to a long straight wire. Can you see why a current develops in the wire with the same frequency as the wave? Qualitatively, how does the current depend on the orientation of the wire and the polarization of the light? This is one of the basic ingredients in both transmission and reception of broadcast radio, television, etc.
4.3 Polarized sunglasses are designed to transmit only one polarization of light – the polarization with electric fields vertical – since glare arising from reflections on horizontal snow and water surfaces is preferentially polarized with horizontal electric fields. LCD displays such as the view screens on many digital cameras (and laptops, etc.) also emit polarized light. Examine the image on a digital camera screen as you rotate it through 90° while wearing polarizing sunglasses. Explain what you observe.
Problems
4.1 Sound waves travel in air at roughly 340 m/s. The human ear can hear frequencies ranging from 20 Hz to 20 000 Hz. Determine the wavelengths of the corresponding sine wave modes and compare to human-scale physical systems.
4.2 A violin A-string of length m with total mass 0.23 g has a fundamental frequency (for the lowest mode) of 440 Hz. Compute the tension on the string. If the string vibrates at the fundamental frequency with maximum amplitude 2 mm, what is the energy of the vibrational motion?
4.3 [T] Derive the equation of motion for the string (4.9) from a microscopic model. Assume a simple model of a string as a set of masses spaced evenly on the x axis at regular intervals of , connected by springs of spring constant . Compute the leading term in the force on each mass and take the limit to get the string wave equation, where . In the same limit show that the energy density of the string is given by eq. (4.10).
4.4 [T] Show that the energy density on a string , defined in eq. (4.10), obeys the conservation law , where is the energy flux, the energy per unit time passing a point x. For the traveling wave , find and and show that energy flows to the right (for ) as the wave passes a point x. Show that the total energy passing each point is equal to the total energy in the wave.
4.5 Compute the maximum energy flux possible for electromagnetic waves in air given the constraint that the electric field cannot exceed the breakdown field described in Example 3.2.
4.6 The strongest radio stations in the US broadcast at a power of 50 kW. Assuming that the power is broadcast uniformly over the hemisphere above Earth’s surface, compute the strength of the electric field in these radio waves at a distance of 100 km.
4.7 [T] Derive the wave equation for B analogous to eq. (4.20).
4.8 Suppose an electromagnetic plane wave is absorbed on a surface oriented perpendicular to the direction of propagation of the wave. Show that the pressure exerted by the radiation on the surface is , where W is the power absorbed per unit area. Solar radiation at the top of Earth’s atmosphere has an energy flux W/m. What is the pressure of solar radiation when the Sun is overhead? What is the total force on Earth exerted by solar radiation?
4.9 It has been proposed that solar collectors could be deployed in space, and that the collected power could be beamed to Earth using microwaves. A potential limiting factor for this technology would be the possible hazard of human exposure to the microwave beam. One proposal involves a circular receiving array of diameter 10 km for a transmitted power of 750 MW. Compute the energy flux in this scenario and compare to the energy flux of solar radiation.
4.10 [T,H] Consider two electromagnetic plane waves (see eq. (4.21)) one with amplitude and wave vector and the other with amplitude and wave vector . These waves are said to add coherently if the average energy density u in the resulting wave is proportional to or incoherently if the average energy density is proportional to . Show that two electromagnetic plane waves are coherent only if they are propagating in the same direction with the same frequency and the same polarization.
4.11 [T] Derive eq. (4.23) by taking the time derivative of eq. (4.22) and using Maxwell’s equations. [Hint: see eq. (B.23).]
4.12 [T] A string of length L begins in the configuration with no initial velocity. Write the exact time-dependent solution of the string . Compute the contribution to the energy from each mode involved.
4.13 [TH] A string of length L is initially stretched into a “zigzag” profile, with linear segments of string connecting the points . Compute the Fourier series coefficients and the time-evolution of the string . Compute the total energy in the tension of the initially stretched string. Compute the energy in each mode and show that the total energy agrees with the energy of the stretched string.
4.14 [T] What is the pressure exerted by a beam of light on a perfect mirror from which it reflects at normal (perpendicular) incidence? Generalize this to light incident at an angle θ to the normal on an imperfect mirror (which reflects a fraction of the light incident at angle θ).
4.15 [T] Consider a cylindrical resistor of cross-sectional area A and length L. Assume that the electric field E and current density j are uniform within the resistor. Prove that the integrated power transferred from electromagnetic fields into the resistor is equal to IV. Compute the electric and magnetic fields on the surface of the resistor, and show that the power transfer is also given by the surface integral of the Poynting vector, so that all energy dissipated in the resistor is transferred in through electric and magnetic fields.
4.16 As stated in the text, the dispersion relation relating the wave number and angular frequency of ocean surface waves is , where m/s. Compute the wavelength and speed of propagation (phase velocity) for ocean surface waves with periods 6 s and 12 s.
4.17 [T] A wave satisfying eq. (4.2) passes from one medium in which the phase velocity for all wavelengths is to another medium in which the phase velocity is . The incident wave gives rise to a reflected wave that returns to the original medium and a refracted wave that changes direction as it passes through the interface. Suppose that the interface is the plane and the incoming wave is propagating in a direction at an angle to the normal . Prove the law of specular reflection, which states that the reflected wave propagates at an angle with respect to the normal . Also prove Snell’s law, which states that the wave in the second medium propagates at an angle from the normal , where . Use the fact that the wave must be continuous across the interface at .
4.18 A wave travels to the right on a string with constant tension τ and a mass density that slowly increases from ρ on the far left to on the far right. The mass density changes slowly enough that its only effect is to change the speed with which the wave propagates. The waveform on the far left is and on the far right is . Find the relation between and k and then use conservation of energy to find the amplitude on the far right. You may find it helpful to use the result of Problem 4.4 ().
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1 Note that the decomposition of into right- and left-moving parts requires knowledge of both the wave form and the time derivative since the wave equation is a second-order differential equation.
CHAPTER 5
Thermodynamics I: Heat and Thermal Energy
Thermal energy has played a central role in energy systems through all of human history. Aside from human and animal power, and limited use of wind and hydropower, most energy put to human use before 1800 AD came in the form of heat from wood and other biomass fuels that were burned for cooking and warmth in pre-industrial societies (and still are in many situations). The development of the steam engine in the late eighteenth century enabled the transformation of thermal energy into mechanical energy, vastly increasing the utility of thermal energy. Humankind has developed ever-increasing reliance on the combustion of coal and other fossil fuels as sources of thermal energy that can be used to power mechanical devices and to generate electricity. Today, over 90% of the world's energy relies either directly or in an intermediate stage on thermal energy, the major exception being hydropower.
Thermal energy, and its conversion into mechanical and electrical energy, is an important theme in this book. The scientific study of the transformation of heat into mechanical energy and vice versa, which began soon after the discovery of the steam engine, led to the discovery of energy conservation and many fundamental aspects of energy physics. Thermal energy belongs to a rich and surprisingly subtle subject – thermodynamics – that we develop systematically beginning in this chapter and continuing in §8.
In contrast to mechanical kinetic and potential energy, thermal energy involves disordered systems, and a new concept, entropy, is needed to provide a quantitative measure of disorder. The definition of entropy (as well as the precise meaning of temperature) can be best appreciated with the help of some basic knowledge of quantum physics. We introduce quantum mechanics in §7. In §8 we define and explore the concepts of entropy and free energy, leading to the fundamental result that there is a physical limit on the efficiency with which thermal energy can be converted to mechanical energy. We apply these ideas to
Reader’s Guide
In this chapter, we take a first look at thermal energy and the associated concepts of heat and temperature. We assume minimal scientific background in this area beyond everyday experience. This chapter initiates a systematic treatment of these subjects, and introduces many related ideas including internal energy, thermodynamic equilibrium, thermal equilibrium, state functions, enthalpy, heat capacity, phase transitions, and the first law of thermodynamics. These concepts play a central role in engines, power plants, and energy storage. As always, we follow the role of energy and energy applications as we navigate a passage through a large and varied subject.
Prerequisites: §2 (Mechanics).
The material in this chapter forms a basis for later chapters on various aspects of thermodynamics, particularly §6 (Heattransfer), §8 (Entropy and temperature), §10 (Heat engines), and §12 (Phase-change energy conversion).
chemical reactions in §9 , to engines in §10 and §11, and to power generation in §12 and §13 . As illustrated in Figure 1.2, morethan half of the thermal energy released from fossil fuel combustion and nuclear reactors is lost in the current US energy stream. A clear understanding of thermodynamics and entropy is needed to distinguish between the fraction of this energy loss that is unavoidable and the fraction that is due to the less-than-optimal performance of existing power systems.
Figure 5.1 At the Nesjavellir geothermal power station in Iceland, thermal energy originating deep within Earth's crust is used to produce 120 MW of electric power and also to heat homes and businesses in the capital, Reykjavík. (Image: G. Ívarsson)
While thermal energy is often harnessed as a means of powering mechanical devices or generating electricity, it is also an end in itself. Indeed, more than a third of US energy use currently goes to heating (and cooling) buildings, water, food, and other material goods. After developing some basic notions of heat, heat content, and heat capacity in this chapter, in §6 we develop tools for studying heat transfer in materials and analyzing thermal energy requirements in situations such as the heating of a building in a cold climate.
We begin in §5.1 with a general introduction to some of the central concepts underlying thermodynamics. Precise definitions of some quantities are left to later chapters. Ideal gases are introduced as a simple paradigm for thermodynamic systems. In §5.2, we describe the transformation of thermal energy to mechanical energy. This leads to the formulation of the first law of thermodynamics in §5.3. The last part of the chapter describes the addition of heat to materials through heat capacity (§5.4), enthalpy (§5.5), and phase transitions (§5.6). For a more detailed introduction to thermal energy, heat, and other concepts introduced in this chapter, see [19] or [20]. A more sophisticated treatment of thermal physics is given in [21], incorporating entropy early in the discussion.
5.1What is Heat?
5.1.1 Thermal Energy and Internal Energy
Heat is surely one of the most basic concepts in human experience. From the time our early ancestors learned to control fire, humankind has experimented with heat, and tried to understand the precise nature of the distinction between that which is hot and that which is cold. Now that we understand materials at a microscopic level, we know that the notions of heat and temperature refer to processes and properties that are most clearly described in terms of energy.
Thermal energy refers to the collective energy contained in the relative motions of the large number of microscopic particles comprising a macroscopic whole. A gas of N identical molecules, where N is a very large number, confined to a volume V (see Figure 5.2) provides a simple example of a system with thermal energy. The molecules move about randomly, colliding with one another and with the walls of the container. The kinetic energy associated with the motion of these molecules relative to the fixed walls of the container is thermal energy. Unless the gas is monatomic, like helium or argon, the molecules will also be rotating and vibrating, and these random motions also contribute to the thermal energy. In a solid the atoms are locked into place in a fixed structure, such as a crystal. Though they are not free to move about or to rotate, they can still vibrate. Their kinetic and potential energies of vibration constitute the thermal energy of the solid.
Figure 5.2 Thermal energy carried by the motion of N molecules in a volume V. For a typical macroscopic system, .
Qualitatively, temperature is a relative measure of the amount of thermal energy in a system (or part of a system). Increasing the temperature of a system – making it hotter – generally requires adding thermal energy. Heat refers to a transfer of thermal energy from one system to another. To proceed quantitatively, it is useful to define the notion of thermal energy, and the closely related concept of total internal energy, more precisely.
Internal energy The internal energy U of any physical system is the sum total of all contributions to the energy of the system considered as an isolated whole. Note that U does not include kinetic or potential energy associated with the motion of the system as a whole relative to external objects such as Earth. So, for example, the collective kinetic energy of wind or a mass of water due to its motion at a fixed velocity relative to Earth's surface is not included in its internal energy. The internal energy U does, however, include contributions from chemical and/or nuclear binding and from rest mass energies of the constituent particles.
Internal and Thermal Energy
The internal energy U of any physical system is the sum total of all contributions to the energy of the system considered as an isolated whole. It does not include the kinetic energy of bulk motion or the potential energy of the center of mass.
Thermal energy is that contribution to the internal energy of a system beyond the energy that the system would have if cooled to a temperature of absolute zero.
Thermal energy Thermal energy, on the other hand, is that contribution to the internal energy of a system beyond the energy that the system would have if cooled to a temperature of absolute zero. Absolute zero is defined to be a state from which no further energy can be removed without changing the nature of the constituents or their chemical or nuclear binding. Thus, thermal energy does not include particle rest masses, nuclear binding energy, or chemical bond energies, but does include kinetic and potential energies associated with the relative motion of the constituent molecules, their rotations and vibrations, as well as the energy needed to separate molecules into a liquid or gaseous state, if applicable.
Note that the internal energy and thermal energy of a system only differ by an additive constant, namely the internal energy of the system at absolute zero. Often we are only interested in energy differences, so we can without confusion use the same symbol U to denote both quantities. Indeed, in some elementary physics texts internal and thermal energy are equated. In this chapter we do not consider chemical or nuclear reactions, so U can unambiguously refer to thermal energy. As soon as chemical and/or nuclear bonds in the system can change, however, as in §9 or §16, the notion of thermal energy becomes ambiguous, and one should always use internal energy, not thermal energy, for a consistent description of the system.
Now that we have a definition of thermal energy, we can give a useful definition of heat. Heat is thermal energy that is transferred from one system (or part of a system) to another system (or part of a system). Note that modern definitions of heat always invoke a transfer of energy from one system to another, so that there is no meaning to the “heat content” of a single system. Our common experience is that heat always flows from regions at higher temperature to regions at lower temperature, so temperature can be thought of as the capacity of a system to generate heat.
