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Poynting's Theorem and the Wave Equation

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Poynting's Theorem and the Wave Equation Chapter 18: Poynting’s Theorem and the Wave Equation Chapter Learning Objectives: After completing this chapter the student will be able to: Use Poynting’s theorem to determine the direction and magnitude of power flow in an electromagnetic system. Use Maxwell’s Equations to derive a general homogeneous wave equation for the electric and magnetic field. Derive a simplified wave equation assuming propagation in a vacuum and an electric field polarized in only one direction. Use Maxwell’s Equations to derive the speed of light in a vacuum. You can watch the video associated with this chapter at the following link: Historical Perspective: John Henry Poynting (1852-1914) was an English physicist who did work in electromagnetic energy flow, elasticity, and astronomy. He coined the term “Greenhouse Effect.” Both the Poynting Vector and Poynting’s Theorem are named in his honor. Photo credit: https://upload.wikimedia.org/wikipedia/commons/5/5f/John_Henry_Poynting.jpg, [Public domain], via Wikimedia Commons. 1 18.1 Poynting’s Theorem With Maxwell’s Equations, we now have the tools necessary to derive Poynting’s Theorem, which will allow us to perform many useful calculations involving the direction of power flow in electromagnetic fields. We will begin with Faraday’s Law, and we will take the dot product of H with both sides: (Copy of Equation 16.24) (Equation 18.1) Next, we will start with Ampere’s Law and will take the dot product of E with both sides: (Copy of Equation 17.13) (Equation 18.2) Now, let’s subtract both sides of Equation 18.2 from both sides of Equation 18.1: (Equation 18.3) We can now apply the following mathematical identity to the left side of Equation 18.3: (Equation 18.4) This substitution yields: (Equation 18.5) Distributing the E across the right side gives: (Equation 18.6) 2 Now let’s concentrate on the first time on the right side. Applying a constitutive equation, we find: (Equation 18.7) Let’s consider for a moment what the derivative of H2 would yield: (Equation 18.8) Dividing this equation by 2, we find: (Equation 18.9) Because H∙H =H2. The dot product of a vector with itself is the magnitude of the vector squared. A similar analysis of the second term on the right side yields: (Equation 18.10) Substituting Equations 18.9 and 18.10 into Equation 18.6 and applying two constitutive relations, we obtain: (Equation 18.11) Taking a volume integral of both sides, we find: (Equation 18.12) Applying Ohm’s Law (J=E) to the right side, we find: (Equation 18.13) Applying the divergence theorem to the left side gives: (Equation 18.14) This equation is the integral form of Poynting’s Theorem. Each of the terms in this equation has a physical significance. Let’s start with the left side. The factor E x H is called the Poynting 3 vector, and it represents the direction in which energy is flowing. We represent it by the variable S. (Equation 18.15) The Poynting vector can be used to determine the net power flowing out of an enclosed space. S will show the direction of power flow, which can be difficult to determine from just E and H. Note that if the integral of S is positive, net power is leaving the region, while if the integral of S is negative, power is flowing into the space. The first term on the right side represents the total energy stored within the enclosed surface. We can simplify this expression by making the following definition: (Equation 18.16) The first term in this expression represents the energy stored in the magnetic fields (such as in an inductor), and the second term represents the energy stored in the electric fields (such as in a capacitor). In many cases, one of these terms will be zero, simplifying the calculation. The second term on the right represents the energy being dissipated within the enclosed surface. This is energy that is being converted from electromagnetic potential energy into (typically) heat energy such as in a resistor. We represent this quantity by the variable pL. The interaction among S, w, and pL is illustrated in Figure 18.1. Energy Flowing Energy Stored B E Figure 18.1. Graphical Representation of Poynting’s Theorem 4 Poynting’s Theorem can also be represented in point form: (Equation 18.17) Example 18.1: Use Poynting’s Theorem to analyze the power flowing in a capacitor connected to a voltage source, as shown below: a ~ V d E H 5 Example 18.2: Use Poynting’s Theorem to analyze the power flowing in an inductor connected to a current source, as shown below: I E E E H a d 18.2 Time-Harmonic Electromagnetic Fields As we will shortly see, electromagnetic fields often take a harmonic form, which simply means that they have a sinusoidal dependence on time. As you are no doubt familiar, electrical and computer engineers have a special affinity to waves with sinusoidal dependence in time, because we can use phasors to analyze them. Recall that the process for working with phasors is to replace the time dependence with ejt, replace any time derivatives with multiplication by j, solve the problem using complex numbers, and at the end we take the real part of the phasor solution and express it as a cosine in the time domain. Given these guidelines, we can represent time-harmonic E and H fields as: 6 (Equation 18.18) (Equation 18.19) Notice how the time dependence of the function is completely contained in the ejt term on the right side. As a result, E(x,y,z) and H(x,y,z) are often referred to as E(r) and H(r). We can also rewrite Maxwell’s Equations for the special case where the electric and magnetic fields are both harmonic. Doing so, we use E(r) and H(r), and we replace time derivatives with j. (Equation 18.20) (Equation 18.21) (Equation 18.22) (Equation 18.23) Let’s give some special attention to the phasor form of Ampere’s Law, Equation 18.21. We can use Ohm’s Law to replace J(r) with E(r), giving: (Equation 18.24) Factoring the E(r) on the right side: (Equation 18.25) Recalling that the E(r) term was originally conduction current, and jE(r) was originally displacement current, we can pull those two terms out and assign them to Jc and Jd, respectively: (Equation 18.26) (Equation 18.27) Remember that in a very good conductor, Jc >> Jd, while in a very good insulator, Jd >> Jc. 7 We can also notice that displacement current increases as the frequency of the harmonic waves increase because of the factor. This means that even very good conductors will eventually exhibit a displacement current that exceeds the conduction current, meaning that they will no longer technically be conductors. Example 18.3: Determine the frequency at which copper (=6x107 S/m) behaves more as an insulator than as a conductor. We can also calculate the average Poynting vector when E(r) and H(r) are in phasor form: (Equation 18.28) This equation will give the correct direction of energy flow, and it will give the average power over the course of one harmonic cycle. It is important to remember that the complex conjugate H*(r) requires that the imaginary component (in rectangular form) or the angle (in polar form) of the phasor representing H(r) must have an extra negative sign added. Example 18.4: Given the values of E(r) and H(r) below, calculate Sav. 18.3 Deriving the Wave Equation from Maxwell’s Equations Now we embark on the derivation that units electricity, magnetism, and light. We will also see that the speed of light has been hiding in Maxwell’s equations all this time. Let’s first begin with a fresh set of Maxwell’s equations (not in harmonic form), using the constitutive equations to eliminate D, B, and J so that the equations are expressed only in terms of E and H. We will further assume that we are working in a region with no charge density, so v=0. This will simplify the right side of Gauss’s Law. 8 (Equation 18.29) (Equation 18.30) (Equation 18.31) (Equation 18.32) Now, let’s take the curl of both sides of Faraday’s Law (Equation 18.29): (Equation 18.33) Next, substitute Ampere’s Law (Equation 18.30) into the right side of this equation: (Equation 18.34) Distributing the time derivative on the right side gives: (Equation 18.35) Considering the left side of this equation, we have seen before that the curl of the curl of a function can be rewritten as: (Equation 18.36) Where the first term on the right side can be set to zero if we choose the Coulomb gauge. Substituting Equation 18.36 into the left side of Equation 18.35, we find: (Equation 18.37) Rearranging the terms gives: (Equation 18.38) 9 Equation 18.38 is called the general homogeneous three-dimensional vector wave equation. Its solution will have the form of a wave that may be polarized in any direction and which may decay due to the third term. A similar derivation (beginning with taking the curl of both sides of Ampere’s Law and substituting Faraday’s Law this time) yields the same equation for H: (Equation 18.39) This is a very interesting result—in spite of the differences in Faraday’s and Ampere’s Laws, they can be used together to derive two equations that describe identical behavior for electric and magnetic fields.
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