6.013 Electromagnetics and Applications, Chapter 2: Introduction to Electrodynamics

Chapter 2: Introduction to Electrodynamics 2.1 Maxwell’s differential equations in the time domain Whereas the Lorentz force law characterizes the observable effects of electric and magnetic fields on charges, Maxwell’s equations characterize the origins of those fields and their relationships to each other. The simplest representation of Maxwell’s equations is in differential form, which leads directly to waves; the alternate integral form is presented in Section 2.4.3. The differential form uses the vector del operator ∇: ∇ ≡ x̂ ∂ + ŷ ∂ + ẑ ∂ ∂x ∂y ∂z (2.1.1) where x̂ , ŷ , and ẑ are defined as unit vectors in cartesian coordinates. Relations involving ∇ are summarized in Appendix D. Here we use the conventional vector dot product1 and cross product2 of ∇ with the electric and magnetic field vectors where, for example: E = x̂E x + ŷE y + ẑE z ∇•E ≡ (2.1.2) ∂E x ∂E y ∂E z + + ∂x ∂y ∂z (2.1.3) We call ∇ • E the divergence of E because it is a measure of the degree to which the vector field⎯E diverges or flows outward from any position. The cross product is defined as: ∂E y ⎞ ⎛ ∂E ∂E ⎞ ⎛ ∂E y ∂E x ⎞ ⎛ ∂E ∇ × E ≡ x̂ ⎜ z − + ŷ ⎜ x − z ⎟ + ẑ ⎜ − ⎟ ∂z ⎠ ∂x ⎠ ⎝ ∂x ∂y ⎟⎠ ⎝ ∂z ⎝ ∂y (2.1.4) xˆ yˆ zˆ = det ∂ ∂x ∂ ∂y ∂ ∂z Ey Ez Ex which is often called the curl of E . Figure 2.1.1 illustrates when the divergence and curl are zero or non-zero for five representative field distributions. 1 2 The dot product of A and B can be defined as A • B = A x B x + A y B y + A z B z = A B cos θ , where θ is the angle between the two vectors. The cross product of A and B can be defined as A × B = x̂ ( A y B z − A z B y ) + ŷ ( A z B x − A x B z ) + ẑ ( A x B y − A y B x ) ; its magnitude is A • B sin θ . Alternatively, A × B = det|[Ax,Ay,Az],[Bx,By,Bz],[ x̂, y, ˆ zˆ ]|. - 23 - curl E = 0 curl E ≠ 0 E div E ≠ 0 div E = 0 Figure 2.1.1 Fields with zero or non-zero divergence or curl. The differential form of Maxwell’s equations in the time domain are: ∇ × E = − ∂B ∂t Faraday’s Law (2.1.5) Ampere’s Law (2.1.6) ∇•D = ρ Gauss’s Law (2.1.7) ∇•B = 0 Gauss’s Law (2.1.8) [volts/meter; Vm-1] (2.1.9) [amperes/meter; Am-1] (2.1.10) B magnetic flux density [Tesla; T] (2.1.11) D electric displacement [coulombs/m2; Cm-2] (2.1.12) J electric current density [amperes/m2; Am-2] (2.1.13) ρ electric charge density [coulombs/m3; Cm-3] (2.1.14) ∇ × H = J + ∂D ∂t The field variables are defined as: E electric field H magnetic field These four Maxwell equations invoke one scalar and five vector quantities comprising 16 variables. Some variables only characterize how matter alters field behavior, as discussed later in Section 2.5. In vacuum we can eliminate three vectors (9 variables) by noting: - 24 - D = εo E (constitutive relation for D ) (2.1.15) B = μo H (constitutive relation for B ) (2.1.16) J = ρv = σE (constitutive relation for J ) (2.1.17) where εo = 8.8542×10-12 [farads m-1] is the permittivity of vacuum, μo = 4π×10-7 [henries m-1] is the permeability of vacuum3, v is the velocity of the local net charge density ρ, and σ is the conductivity of a medium [Siemens m-1]. If we regard the electrical sources ρ and J as given, then the equations can be solved for all remaining unknowns. Specifically, we can then find E and H , and thus compute the forces on all charges present. Except for special cases we shall avoid solving problems where the electromagnetic fields and the motions of ρ are interdependent. The constitutive relations for vacuum, D = ε0 E and B = μ0 H , can be generalized to D = εE , B = μH , and J = σE for simple media. Media are discussed further in Section 2.5. Maxwell’s equations require conservation of charge. By taking the divergence of Ampere’s law (2.1.6) and noting the vector identity ∇ • ( ∇ × A ) = 0 , we find: ∇ • ( ∇ × H ) = 0 = ∇ • ∂D + ∇ • J ∂t (2.1.18) Then, by reversing the sequence of the derivatives in (2.1.18) and substituting Gauss’s law ∇ • D = ρ (2.1.7), we obtain the differential expression for conservation of charge: ∇•J = − ∂ρ ∂t (conservation of charge) (2.1.19) The integral expression can be derived from the differential expression by using Gauss’s divergence theorem, which relates the integral of ∇ • G over any volume V to the integral of G • n̂ over the surface area A of that volume, where the surface normal unit vector n̂ points outward: ∫∫∫V ∇ • G dv = ∫∫ A G • n̂ da (Gauss’s divergence theorem) (2.1.20) Thus the integral expression for conservation of charge is: d ρ dv = − ∫∫ J • n̂ da A dt ∫∫∫V 3 (conservation of charge) The constant 4π × 10-7 is exact and enters into the definition of an ampere. - 25 - (2.1.21) which says that if no net current⎯J flows through the walls A of a volume V, then the total charge inside must remain constant. Example 2.1A If the electric field in vacuum is E = x̂E o cos ( ωt − ky ) , what is H ? Solution: From Faraday’s law (2.1.5): μo( ∂H ∂t ) = -(∇× E ) = zˆ ∂Ex/∂y = ẑ kEo sin (ωt-ky), using (2.1.4) for the curl operator. Integration of this equation with respect to time yields: H = −ẑ ( kEo μo ω) cos ( ωt − ky ) . Example 2.1B Does the electric field in vacuum E = x̂E o cos ( ωt − kx ) satisfy Maxwell’s equations? Under what circumstances would this⎯E satisfy the equations? Solution: This electric field does not satisfy Gauss’s law for vacuum, which requires ∇•D = ρ = 0. It satisfies Gauss’s law only for non-zero charge density: ρ = ∇ • D = εo ∂E x ∂x = ∂ [ εo E o cos ( ωt − kx )] ∂x = kεo Eo sin ( ωt − kx ) ≠ 0 . To satisfy the remaining Maxwell equations and conservation of charge (2.1.19) there must also be a current J ≠ 0 corresponding to ρ : J = σE = xˆσE o cos ( ωt − kx ) , where (2.1.17) simplified the computation. 2.2 Electromagnetic waves in the time domain Perhaps the greatest triumph of Maxwell’s equations was their ability to predict in a simple way the existence and velocity of electromagnetic waves based on simple laboratory measurements of the permittivity and permeability of vacuum. In vacuum the charge density ρ =⎯J = 0, and so Maxwell’s equations become: ∇ × E = −μo ∂H ∂t (Faraday’s law in vacuum) (2.2.1) ∇ × H = εo ∂E ∂t (Ampere’s law in vacuum) (2.2.2) ∇•E = 0 (Gauss’s law in vacuum) (2.2.3) ∇•H = 0 (Gauss’s law in vacuum) (2.2.4) We can eliminate H from these equations by computing the curl of Faraday’s law, which introduces ∇ × H on its right-hand side so Ampere’s law can be substituted: - 26 - ( ) ∇ × ( ∇ × E ) = −μo ∂ ∇ × H = −μo εo ∂E ∂t ∂t 2 2 (2.2.5) Using the well known vector identity (see Appendix D): ∇ × ( ∇ × A ) = ∇ ( ∇ • A ) − ∇2 A (“well-known vector identity”) (2.2.6) and then using (2.2.3) to eliminate ∇ • E , (2.2.5) becomes the electromagnetic wave equation, often called the Helmholtz wave equation: 2 ∇ 2 E − μo εo ∂ E = 0 ∂t 2 (Helmholtz wave equation) (2.2.7) where: 2 2 ⎞ ⎛ 2 ∇ 2 E ≡ ⎜ ∂ + ∂ + ∂ ⎟ x̂E x + ŷE y + ẑE z ⎜ ∂x 2 ∂y 2 ∂z 2 ⎟ ⎝ ⎠ ( ) (2.2.8) The solutions to this wave equation (2.2.7) are any fields E ( r, t ) for which the second ( ) spatial derivative ( ∇ 2 E ) equals a constant times the second time derivative ∂ 2 E ∂t 2 . The position vector r ≡ x̂x + ŷy + ẑz . The wave equation is therefore satisfied by any arbitrary E ( r, t ) having identical dependence on space and time within a constant multiplier. For example, arbitrary functions of the arguments (z - ct), (z + ct), or (t ± z/c) have such an identical dependence and are among the valid solutions to (2.2.7), where c is some constant to be determined. One such solution is: E ( r, t ) = E ( z − ct ) = xˆE x ( z − ct ) (2.2.9) where the arbitrary function Ex(z - ct) might be that illustrated in Figure 2.2.1 at time t = 0 and again at some later time t. Note that as time advances within the argument (z - ct), z must advance with ct in order for that argument, or⎯E at any point of interest on the waveform, to remain constant. Ex(z,t) t>0 t=0 ⎯z z = ct Figure 2.2.1 Arbitrary electromagnetic wave propagating in the +z direction. - 27 - We can test this candidate solution (2.2.9) by substituting it into the wave equation (2.2.7), yielding: ∇ 2 E ( z − ct ) = ∂ 2 [ E ( z − ct )] ∂z 2 ≡ E '' ( z − ct ) 2[ ∂ E ( z − ct )] 2 = μo εo = μo εo ( −c ) E '' ( z − ct ) 2 ∂t (2.2.10) where we define A '(q) as the first derivative of A with respect to its argument q and A ''(q) as its second derivative. Equation (2.2.10) is satisfied if: c= 1 (μo εo [m/s] (2.2.11) where we define c as the velocity of light in vacuum: c = 2.998×108 [m s-1] (velocity of light) (2.2.12) Figure 2.2.1 illustrates how an arbitrary E(z, t) can propagate by translating at velocity c. However, some caution is warranted when E(z, t) is defined. Although our trial solution (2.2.9) satisfies the wave equation (2.2.7), it may not satisfy Gauss’s laws. For example, consider the case where: E ( z, t ) = ẑE z ( z − ct ) (2.2.13) Then Gauss’s law ∇ • E = 0 is not satisfied: ∇ • E = ∂ E z ≠ 0 for arbitrary E ( z ) ∂z (2.2.14) In contrast, if E(z, t) is oriented perpendicular to the direction of propagation (in the x̂ and/or ŷ directions for z-directed propagation), then all Maxwell’s equations are satisfied and the solution is valid. In the case E ( z, t ) = ŷE y ( z − ct ) , independent of x and y, we have a uniform plane wave because the fields are uniform with respect to two of the coordinates (x,y) so that ∂ E ∂x = ∂ E ∂y = 0 . Since this electric field is in the y direction, it is said to be y-polarized; by convention, polarization of a wave refers to the direction of its electric vector. Polarization is discussed further in Section 2.3.4. Knowing E ( z, t ) = ŷE y( z − ct ) for this example, we can now find H(z, t) using Faraday’s law (2.2.1): - 28 - ∂H = − ( ∇ × E ) ∂t μo (2.2.15) We can evaluate the curl of E using (2.1.4) and knowing E x = E z = ∂ = ∂ = 0 : ∂x ∂y ∂E y ∂E y ⎞ ⎛ ∂E ∂E ⎞ ⎛ ∂E y ∂E x ⎞ ⎛ ∂E + ŷ ⎜ x − z ⎟ + ẑ ⎜ = − xˆ − ∇ × E = x̂ ⎜ z − ⎟ ⎟ ∂z ⎠ ∂x ⎠ ⎝ ∂x ∂y ⎠ ∂z ⎝ ∂z ⎝ ∂y (2.2.16) Then, by integrating (2.2.15) over time it becomes: H ( z, t ) = − ∫ t −∞ ( ∇ × E ) dt = x̂ μo 1 t ∂E y ( z − ct ) dt μo ∫−∞ ∂z ε = −x̂ 1 E y ( z − ct ) = −x̂ o E y ( z − ct ) cμ o μo H(z, t) = E ( z, t ) εo ẑ × E ( z, t ) = ẑ × μo ηo (2.2.17) (2.2.18) where we used the velocity of light c = 1/ ε oμ o , and defined ηo = μ o /ε o . Thus E and H in a uniform plane wave are very simply related. Their directions are orthogonal to each other and to the direction of propagation, and the magnitude of the electric field is (μo/εo)0.5 times that of the magnetic field; this factor ηo = μ o /ε o is known as the characteristic impedance of free space and equals ~377 ohms. That is, for a single uniform plane wave in free space, E H = ηo = μ o /ε o ≅ 377 [ohms] (2.2.19) Electromagnetic waves can propagate in any arbitrary direction in space with arbitrary time behavior. That is, we are free to define x̂ , ŷ , and ẑ in this example as being in any three orthogonal directions in space. Because Maxwell’s equations are linear in field strength, superposition applies and any number of plane waves propagating in arbitrary directions with arbitrary polarizations can be superimposed to yield valid electromagnetic solutions. Exactly which superposition is the valid solution in any particular case depends on the boundary conditions and the initial conditions for that case, as discussed later in Chapter 9 for a variety of geometries. - 29 - Example 2.2A Show that E = ŷE o ( t + z c ) satisfies the wave equation (2.2.7). In which direction does this wave propagate? 2 ⎞ ⎛ Solution: ⎜ ∇ 2 − ∂ ⎟ E = ŷ 1 ⎡⎣ E o" ( t + z c ) − E o" ( t + z c ) ⎤⎦ = 0 ; Q.E.D4. Since the argument c2∂t 2 ⎠ c2 ⎝ (t + z/c) remains constant as t increases only if z/c decreases correspondingly, the wave is propagating in the -z direction. 2.3 Maxwell’s equations, waves, and polarization in the frequency domain 2.3.1 Sinusoidal waves Linear systems are easily characterized by the magnitude and phase of each output as a function of the frequency at which the input is sinusoidally stimulated. This simple characterization is sufficient because sinusoids of different frequencies can be superimposed to construct any arbitrary input waveform5, and the output of a linear system is the superposition of its responses to each superimposed input. Systems with multiple inputs and outputs can be characterized in the same way. Nonlinear systems are more difficult to characterize because their output frequencies generally include harmonics of their inputs. Fortunately free space is a linear system, and therefore it is fully characterized by its response to sinusoidal plane waves. For example, the arbitrary z-propagating x-polarized uniform plane wave of (2.2.9) and Figure 2.2.1 could be sinusoidal and represented by: E ( r, t ) = x̂E o cos ⎡⎣ k ( z − ct ) ⎦⎤ (2.3.1) H ( r, t ) = ŷ ε o /μ o E o cos ⎡⎣ k ( z − ct ) ⎤⎦ (2.3.2) where the wave amplitude Eo is a constant and the factor k is related to frequency, as shown below. It is more common to represent sinusoidal waves using the argument (ωt - kz) so that their frequency and spatial dependences are more evident. The angular frequency ω is simply related to frequency f [Hz]: ω = 2πf [radians s-1] 4 (angular frequency) (2.3.3) Q.E.D. is the abbreviation for the Latin phrase “quod erat demonstratum” or “that which was to be demonstrated.” 5 The Fourier transform pair (10.4.17) and (10.4.18) relate arbitrary pulse waveforms h(t) to their corresponding spectra H(f), where each frequency f has its own magnitude and phase represented by H(f). - 30 - and the spatial frequency k, often called the wavenumber, is simply related to ω and wavelength λ [m], which is the length of one period in space: k = 2π/λ = ω/c [radians m-1] (wave number) (2.3.4) The significance and dimensions of ω and k are directly analogous; they are radians s-1 and radians m-1, respectively. Therefore we can alternatively represent the wave of (2.3.1) and (2.3.2) as: E ( r, t ) = x̂Eo cos ( ωt − kz ) ⎡⎣ v m-1 ⎤⎦ (2.3.5) H ( z, t ) = ŷ ε o /μ o E o cos ( ωt − kz ) ⎡⎣A m-1 ⎤⎦ (2.3.6) Figure 2.3.1 suggests the form of this wave. Its wavelength is λ, the length of one cycle, where: λ = c/f [m] (wavelength) (2.3.7) The figure illustrates how these electric and magnetic fields are in phase but orthogonal to each other and to the direction of propagation. When the argument (ωt – kz) equals zero, the fields are maximum, consistent with cos(ωt - kz). λ H ( z, 0 ) E ( z, 0 ) z x direction of propagation y Figure 2.3.1 +z propagating y-polarized uniform plane wave of wavelength λ. This notation makes it easy to characterize uniform plane waves propagating in other directions as well. For example: E ( r, t ) = x̂Eo cos ( ωt + kz ) (x-polarized wave in -z direction) - 31 - (2.3.8) 2.3.2 E ( r, t ) = ŷEo cos ( ωt − kz ) (y-polarized wave in +z direction) (2.3.9) E ( r, t ) = ŷE o cos ( ωt − kx ) (y-polarized wave in +x direction) (2.3.10) E ( r, t ) = ẑEo cos ( ωt + kx ) (z-polarized wave in -x direction) (2.3.11) Maxwell’s equations in the complex-frequency domain Electromagnetic fields are commonly characterized in the frequency domain in terms of their magnitudes and phases as a function of position r for frequency f. For example, the x̂ component of a general sinusoidally varying E might be: E(r,t ) = x̂E x(r )cos⎣⎡ωt + φ(r )⎤⎦ (2.3.12) This might become E(r,t ) = x̂ E x cos(ωt − kz) for a uniform plane wave propagating in the +z direction. It is generally more convenient to express phase using complex notation (see Appendix B). The x-component of the wave of (2.3.12) can also be represented as: { E x (r,t ) = R e E x (r )e } = x̂R e{Ex(r)e jωt } j(ωt + φx (r )) (2.3.13) where the spatial and frequency parts of E x (r,t ) have been separated and E x(r ) = E x(r ) e jφx (r) is called a phasor. The simplicity will arise later when we omit R e {[ ] e jωt } from our expressions as “understood”, so only the phasors remain. The underbar under E x indicates E x is not a function of time, but rather is a complex quantity with a real part and an imaginary part, where: () E x(r ) = R e{E x(r )} + jIm {E x(r )} = E x(r ) e jφx r and φx(r ) = tan −1(Im{E x(r )} Re{E x(r )}) . (2.3.14) A general vector can also be a phasor, e.g., E(r ) = x̂E x(r) + yˆE y(r) + ẑ E z(r ) , where E(r,t ) = R e {E(r)e jωt} . We can use such phasors to simplify Maxwell’s equations. For example, we can express Faraday’s law (2.2.1) as: { ∇ × R e {E(r)e jωt} = −∂R e{B(r )e jωt} ∂t = R e {∇ × E(r)e jωt} = R e − jωB(r )e jωt - 32 - } (2.3.15) The other Maxwell equations can be similarly transformed, which suggests that the notation R e {[ ] e jωt } can be omitted and treated as understood. For example, removing this redundant notation from (2.3.15) results in: ∇ × E = − jωB . Any problem solution expressed as a phasor, e.g. E ( r ) , can be converted back into a time-domain expression by the operator R e {[ ] e jωt } . These omissions of the understood notation result in the complex or time-harmonic Maxwell equations: ∇ × E = − jωB (Faraday’s law) (2.3.16) ∇ × H = J + jωD (Ampere’s law) (2.3.17) ∇•D = ρ (Gauss’s law) (2.3.18) ∇•B = 0 (Gauss’s law) (2.3.19) Note that these equations are the same as before [i.e., (2.2.1–4)], except that we have simply replaced the operator ∂/∂t with jω and placed an underbar under all variables, signifying that they are now phasors. We can immediately derive the time-harmonic equation for conservation of charge (2.1.19) by computing the divergence of (2.3.17), noting that ∇ • ( ∇ × A ) = 0 for any A , and substituting ∇ • D = ρ (2.3.18): ∇ • J + jωρ = 0 (2.3.20) Example 2.3A Convert the following expressions into their time-domain equivalents: jω∇ × Q = Rj , Re − jkz , and E = x̂3 + ŷj4 . Solution: −ω ( ∇ × Q ) sin(ωt) = −R sin ωt , R cos ( ωt − kz ) , and 3 x̂ cos ωt − 4 yˆ sin ωt . Example 2.3B Convert the following expressions into their complex frequency-domain equivalents: A cos ( ωt + kz ) , and Bsin ( ωt + φ ) . Solution: Ae+ jkz , and − jBe jφ = − jBcos φ + Bsin φ . - 33 - 2.3.3 Sinusoidal uniform plane waves We can readily derive from Maxwell’s equations the time-harmonic Helmholtz wave equation for vacuum (2.2.7) by substituting jω for ∂/∂t or, as we did earlier, by taking the curl of Faraday’s law, using the well known vector identity (2.2.6) and Gauss’s law, replacing B by μ o H , and using Ampere’s law to replace ∇× H . In both cases the Helmholtz wave equation becomes: ( ∇2 + ω2μoεo ) E = 0 (wave equation) (2.3.21) As before, the solution E ( r ) to the wave equation can be any arbitrary function of space ( r ) such that its second spatial derivative ( ∇ 2 E ) equals a constant ( −ω2εoμo ) times that same function E(r) . One solution with these properties is the time-harmonic version of the timedomain expression E ( r, t ) = ŷEo cos ( ωt − kz ) : E ( r ) = yˆE o e − jkz ⎡⎣ v m -1 ⎦⎤ (2.3.22) Substituting (2.3.22) into the wave equation (2.3.21) yields: ( ⎡⎣∂ 2 ∂z2 ⎤⎦ + ω2μoεo ) E = ([− jk ]2 + ω2μoεo ) E = 0 (2.3.23) which is satisfied if the wavenumber k is: k = ω μ o ε o = ω = 2πf = 2π ⎡⎣radians m-1⎤⎦ c c λ (2.3.24) It is now an easy matter to find the magnetic field that corresponds to (2.3.22) by using Faraday’s law (2.3.16), B = μo H , and the definition of the “∇×” operator (2.1.1): H (r) = − (∇ × E ) jωμo = − jkz 1 xˆ∂E y = − xˆkE o e jωμo ∂z ωμo = −x̂ 1 E o e− jkz ⎡⎣Am-1 ⎤⎦ ηo (2.3.25) As before, E and H are orthogonal to each other and to the direction of propagation, and E = ηo H . As another example, consider a z-polarized wave propagating in the -x direction; then: - 34 - E ( r ) = ẑE o e+ jkx , H ( r ) = ŷE o e jkx ηo (2.3.26) It is easy to convert phasor expressions such as (2.3.26) into time-domain expressions. We simply divide the phasor expressions into their real and imaginary parts, and note that the real part varies as cos(ωt - kz) and the imaginary part varies as sin(ωt - kz). Thus the fields in (2.3.22) could be written instead as a real time-domain expression: E ( r, t ) = ŷEo cos ( ωt − kz ) (2.3.27) Had the electric field solution been instead the phasor ŷjE o e− jkz , the time domain expression R e {E ( r ) e jωt } would then be: E ( r, t ) = − ŷEo sin ( ωt − kz ) (2.3.28) The conversion of complex phasors to time-domain expressions, and vice-versa, is discussed further in Appendix B. 2.3.4 Wave polarization Complex notation simplifies the representation of wave polarization, which characterizes the behavior of the sinusoidally varying electric field vector as a function of time. It is quite distinct from the polarization of media discussed in Section 2.5.3. Previously we have seen waves for which the time-varying electric vector points only in the ±x, ±y, or ±z directions, corresponding to x, y, or z polarization, respectively. By superimposing such waves at the same frequency and propagating in the same direction we can obtain any other desired time-harmonic polarization. Linear polarization results when the oscillating electric vector points only along a single direction in the plane perpendicular to the direction of propagation, while elliptical polarization results when the x and y components of the electric vector are out of phase so that the tip of the electric vector traces an ellipse in the same plane. Circular polarization results only when the phase difference between x and y is 90 degrees and the two amplitudes are equal. These various polarizations for +ẑ propagation are represented below at z = 0 in the time domain and as phasors, and in Figure 2.3.2. E ( t ) = ŷE o cos ωt E = ŷE o (y-polarized) (2.3.29) E ( t ) = x̂E o cos ωt E = x̂E o (x-polarized) (2.3.30) E ( t ) = ( xˆ + yˆ ) E o cos ωt E = ( xˆ + yˆ ) E o (45°-polarized) (2.3.31) E ( t ) = E o ( x̂ cos ωt + yˆ sin ωt ) E = ( xˆ − jyˆ ) E o (right-circular) (2.3.32) E ( t ) = E o ( x̂ cos ωt + 1.5 ŷ sin ωt ) E = ( x̂ − 1.5 jŷ ) E o (right-elliptical) (2.3.33) - 35 - ( E ( t ) = Eo [ x̂ cos ωt + ŷ cos ( ωt + 20° )] x̂ x̂ y-polarized x-polarized ŷ ŷ x̂ ) E = x̂ + e0.35 j ŷ Eo rightcircular x̂ (left-elliptical) x̂ 45o-polarized ŷ right-elliptical polarization left-elliptical x̂ polarization ŷ ŷ (2.3.34) ŷ Figure 2.3.2 Polarization ellipses for +z-propagating plane waves (into the page). The Institute of Electrical and Electronics Engineers (IEEE) has defined polarization as right-handed if the electric vector traces a right-handed ellipse in the x-y plane for a wave propagating in the +z direction, as suggested in Figure 2.3.3. That is, for right-handed polarization the fingers of the right hand circle in the direction taken by the electric vector while the thumb points in the direction of propagation. This legal definition is opposite that commonly used in physics, where that alternative definition is consistent with the handedness of the “screw” formed by the instantaneous three-dimensional loci of the tips of the electric vectors comprising a wave. Wave propagation x̂ ŷ ẑ Figure 2.3.3 IEEE definition of right-handed polarization. Example 2.3C If E = E o e− jkz , what polarizations correspond to: Eo = ŷ , E o = x̂ + 2 ŷ , and E o = x̂ − jŷ ? Solution: y polarization, linear polarization at angle tan-12 relative to the x-z plane, and rightcircular polarization. - 36 - 2.4 Relation between integral and differential forms of Maxwell’s equations 2.4.1 Gauss’s divergence theorem Two theorems are very useful in relating the differential and integral forms of Maxwell’s equations: Gauss’s divergence theorem and Stokes theorem. Gauss’s divergence theorem (2.1.20) states that the integral of the normal component of an arbitrary analytic vector field A over a surface S that bounds the volume V equals the volume integral of ∇ • A over V. The theorem can be derived quickly by recalling (2.1.3): ∇•A ≡ ∂A x ∂A y ∂A z + + ∂x ∂y ∂z (2.4.1) Therefore ∇ • A at the position xo, yo, zo can be found using (2.4.1) in the limit where Δx, Δy, and Δz approach zero: { ∇ • A = lim ⎡A x ( x o + Δx 2 ) − A x ( x o − Δx 2 ) ⎤⎦ Δx Δi →0 ⎣ + ⎡A ⎣ y ( yo + Δy 2 ) − A y ( yo − Δy 2 ) ⎤⎦ Δy (2.4.2) } + ⎡A ⎣ z ( zo + Δz 2 ) − A z ( z o − Δz 2 ) ⎤⎦ Δz { = lim ΔyΔz ⎡A ⎣ x ( x o + Δx 2 ) − A x ( x o − Δx 2 ) ⎦⎤ Δi →0 +ΔxΔz ⎡⎣ A y ( yo + Δy 2 ) − A y ( yo − Δy 2 ) ⎦⎤ (2.4.3) } +ΔxΔy ⎣⎡A z ( zo + Δz 2 ) − A z ( z o − Δz 2 ) ⎤⎦ ΔxΔyΔz = lim Δv → 0 { ∫∫ Sc } A • nˆ da Δv (2.4.4) where n̂ is the unit normal vector for an incremental cube of dimensions Δx, Δy, Δz; da is its differential surface area; Sc is its surface area; and Δv is its volume, as suggested in Figure 2.4.1(a). We may now stack an arbitrary number of such infinitesimal cubes to form a volume V such as that shown in Figure 2.4.1(b). Then we can sum (2.4.4) over all these cubes to obtain: lim Δv → 0 ∑ ( ∇ • A ) Δvi = Δvlim ∑ { ∫∫Sc A • n̂ dai } →0 i i - 37 - (2.4.5) (a) surface Sc (b) n̂ da V Δy -Az(zo - Δz/2) surface Sc ith cube Az(zo + Δz/2) Δx Δv n̂ da Δz Figure 2.4.1 Derivation of Gauss’s divergence theorem. Since all contributions to ∑i { ∫∫S A • n̂ dai } from interior-facing adjacent cube faces cancel, the only remaining contributions from the right-hand side of (2.4.5) are from the outer surface of the volume V. Proceeding to the limit, we obtain Gauss’s divergence theorem: ∫∫∫V ( ∇ • A ) dv = ∫∫S ( A • n̂ ) da 2.4.2 (2.4.6) Stokes’ theorem Stokes’ theorem states that the integral of the curl of a vector field over a bounded surface equals the line integral of that vector field along the contour C bounding that surface. Its derivation is similar to that for Gauss’s divergence theorem (Section 2.4.1), starting with the definition of the z component of the curl operator [from Equation (2.1.4)]: ⎛ ∂A y ∂A x ⎞ − ∂y ⎟⎠ ⎝ ∂x ( ∇ × A )z ≡ ẑ ⎜ = ẑ lim Δx, Δy→0 (2.4.7) {⎡A ⎣ y ( x o + Δx 2 ) − A y ( x o − Δx 2 ) ⎤⎦ Δx (2.4.8) } − ⎡A ⎣ x ( yo + Δy 2 ) − A x ( yo − Δy 2 ) ⎤⎦ Δy = zˆ lim Δx, Δy→0 {Δy ⎡⎣A y ( xo + Δx 2) − A y ( xo − Δx 2)⎤⎦ ΔxΔy } − Δx ⎡⎣ A x ( yo + Δy 2 ) − A x ( yo − Δy 2 ) ⎤⎦ ΔxΔy (2.4.9) Consider a surface in the x-y plane, perpendicular to ẑ and n̂ , the local surface normal, as illustrated in Figure 2.4.2(a). - 38 - (a) (b) ẑ,nˆ d⎯s da = ΔxΔy n̂ ds da Δy Δx area A contour C Figure 2.4.2 Derivation of Stokes’ theorem. Then (2.4.9) applied to ΔxΔy becomes: ΔxΔy ( ∇ × A ) • nˆ = ∫C A • ds (2.4.10) where ds is a vector differential length [m] along the contour C bounding the incremental area defined by ΔxΔy = da. The contour C is transversed in a right-hand sense relative to n̂ . We can assemble such infinitesimal areas to form surfaces of arbitrary shapes and area A, as suggested in Figure 2.4.2(b). When we sum (2.4.10) over all these infinitesimal areas da, we find that all contributions to the right-hand side interior to the area A cancel, leaving only the contributions from contour C along the border of A. Thus (2.4.10) becomes Stokes’ theorem: ∫∫A ( ∇ × A ) • n̂ da = ∫C A • ds (2.4.11) where the relation between the direction of integration around the loop and the orientation of n̂ obey the right-hand rule (if the right-hand fingers curl in the direction of ds , then the thumb points in the direction n̂ ). 2.4.3 Maxwell’s equations in integral form The differential form of Maxwell’s equations (2.1.5–8) can be converted to integral form using Gauss’s divergence theorem and Stokes’ theorem. Faraday’s law (2.1.5) is: ∇ × E = − ∂B ∂t (2.4.12) Applying Stokes’ theorem (2.4.11) to the curved surface A bounded by the contour C, we obtain: ∂B ∫∫A ( ∇ × E ) • n̂ da = ∫C E • ds = −∫∫A ∂t • n̂ da This becomes the integral form of Faraday’s law: - 39 - (2.4.13) ∂ ∫C E • ds = − ∂t ∫∫A B • nˆ da (Faraday’s Law) (2.4.14) A similar application of Stokes’ theorem to the differential form of Ampere’s law yields its integral form: ⎡ ∫ C H • ds = ∫∫A ⎢⎣J + ∂ D ⎤ • n̂ da ∂t ⎥⎦ (Ampere’s Law) (2.4.15) Gauss’s divergence theorem (2.1.20) can be similarly applied to Gauss’s laws to yield their integral form: ∫∫∫V ( ∇ • D ) dv = ∫∫∫V ρ dv = ∫∫ A ( D • n̂ ) da (2.4.16) This conversion procedure thus yields the integral forms of Gauss’s laws. That is, we can integrate D • n̂ and B • n̂ in the differential equations over the surface A that bounds the volume V: ∫∫ A ( D • n̂ ) da = ∫∫∫V ρ dv (Gauss’s Law for charge) (2.4.17) (Gauss’s Law for⎯B) (2.4.18) ∫∫ A ( B • n̂ ) da = 0 Finally, conservation of charge (1.3.19) can be converted to integral form as were Gauss’s laws: ∂ρ ∫∫ A ( J • n̂ ) da = −∫∫∫V ∂t dv Faraday's law (conservation of charge) Gauss's law for charge ∂B ∂t ⎯D ρ ⎯E Ampere's law Gauss's law for⎯B ⎯B ∂D ∂t ⎯J zero ⎯H Figure 2.4.3 Maxwell’s equations in sketch form. - 40 - (2.4.19) The four sketches of Maxwell’s equations presented in Figure 2.4.3 may facilitate memorization; they can be interpreted in either differential or integral form because they capture the underlying physics. Example 2.4A Using Gauss’s law, find E at distance r from a point charge q. Solution: The spherical symmetry of the problem requires E to be radial, and Gauss’s law requires ∫A εo E • r̂ dA = ∫V ρ dv = q = 4πr 2 εo E r , so E = r̂E r = r̂q 4πεo r 2 . Example 2.4B What is⎯H at r = 1 cm from a line current I = ẑ [amperes] positioned at r = 0? Solution: Because the geometry of this problem is cylindrically symmetric, so is the solution. Using the integral form of Ampere’s law (2.4.15) and integrating in a right-hand sense around a circle of radius r centered on the current and in a plane orthogonal to it, we obtain 2πrH = I, so⎯H = θ̂ 100/2π [A m-1]. 2.5 Electric and magnetic fields in media 2.5.1 Maxwell’s equations and media The great success of Maxwell’s equations lies partly in their simple prediction of electromagnetic waves and their simple characterization of materials in terms of conductivity σ [Siemens m-1], permittivity ε [Farads m-1], and permeability μ [Henries m-1]. In vacuum we find σ = 0, ε = εo, and μ = μo, where εo = 8.8542×10-12 and μo = 4π×10-7. For reference, Maxwell’s equations are: ∇ × E = − ∂B ∂t (2.5.1) ∇ × H = J + ∂D ∂t (2.5.2) ∇•D = ρ (2.5.3) ∇•B = 0 (2.5.4) The electromagnetic properties of most media can be characterized by the constitutive relations: D = εE (2.5.5) - 41 - B = μH (2.5.6) J = σE (2.5.7) In contrast, the nano-structure of media can be quite complex and requires quantum mechanics for its full explanation. Fortunately, simple classical approximations to atoms and molecules suffice to understand the origins of σ, ε, and μ, as discussed below in that sequence. 2.5.2 Conductivity Conduction in metals and n-type semiconductors6 involves free electrons moving many atomic diameters before they lose momentum by interacting with atoms or other particles. Acceleration induced by the small applied electric field inside the conductor restores electron velocities to produce an equilibrium current. The total current density J [A m-2] is proportional to the product of the average electron velocity v [m s-1] and the number density n [m-3] of free electrons. A related conduction process occurs in ionic liquids, where both negative and positive ions can carry charge long distances. In metals there is approximately one free electron per atom, and in warm n-type semiconductors there is approximately one free electron per donor atom, where the sparse donor atoms are easily ionized thermally. Since, for non-obvious reasons, the average electron velocity v is proportional to E , therefore J = −en3 v = σE , as stated in (2.5.7). As the conductivity σ approaches infinity the electric field inside a conductor approaches zero for any given current density J . Warm donor atoms in n-type semiconductors can be easily ionized and contribute electrons to the conduction band where they move freely. Only certain types of impurity atoms function as donors--those that are most easily ionized. As the density of donor atoms approaches zero and as temperature declines, the number of free electrons and the conductivity approach very low values that depend on temperature and any alternative ionization mechanisms that are present. In p-type semiconductors the added impurity atoms readily trap nearby free electrons to produce a negative ion; this results in a corresponding number of positively ionized semiconductor atoms that act as “holes”. As a result any free electrons typically move only short distances before they are trapped by one of these holes. Moreover, the threshold energy required to move an electron from a neutral atom to an adjacent positive ion is usually less than the available thermal energy, so such transfers occur rapidly, causing the hole to move quickly from 6 n-type semiconductors (e.g., silicon, germanium, gallium arsenide, indium phosphide, and others) are doped with a tiny percentage of donor atoms that ionize easily at temperatures of interest, releasing mobile electrons into the conduction band of the semiconductor where they can travel freely across the material until they recombine with another ionized donor atom. The conduction band is not a place; it refers instead to an energy and wave state of electrons that enables them to move freely. The conductivity of semiconductors therefore increases with temperature; they become relatively insulating at low temperatures, depending on the ionization potentials of the impurity atoms. - 42 - place to place over long distances. Thus holes are the dominant charge carriers in p-type semiconductors, whereas electrons dominate in n-type semiconductors. More broadly, semiconductors have a conduction band in which free electrons can propagate long distances; this band is separated by an energy of one or a few electron volts from the valence band in which electrons cannot move. The conduction band is not a location, it is a family of possible electron wave states. When electrons are excited from the valence band to the conduction band by some energetic process, they become free to move in response to electric fields. Semiconductor conductivity is approximately proportional to the number of free electrons or holes produced by the scarce impurity atoms, and therefore to the doping density of those impurity atoms. Easily ionized impurity atoms are the principal mechanism by which electrons enter the conduction band, and impurities that readily trap adjacent electrons are the principal mechanism by which holes enter and move in the valence band. Semiconductors are discussed further in Section 8.2.4. The current leakage processes in insulators vaguely resemble electron and hole conduction in semiconductors, and can include weak surface currents as well as bulk conduction; microscopic flaws can also increase conductivity. The conductivities of typical materials are listed in Table 2.5.1. Table 2.5.1 Nominal conductivities σ of common materials [Siemens m-1]. paraffin glass dry earth distilled water 10-14 – 10-16 10-12 10-4 – 10-5 2×10-4 sea water iron copper silver 3-5 107 5.8×107 6.14×107 In some exotic materials the conductivity is a function of direction and can be represented by the 3×3 matrix σ ; such materials are not addressed here, but Section 2.5.3 addresses similar issues in the context of permittivity ε. Some materials exhibit superconductivity, or infinite conductivity. In these materials pairs of electrons become loosely bound magnetically and move as a unit called a Cooper pair. Quantum mechanics prevents these pairs from colliding with the lattice and losing energy. Because the magnetic binding energy for these pairs involves electron spins, it is quite small. Normal conductivity returns above a threshold critical temperature at which the pairs are shaken apart, and it also returns above some threshold critical magnetic field at which the magnetic bonds coupling the electrons break as the electron spins all start to point in the same direction. Materials having critical temperatures above 77K (readily obtained in cryogenic refrigerators) are difficult to achieve. The finite number of such pairs at any temperature and magnetic field limits the current to some maximum value; moreover that current itself produces a magnetic field that can disrupt pairs. Even a few pairs can move so as to reduce electric fields to zero by shortcircuiting the normal electrons until the maximum current carrying capacity of those pairs is exceeded. If the applied fields have frequency f > 0, then the Cooper pairs behave much like collisionless electrons in a plasma and therefore the applied electric field can penetrate that plasma to its skin depth, as discussed in Section 9.8. Those penetrating electric fields interact - 43 - with a small number of normal electrons to produce tiny losses in superconductors that increase with frequency. 2.5.3 Permittivity The permittivity εo of free space is 8.854×10-12 farads/meter, where⎯D = εo E . The permittivity ε of any material deviates from εo for free space if applied electric fields induce electric dipoles in the medium; such dipoles alter the applied electric field seen by neighboring atoms. Electric fields generally distort atoms because E pulls on positively charged nuclei ( f = qE [N]) and repels the surrounding negatively charged electron clouds. The resulting small offset d of each atomic nucleus of charge +q relative to the center of its associated electron cloud produces a tiny electric dipole in each atom, as suggested in Figure 2.5.1(a). In addition, most asymmetric molecules are permanently polarized, such as H2O or NH3, and can rotate within fluids or gases to align with an applied field. Whether the dipole moments are induced, or permanent and free to rotate, the result is a complete or partial alignment of dipole moments as suggested in Figure 2.5.1(b). These polarization charges generally cancel inside the medium, as suggested in Figure 2.5.1(b), but the immobile atomic dipoles on the outside surfaces of the medium are not fully cancelled and therefore contribute the surface polarization charge ρsp. (c) (a) electron cloud - + ⎯E +q d - surface polarization charge σsp: negative positive (b) + - + - + - - dielectric nucleus - + + ε - + + metal plates + εo + ⎯E + + ε + ⎯P + + --+ + surface charge σs (d) ⎯E positive bound charge + polarized d atoms +nq [C m-3] ρp = 0 + -nq [C m-3] ⎯P medium z negative bound charge Figure 2.5.1 Polarized media. Figure 2.5.1(c) suggests how two charged plates might provide an electric field E that polarizes a dielectric slab having permittivity ε > εo. [The electric field E is the same in vacuum - 44 - as it is inside the dielectric (assuming no air gaps) because the path integral of E • ds from plate to plate equals their voltage difference V in both cases. The electric displacement vector De = εE and therefore differs.] We associate the difference between Do = εo E (vacuum) and Dε = εE (dielectric) with the electric polarization vector P , where: D = εE = εo E + P = εo E (1 + χ ) (2.5.8) The polarization vector P is defined by (2.5.8) and is normally parallel to E in the same direction, as shown in Figure 2.5.1(c); it points from the negative surface polarization charge to the positive surface polarization charge (unlike E , which points from positive charges to negative ones). As suggested in (2.5.8), P = Eεo χ , where χ is defined as the dimensionless susceptibility of the dielectric. Because nuclei are bound rather tightly to their electron clouds, χ is generally less than 3 for most atoms, although some molecules and crystals, particularly in fluid form, can exhibit much higher values. It is shown later in (2.5.13) that P simply equals the product of the number density n of these dipoles and the average vector electric dipole moment of each atom or molecule, p = qd , where d is the offset (meters) of the positive charge relative to the negative charge: P = nqd ⎡⎣C m-2 ⎤⎦ (2.5.9) Gauss’s law relates D to charge density ρ [C m-2], but we now have two types of density: that of free charges ρf, including ions and surface charges on conductors, and that of any locally un-neutralized polarization charge ρp bound within charge-neutral atoms or molecules. Gauss’s law says: ∇ • D = ρf (2.5.10) where ρf is the free charge density [C m-3]. We can derive a relation similar to (2.5.10) for P by treating materials as distributed bound positive and negative charges with vacuum between them; the net bound charge density is designated the polarization charge density ρp [C m-3]. Then in the vacuum between the charges D = ε o E and (2.5.10) becomes: ε o ∇ • E = ρf + ρ p (2.5.11) From ∇ • (2.5.8), we obtain ∇ • D = εo∇ • E + ∇ • P = ρf . Combining this with (2.5.11) yields: ∇ • P = −ρp (2.5.12) - 45 - The negative sign in (2.5.12) is consistent with the fact that P , unlike E , is directed from negative to positive polarization charge, as noted earlier. Outside a polarized dielectric the polarization P is zero, as suggested by Figure 2.5.1(d). Note that the net polarization charge density is ±nq for only an atomic-scale distance d at the boundaries of the dielectric, where we model the positive and negative charge distributions within the medium as continuous uniform rectilinear slabs of density ± nq [C m-3]. These two slabs are offset by the distance d. If P is in the z direction and arises from n dipole moments p = qd per cubic meter, where d is the offset [m] of the positive charge relative to the negative charge, then (2.5.12) can be integrated over a volume V that encloses a unit area of the left-hand face of a polarized dielectric [see Figure 2.5.1(d)] to yield the polarization vector P inside the dielectric: P = ∫ ∇ • P dv = − ∫ ρp dv = nqd V V (2.5.13) The first equality of (2.5.13) involving⎯P follows from Gauss’s divergence theorem: ∫V∇ • P dv = ∫A P • zˆ da = Pz if A = 1. Therefore, P = nqd , proving (2.5.9). When the electric displacement⎯D varies with time it becomes displacement current, ∂ D /∂t [A/m2], analogous to⎯J, as suggested by Ampere's law: ∇ ×⎯H =⎯J + ∂ D /∂t. For reference, Table 2.5.2 presents the dielectric constants ε/εo for some common materials near 1 MHz, after Von Hippel (1954). Table 2.5.2 Dielectric constants ε/εo of common materials near 1 MHz. vacuum fir wood Teflon, petroleum vaseline paper polystyrene sandy soil 1.0 1.8 – 2.0 2.1 2.16 2–3 2.55 2.59 fused quartz ice pyrex glass aluminum oxide ethyl alcohol water titaniumdioxide 3.78 4.15 5.1 8.8 24.5 81.0 100.0 Most dielectric materials are lossy because oscillatory electric fields dither the directions and magnitudes of the induced electric dipoles, and some of this motion heats the dielectric. This effect can be represented by a frequency-dependent complex permittivity ε , as discussed further in Section 9.5. Some dielectrics have direction-dependent permittivities that can be represented by ε ; such anisotropic materials are discussed in Section 9.6. Lossy anisotropic materials can be characterized by ε . Some special dielectric media are spontaneously polarized, even in the absence of an externally applied E . This occurs for highly polarizable media where orientation of one electric dipole in the media can motivate its neighbors to orient themselves similarly, forming domains - 46 - of atoms, molecules, or crystal unit cells that are all polarized the same. Such spontaneously polarized domains are illustrated for magnetic materials in Figure 2.5.2. As in the case of ferromagnetic domains, in the absence of externally applied fields, domain size is limited by the buildup of stored field energy external to the domain; adjacent domains are oriented so as to largely cancel each other. Such ferroelectric materials have large effective values of ε, although D saturates if E is sufficient to produce ~100% alignment of polarization. They can also exhibit hysteresis as do the ferromagnetic materials discussed in Section 2.5.4. Example 2.5A What are the free and polarization charge densities ρf and ρp in a medium having conductivity σ = σo (1 + z ) , permittivity ε = 3εo , and current density J = zˆJ o ? Solution: J = σE , so E = zˆJ o (1 + z ) σo = εo E + P . From (2.5.10) ρf = ∇ • D = ( ∂ ∂z ) ⎣⎡3εo J o (1 + z ) σo ⎦⎤ = 3εo J o σo From (2.5.12) ρp = −∇ • P = −∇ • ( ε − εo ) E = − ( ∂ ∂z ) 2εo J o (1 + z ) ⎡⎣ C m -3 ⎤⎦ . σo = 2εo J o σo ⎡⎣C m-3 ⎤⎦ . 2.5.4 Permeability The permeability μo of free space is 4π10-7 Henries/meter by definition, where⎯B = μ H . The permeability μ of matter includes the additional contributions to B from atomic magnetic dipoles aligning with the applied H . These magnetic dipoles and their associated magnetic fields arise either from electrons orbiting about atomic nuclei or from spinning charged particles, where such moving charge is current. All electrons and protons have spin ±1/2 in addition to any orbital spin about the nucleus, and the net spin of an atom or nucleus can be non-zero. Their magnetic fields are linked to their equivalent currents by Ampere’s law, ∇ × H = J + ∂D ∂t . Quantum theory describes how these magnetic moments are quantized at discrete levels, and for some devices quantum descriptions are necessary. In this text we average over large numbers of atoms, so that B = μH accurately characterizes media, and quantum effects can be ignored. In any medium the cumulative contribution of induced magnetic dipoles to B is characterized by the magnetization M , which is defined by: B = μH = μo ( H + M ) = μo H (1 + χm ) (2.5.14) where χm is the magnetic susceptibility of the medium. Because of quantum effects χm for diamagnetic materials is slightly negative so that μ < μo; examples include silver, copper, and water, as listed in Table 2.5.3. The table also lists representative paramagnetic materials, which have slightly positive magnetic susceptibilities, and ferromagnetic materials, which have very large susceptibilities (e.g., cobalt, etc.). - 47 - Table 2.5.3 Approximate relative permeabilities μ/μo of common materials. bismuth silver lead copper water vacuum air 0.99983 0.99998 0.999983 0.999991 0.999991 1.000000 1.0000004 aluminum cobalt nickel mild steel iron mumeltal supermalloy 1.00002 250 600 2000 5000 100,000 1,000,000 The sharp difference between normal materials with μ ≅ μo and ferromagnetic materials having μ >> μo is due to the spontaneous alignment of atomic magnetic dipoles in the same direction so as to increase the applied field, reorienting the remaining dipoles. That is, if the susceptibility of a material is above some threshold, then the atomic magnetic dipoles spontaneously align over regions of size limited by grain structure or energy considerations, as suggested in Figure 2.5.2(a and b). These regions of nearly perfect alignment are called magnetic domains. These domains are normally quite small (perhaps micron-size) so as to minimize the stored magnetic energy μH2. In this regime, if only energy considerations control domain size, then the sizes of those domains oriented in the general direction of the applied magnetic field grow as that field increases, while other domains shrink, as suggested in Figure 2.5.2(c). Since domain walls cannot easily move across grain walls, the granular structure of the material can be engineered to control magnetic properties. If domain walls move easily, the magnetic susceptability χm is large. (a) (b) ⎯H = 0 atoms or crystal unit cells magnetic domains magnetic domain (c) ⎯H > 0 ⎯M domain wall Figure 2.5.2 Magnetic domains in ferromagnetic materials. At sufficiently high magnetic fields all domains will expand to their maximum size and/or rotate in the direction of H . This corresponds to the maximum value of M and magnetic saturation. The resulting typical non-linear behavior of the magnetization curve relating B and H for ferromagnetic materials is suggested in Figure 2.5.3(a). The slope of the B vs. H curve is - 48 - ~μ near the origin and ~μo beyond saturation. If the domains resist change and dissipate energy when doing so, the hysteresis curve of Figure 2.5.3(b) results. It can be shown that the area enclosed by the figure is the energy dissipated per cycle as the applied H oscillates. (a) (b) B slope ≅ μo slope ≅ μ 0 B Br -Hc H 0 H Figure 2.5.3 Magnetization curve and hysterisis loop for a ferromagnetic material. Hard magnetic materials have large values of residual flux density Br and magnetic coercive force or coercivity Hc, as illustrated. Br corresponds to the magnetic strength B of this permanent magnet with no applied H. The magnetic energy density Wm = B • H 2 ≅ 0 inside permanent magnets because H = 0 , while Wm = μo H 2 2 [J m-3] outside. To demagnetize a permanent magnet we can apply a magnetic field H of magnitude Hc, which is the field strength necessary to drive B to zero. If we represent the magnetic dipole moment of an atom by m, where m for a current loop of magnitude I and area n̂A is n̂IA [A m]7, then it can be shown that the total magnetization M of a medium is n m [A m-1] via the same approach used to derive P = np (2.5.13) for the polarization of dielectrics; n is the number of dipoles per m3. Example 2.5B Show how the power dissipated in a hysteretic magnetic material is related to the area circled in Figure 2.5.3(b) as H oscillates. For simplicity, approximate the loop in the figure by a rectangle bounded by ±Ho and ±Bo. Solution: We seek the energy dissipated in the material by one traverse of this loop as H goes from +Ho to -Ho and back to +Ho. The energy density Wm = BH/2 when B = 0 at t = 0 is Wm = 0; Wm→BoHo/2 [J m-3] as B→Bo. As H returns to 0 while B = Bo, this energy is dissipated and cannot be recovered by an external circuit because any voltage induced in that circuit would be ∝ ∂B/∂t = 0. As H→-Ho, Wm→BoHo/2; this energy can be recovered by an external circuit later as B→0 because ∂B/∂t ≠ 0. As B→-Bo, Wm→BoHo/2, which is lost later as H→0 with ∂B/∂t = 0. The energy stored 7 n̂ is the unit vector normal to the tiny area A enclosed by the current I, using the right-hand-rule. - 49 - as H→Ho with B = -Bo is again recoverable as B→0 with H = Ho. Thus the minimum energy dissipated during one loop traverse is vBoHo[J], where v is material volume. If the drive circuits do not recapture the available energy but dissipate it, the total energy dissipated per cycle is doubled. 2.6 Boundary conditions for electromagnetic fields 2.6.1 Introduction Maxwell’s equations characterize macroscopic matter by means of its permittivity ε, permeability μ, and conductivity σ, where these properties are usually represented by scalars and can vary among media. Section 2.5 discussed media for which ε, μ, and σ are represented by matrices, complex quantities, or other means. This Section 2.6 discusses how Maxwell’s equations strongly constrain the behavior of electromagnetic fields at boundaries between two media having different properties, where these constraint equations are called boundary conditions. Section 2.6.2 discusses the boundary conditions governing field components perpendicular to the boundary, and Section 2.6.3 discusses the conditions governing the parallel field components. Section 2.6.4 then treats the special case of fields adjacent to perfect conductors. One result of these boundary conditions is that waves at boundaries are generally partially transmitted and partially reflected with directions and amplitudes that depend on the two media and the incident angles and polarizations. Static fields also generally have different amplitudes and directions on the two sides of a boundary. Some boundaries in both static and dynamic situations also possess surface charge or carry surface currents that further affect the adjacent fields. 2.6.2 Boundary conditions for perpendicular field components The boundary conditions governing the perpendicular components of E and H follow from the integral forms of Gauss’s laws: ∫∫S ( D • n̂ ) da = ∫∫∫V ρ dv (Gauss’s Law for D ) (2.6.1) ∫∫S ( B • n̂ ) da = 0 (Gauss’s Law for B ) (2.6.2) We may integrate these equations over the surface S and volume V of the thin infinitesimal pillbox illustrated in Figure 2.6.1. The pillbox is parallel to the surface and straddles it, half being on each side of the boundary. The thickness δ of the pillbox approaches zero faster than does its surface area S, where S is approximately twice the area A of the top surface of the box. - 50 - n̂ a n̂ Area A (top surface) ⎯D1,⎯B1 da δ ⎯D2,⎯B2 surface charge density σs pillbox surface S, volume V Figure 2.6.1 Elemental volume for deriving boundary conditions for perpendicular field components. Beginning with the boundary condition for the perpendicular component D⊥, we integrate Gauss’s law (2.6.1) over the pillbox to obtain: ∫∫S ( D • n̂a ) da ≅ ( D1⊥ − D2⊥ ) A = ∫∫∫V ρ dv = ρs A (2.6.3) where ρs is the surface charge density [Coulombs m-2]. The subscript s for surface charge ρs distinguishes it from the volume charge density ρ [C m-3]. The pillbox is so thin (δ → 0) that: 1) the contribution to the surface integral of the sides of the pillbox vanishes in comparison to the rest of the integral, and 2) only a surface charge q can be contained within it, where ρs = q/A = lim ρδ as the charge density ρ → ∞ and δ → 0. Thus (2.6.3) becomes D1⊥ - D2⊥ = ρs, which can be written as: n̂ • ( D1 − D2 ) = ρs (boundary condition for D⊥ ) (2.6.4) where n̂ is the unit vector normal to the boundary and points into medium 1. Thus the perpendicular component of the electric displacement vector D changes value at a boundary in accord with the surface charge density ρs. Because Gauss’s laws are the same for electric and magnetic fields, except that there are no magnetic charges, the same analysis for the magnetic flux density B in (2.6.2) yields a similar boundary condition: n̂ • ( B1 − B2 ) = 0 (boundary condition for B⊥ ) Thus the perpendicular component of B must be continuous across any boundary. - 51 - (2.6.5) 2.6.3 Boundary conditions for parallel field components The boundary conditions governing the parallel components of E and H follow from Faraday’s and Ampere’s laws: ∂ ∫C E • ds = − ∂t ∫∫A B • nˆ da ⎡ ∂D ⎤ ∫C H • ds = ∫∫A ⎢⎣J + ∂t ⎥⎦ • n̂ da (Faraday’s Law) (2.6.6) (Ampere’s Law) (2.6.7) We can integrate these equations around the elongated rectangular contour C that straddles the boundary and has infinitesimal area A, as illustrated in Figure 2.6.2. We assume the total height δ of the rectangle is much less than its length W, and circle C in a right-hand sense relative to the surface normal n̂a . nˆ ds area A surface current density Js E1, H1 δ da n̂ a W medium 1 medium 2 E2 , H2 boundary contour C around area A Figure 2.6.2 Elemental contour for deriving boundary conditions for parallel field components. Beginning with Faraday’s law, (2.6.6), we find: ∂ ∫C E • ds ≅ ( E1// − E 2 // ) W = − ∂t ∫∫A B • n̂a da → 0 (2.6.8) where the integral of B over area A approaches zero in the limit where δ approaches zero too; there can be no impulses in B . Since W ≠ 0, it follows from (2.6.8) that E1// - E2// = 0, or more generally: n̂ × ( E1 − E 2 ) = 0 (boundary condition for E // ) (2.6.9) Thus the parallel component of E must be continuous across any boundary. A similar integration of Ampere’s law, (2.6.7), under the assumption that the contour C is chosen to lie in a plane perpendicular to the surface current Js and is traversed in the right-hand sense, yields: - 52 - ∫C H • ds = ( H1// − H2 // ) W (2.6.10) = ∫∫ ⎡⎢ J + ∂D ⎤⎥ • nˆ da ⇒ ∫∫ J • nˆa da = Js W A⎣ A ∂t ⎦ where we note that the area integral of ∂D ∂t approaches zero as δ → 0, but not the integral over the surface current Js , which occupies a surface layer thin compared to δ. Thus H1// − H 2 // = Js , or more generally: n̂ × ( H1 − H 2 ) = J s (boundary condition for H // ) where n̂ is defined as pointing from medium 2 into medium 1. conducting,⎯Js = 0. (2.6.11) If the medium is non­ A simple static example illustrates how these boundary conditions generally result in fields on two sides of a boundary pointing in different directions. Consider the magnetic fields H1 and H 2 illustrated in Figure 2.6.3, where μ2 ≠ μ1, and both media are insulators so the surface current must be zero. If we are given H1 , then the magnitude and angle of H 2 are determined because H // and B⊥ are continuous across the boundary, where Bi = μi Hi . More specifically, H 2 // = H1// , and: H 2⊥ = B2⊥ μ2 = B1⊥ μ2 = μ1H1⊥ μ2 (2.6.12) H1, B1 θ1 B2 ⊥ = μ 2 H 2 ⊥ H1⊥ , B1⊥ θ2 H1// , B1// H2 H2⊥ H 2 // Figure 2.6.3 Static magnetic field directions at a boundary. It follows that: θ2 = tan −1( H 2// H 2 ⊥ ) = tan −1(μ2 H1// μ1H1⊥ ) = tan −1 ⎡⎣(μ2 μ1)tanθ1⎤⎦ - 53 - (2.6.13) Thus θ2 approaches 90 degrees when μ2 >> μ1, almost regardless of θ1, so the magnetic flux inside high permeability materials is nearly parallel to the walls and trapped inside, even when the field orientation outside the medium is nearly perpendicular to the interface. The flux escapes high-μ material best when θ1 ≅ 90°. This phenomenon is critical to the design of motors or other systems incorporating iron or nickel. If a static surface current Js flows at the boundary, then the relations between B1 and B2 are altered along with those for H1 and H 2 . Similar considerations and methods apply to static electric fields at a boundary, where any static surface charge on the boundary alters the relationship between D1 and D2 . Surface currents normally arise only in non-static or “dynamic” cases. Example 2.6A Two insulating planar dielectric slabs having ε1 and ε2 are bonded together. Slab 1 has E1 at angle θ1 to the surface normal. What are E 2 and θ2 if we assume the surface charge at the boundary ρs = 0? What are the components of E 2 if ρs ≠ 0? Solution: E // is continuous across any boundary, and if ρs = 0, then D⊥ = εi E ⊥ is continuous too, which implies E 2 ⊥ = ( ε1 ε 2 ) E1⊥ . Also, θ1 = tan −1 ( E // E1⊥ ) , and θ2 = tan −1 ( E // E 2 ⊥ ) . It follows that θ2 = tan −1 ⎡⎣( ε 2 ε1 ) tan θ1 ⎤⎦ . If ρs ≠ 0 then E // is unaffected and D 2⊥ = D1⊥ + n̂ρs so that E 2 ⊥ = D 2 ⊥ ε 2 = ( ε1 ε 2 ) E1⊥ + nˆρs ε2 . 2.6.4 Boundary conditions adjacent to perfect conductors The four boundary conditions (2.6.4), (2.6.5), (2.6.9), and (2.6.11) are simplified when one medium is a perfect conductor (σ = ∞) because electric and magnetic fields must be zero inside it. The electric field is zero because otherwise it would produce enormous J = σE so as to redistribute the charges and to neutralize that E almost instantly, with a time constant τ = ε σ seconds, as shown in Equation (4.3.3). It can also be easily shown that B is zero inside perfect conductors. Faraday’s law says ∇ × E = −∂ B ∂t , so if E = 0 , then ∂ B ∂t = 0 . If the perfect conductor were created in the absence of B then B would always remain zero inside. It has further been observed that when certain special materials become superconducting at low temperatures, as discussed in Section 2.5.2, any pre-existing⎯B is thrust outside. The boundary conditions for perfect conductors are also relevant for normal conductors because most metals have sufficient conductivity σ to enable J and ρs to cancel the incident electric field, although not instantly. As discussed in Section 4.3.1, this relaxation process by which charges move to cancel E is sufficiently fast for most metallic conductors that they - 54 - largely obey the perfect-conductor boundary conditions for most wavelengths of interest, from DC to beyond the infrared region. This relaxation time constant is τ = ε/σ seconds. One consequence of finite conductivity is that any surface current penetrates metals to some depth δ = 2 ωμσ , called the skin depth, as discussed in Section 9.2. At sufficiently low frequencies, even sea water with its limited conductivity largely obeys the perfect-conductor boundary condition. The four boundary conditions for fields adjacent to perfect conductors are presented below together with the more general boundary condition from which they follow when all fields in medium 2 are zero: nˆ • B = 0 ⎡from nˆ • ( B1 − B2 ) = 0 ⎤ ⎣ ⎦ (2.6.14) n̂ • D = ρs ⎡from nˆ • ( D1 − D2 ) = ρs ⎤ ⎦ ⎣ (2.6.15) n̂ × E = 0 ⎡from n̂ × ( E1 − E 2 ) = 0 ⎤ ⎣ ⎦ (2.6.16) n̂ × H = Js ⎡ ⎤ ( ) ⎣from n̂ × H1 − H 2 = Js ⎦ (2.6.17) These four boundary conditions state that magnetic fields can only be parallel to perfect conductors, while electric fields can only be perpendicular. Moreover, the magnetic fields are always associated with surface currents flowing in an orthogonal direction; these currents have a numerical value equal to H . The perpendicular electric fields are always associated with a surface charge ρs numerically equal to D ; the sign of σ is positive when D points away from the conductor. Example 2.6B What boundary conditions apply when μ→∞, σ = 0, and ε = εo? Solution: Inside this medium H = 0 and J = 0 because otherwise infinite energy densities, μ|H|2/2, are required; static E and B are unconstrained, however. Since ∇ × H = 0 = J + ∂D ∂t inside, dynamic E and D = 0 there too. Since H // and B⊥ are continuous across the boundary, H // = 0 and H ⊥ can be anything at the boundary. Since E // and D⊥ are continuous (let’s assume ρs = 0 if J = 0 ), static E and D are unconstrained at the boundary while dynamic E = D = 0 there because there is no dynamic electric field inside and no dynamic surface charge. Since only H ⊥ ≠ 0 at the boundary, this is non-physical and such media don’t exist. For example, there is no way to match boundary conditions for an incoming plane wave. This impasse would be avoided if σ ≠ 0, for then dynamic H // and E ⊥ could be non-zero. - 55 - 2.7 Power and energy in the time and frequency domains, Poynting theorem 2.7.1 Poynting theorem and definition of power and energy in the time domain To derive the Poynting theorem we can manipulate Maxwell’s equations to produce products of variables that have the dimensions and character of power or energy. For example, the power dissipated in a resistor is the product of its voltage and current, so the product E • J [W m-3] would be of interest. The dimensions of E and J are volts per meter and amperes per square meter, respectively. Faraday’s and Ampere’s laws are: ∇ × E = − ∂B ∂t (Faraday’s law) (2.7.1) ∇ × H = J + ∂D ∂t (Ampere’s law) (2.7.2) We can produce the product E • J and preserve symmetry in the resulting equation by taking the dot product of E with Ampere’s law and subtracting from it the dot product of H with Faraday’s law, yielding: E • ( ∇ × H ) − H • ( ∇ × E ) = E • J + E • ∂D + H • ∂B ∂t ∂t = −∇ • ( E × H ) (2.7.3) (2.7.4) where (2.7.4) is a vector identity. Equations (2.7.3) and (2.7.4) can be combined to form the Poynting theorem: ∇ • ( E × H ) + E • J + E • ∂D + H • ∂B = 0 ⎡⎣ Wm-3 ⎤⎦ ∂t ∂t (2.7.5) The dimension of E • J and every other term in this equation is W m-3. If D = εE and 2 2 B = μH , then E • ∂D ∂t = ∂ ⎡⎣ε E 2⎦⎤ ∂t and H • ∂ B ∂t = ∂ ⎡⎣μ H 2⎤⎦ ∂t . The factor of one-half arises because we are now differentiating with respect to a squared quantity rather than a single 2 2 quantity, as in (2.7.5). It follows that ε E 2 and μ H 2 have the dimension of J m-3 and represent electric and magnetic energy density, respectively, denoted by We and Wm. The product E • J can represent either power dissipation or a power source, both denoted by Pd. If J = σE , then Pd = σ E [ W m -3 ] , where σ is the conductivity of the medium, as discussed later in Section 3.1.2. To summarize: 2 Pd = J • E ⎡⎣ W m-3 ⎤⎦ (power dissipation density) - 56 - (2.7.6) 2 ⎡ We = 1 ε E ⎣ J m -3 ⎤⎦ 2 2 ⎡ Wm = 1 μ H ⎣ J m-3 ⎤⎦ 2 (electric energy density) (2.7.7) (magnetic energy density) (2.7.8) (Poynting theorem) (2.7.9) Thus we can write Poynting’s theorem in a simpler form: ∇ • ( E × H ) + E • J + ∂ ( We + Wm ) = 0 ⎡⎣ Wm-3 ⎦⎤ ∂t suggesting that the sum of the divergence of electromagnetic power associated with E × H , the density of power dissipated, and the rate of increase of energy storage density must equal zero. The physical interpretation of ∇ • ( E × H ) is best seen by applying Gauss’s divergence theorem to yield the integral form of the Poynting theorem: 1 ∫∫ A ( E × H ) • n̂ da + ∫∫∫V E • J dv + ( ∂ ∂t ) ∫∫∫V 2 ( E • D + H • B) dv = 0 [W] (2.7.10) which can also be represented as: ∂ ∫∫ A ( E × H ) • nˆ da + pd + ∂t ( w e + w m ) = 0 [W] (Poynting theorem) (2.7.11) Based on (2.7.8) and conservation of power (1.1.6) it is natural to associate E × H [W m-2] with the instantaneous power density of an electromagnetic wave characterized by E [V m-1] and H [A m-1]. This product is defined as the Poynting vector: S ≡ E × H ⎡⎣ Wm-2 ⎤⎦ (Poynting vector) (2.7.12) The instantaneous electromagnetic wave intensity of a uniform plane wave. Thus Poynting’s theorem says that the integral of the inward component of the Poynting vector over the surface of any volume V equals the sum of the power dissipated and the rate of energy storage increase inside that volume. Example 2.7A Find S ( t ) and S ( t ) for a uniform plane wave having E = x̂E o cos ( ωt − kz ) . Find the electric and magnetic energy densities We(t,z) and Wm(t,z) for the same wave; how are they related? Solution: H = ẑ × E ηo = ŷE o cos ( ωt − kz ) ηo where ηo = (μo/εo)0.5. S = E × H = ẑ Eo2cos2(ωt – kz)/ηo and S ( t ) = ẑ Eo2/2ηo [Wm-2]. We(t,z) = εoEo2 cos2 (ωt-kz)/2 [Jm-3] and Wm(t,z) = μoEo2 cos2 (ωt – kz)/2ηo2 = εoEo2 cos2 (ωt – kz)/2 = We(t,z); the electric and - 57 - magnetic energy densities vary together in space and time and are equal for a single uniform plane wave. 2.7.2 Complex Poynting theorem and definition of complex power and energy Unfortunately we cannot blindly apply to power and energy our standard conversion protocol between frequency-domain and time-domain representations because we no longer have only a single frequency present. Time-harmonic power and energy involve the products of sinusoids and therefore exhibit sum and difference frequencies. More explicitly, we cannot simply represent the Poynting vector S(t) for a field at frequency f [Hz] by Re {S e jωt } because power has components at both f = 0 and 2f Hz, where ω = 2πf. Nonetheless we can use the convenience of the time-harmonic notation by restricting it to fields, voltages, and currents while representing their products, i.e. power and energy, only by their complex averages. To understand the definitions of complex power and energy, consider the product of two sinusoids, a(t) and b(t), where: a(t) = R e {Ae jωt } = R e {Ae jα e jωt } = A cos ( ωt + α ) , b(t) = B cos ( ωt + β ) a(t)b(t) = ABcos ( ωt + α ) cos ( ωt + β ) = ( AB 2 ) ⎡⎣cos ( α − β ) + cos ( 2ωt + α + β ) ⎤⎦ (2.7.13) (2.7.14) where we used the identity cos γ cos θ = ⎣⎡cos ( γ − θ ) + cos ( γ + θ ) ⎦⎤ 2 . If we time average (2.7.14) over a full cycle, represented by the operator • , then the last term becomes zero, leaving: a(t)b(t) = 1 ABcos ( α − β ) = 1 R e {Ae jα Be− jβ } = 1 R e {AB∗} 2 2 2 (2.7.15) By treating each of the x, y, and z components separately, we can readily show that (2.7.15) can be extended to vectors: { ∗ A(t) • B(t) = 1 R e A • B 2 } (2.7.16) The time average of the Poynting vector S(t) = E(t) × H(t) is: { } ∗ S(t) = 1 R e E × H = 1 Re{S} ⎣⎡ W/m 2 ⎤⎦ (Poynting average power density) 2 2 (2.7.17) where we define the complex Poynting vector as: S = E×H ∗ ⎡⎣W m-2 ⎤⎦ (complex Poynting vector) - 58 - (2.7.18) Note that S is complex and can be purely imaginary. Its average power density is given by (2.7.17). We can re-derive Poynting’s theorem to infer the physical significance of this complex vector, starting from the complex Maxwell equations: ∇ × E = − jωB (Faraday’s law) (2.7.19) ∇ × H = J + jωD (Ampere’s law) (2.7.20) ∗ To see how time-average dissipated power, E • J 2 is related to other terms, we compute the dot product of E and the complex conjugate of Ampere’s law, and subtract it from the dot product ∗ of H and Faraday’s law to yield: ( ∗ ∗ H • (∇ × E ) − E • ∇ × H ) = − jωH∗ • B − E • J∗ + jωE • D∗ (2.7.21) Using the vector identity in (2.7.4), we obtain from (2.7.21) the differential form of the complex Poynting theorem: ( ∗ ∇• E×H ) = −E • J∗ − jω ( H∗ • B − E • D∗ ) (complex Poynting theorem) (2.7.22) The integral form of the complex Poynting theorem follows from the complex differential form as it did in the time domain, by using Gauss’s divergence theorem. The integral form of (2.7.22), analogous to (2.7.10), therefore is: ∗ ∗ ∗ ∗ ∫∫ A ( E × H ) • nˆ da + ∫∫∫V {E • J + jω ( H • B − E • D )} dv = 0 (2.7.23) We can interpret the complex Poynting theorem in terms of physical quantities using (2.7.17) and by expressing the integral form of the complex Poynting theorem as: 1 2 1 ∫∫ A S • n̂ da + 2 ∫∫∫V ⎣⎡ E • J * + 2 jω ( W m − W e ) ⎦⎤ dv = 0 (2.7.24) where the complex energy densities and time-average power density dissipated Pd are: 2 ⎡ ∗ Wm = 1 H • B = 1 μ H ⎣ J/m3 ⎤⎦ 2 2 (2.7.25) 2 ⎡ W e = 1 E • D* = 1 ε E ⎣ J/m3 ⎤⎦ 2 2 (2.7.26) - 59 - 2 Pd = 1 σ E + ∫∫∫ 2ωI m [ W e − W m ] dv ≡ ⎡⎣ W/m3 ⎤⎦ V 2 We recall that the instantaneous magnetic energy density Wm(t) is μ H (2.7.27) 2 2 [J/m3] from (2.7.8), 2 2 and that its time average is μ H 4 because its peak density is μ H 2 and it varies sinusoidally at 2f Hz. If B = μ H = (μ r + jμi )H and D = εE = (ε r + jεi )E , then (2.7.25)-(2.5.27) become: 2 ⎡ Wm (t) = 1 R e {W m } = 1 μ r H ⎣ J/m3 ⎤⎦ 4 2 We ( t) 2 ⎡ = 1 R e {W e } = 1 ε r E ⎣ J/m3 ⎤⎦ 2 4 (magnetic energy density) (2.7.28) (electric energy density) (2.7.29) 2 Pd = 1 σ E + ∫∫∫ 2ωI m [ W e − W m ] dv ≡ ⎣⎡ W/m3 ⎦⎤ (power dissipation density) (2.7.30) V 2 We can now interpret the physical significance of the complex Poynting vector S by restating the real part of (2.7.24) as the time-average quantity: p r + pd = 0 [W] (2.7.31) where the time-average total power radiated outward across the surface area A is: p r = 1 R e ∫∫ {S • n̂} da [W] A 2 (2.7.32) as also given by (2.7.17), and pd is the time-average power dissipated [W] within the volume V, as given by (2.7.27). The flux density or time-average radiated power intensity [W/m2] is therefore Pr = 0.5R e [S] . Note that the dissipated power pd can be negative if there is an external or internal source (e.g., a battery) supplying power to the volume; it is represented by negative * contribution to E • J . The imaginary part of the radiation (2.7.24) becomes: ∫∫ A Im [S• n̂] da + ∫∫∫V ⎜⎝ Im {E • J } + 4ω ⎡⎣ w m (t) ⎛ * − w e (t) ⎤⎦ ⎞⎟dv = 0 [W] ⎠ (2.7.33) The surface integral over A of the imaginary part of the Poynting vector is the reactive power, which is simply related by (2.7.33) to the difference between the average magnetic and electric energies stored in the volume V and to any reactance associated with⎯J. - 60 - 2.7.3 Power and energy in uniform plane waves Consider the +z-propagating uniform time-harmonic plane wave of (2.3.1–2), where: E ( r, t ) = xˆEo cos ( z − ct ) [ V/m] H ( r, t ) = yˆ (2.7.34) εo E cos ( z − ct ) ⎡⎣ A/m 2 ⎤⎦ μo o (2.7.35) The flux density for this wave is given by the Poynting vector S(t) (2.7.12): E o2 S(t) = E × H = zˆ cos 2 ( z − ct ) ⎡⎣ W/m 2 ⎤⎦ ηo (2.7.36) where the characteristic impedance of free space is ηo = μo /εo ohms (2.2.19). The time average of S(t) in (2.7.36) is ẑE o2 2ηo [W/m2]. The electric and magnetic energy densities for this wave can be found from (2.7.7–8): 2 We = 1 ε E = 1 εo E o2 cos 2 ( z − ct ) ⎡⎣ J/m3 ⎤⎦ 2 2 2 Wm = 1 μ H = 1 εo E o2 cos 2 ( z − ct ) ⎡⎣ J/m3 ⎤⎦ 2 2 (electric energy density) (2.7.37) (magnetic energy density) (2.7.38) Note that these two energy densities, We and Wm, are equal, non-negative, and sinusoidal in behavior at twice the spatial frequency k = 2π/λ [radians/m] of the underlying wave, where cos2(z - ct) = 0.5[1 + cos 2(z - ct)], as illustrated in Figure 2.7.1; their frequency f[Hz] is also double that of the underlying wave. They have the same space/time form as the flux density, except with a different magnitude. The complex electric and magnetic fields corresponding to (2.7.34–5) are: E ( r ) = x̂ E o [ V/m ] (2.7.39) H ( r ) = ŷ εo μo E o ⎡⎣ A/m2 ⎤⎦ (2.7.40) The real and imaginary parts of the complex power for this uniform plane wave indicate the time average and reactive powers, respectively: 2 2 ⎡ ∗ S(t) = 1 R e [S] = 1 R e ⎣⎡ E × H ⎤⎦ = ẑ 1 E o =ẑ 1 ηo H o ⎣W/m 2 ⎤⎦ 2 2 2ηo 2 - 61 - (2.7.41) { Im {S ( r )} = Im E × H ∗ }=0 (2.7.42) If two superimposed plane waves are propagating in opposite directions with the same polarization, then the imaginary part of the Poynting vector is usually non-zero. Positive reactive power flowing into a volume is generally associated with an excess of time-average magnetic energy storage over electric energy storage in that volume, and vice-versa, with negative reactive power input corresponding to excess electric energy storage. E ( r, t ) = xˆE o cos ( z − ct ) λ z S ( r, t ) = ẑ E o2 ηo cos 2 ( z − ct ) ( 0 ) z 0 Figure 2.71 Electric field, electric and magnetic storage, and wave intensity for a uniform plane wave. Example 2.7B Two equal x-polarized plane waves are propagating along the z axis in opposite directions such that E ( t ) = 0 at z = 0 for all time t. What is S ( z ) ? Solution: E ( z ) = x̂E o ( e− jkz − e+ jkz ) = 0 at z = 0; and H ( z ) = ŷ ( Eo ηo ) ( e− jkz + e+ jkz ) , so ( ) ( ) * S is pure S = E × H = ẑ E o 2 ηo ( e− j2kz − e+ j2kz ) = − z2 ˆ j E o 2 ηo sin ( 2kz ) . imaginary and varies sinusoidally along z between inductive and capacitive reactive power, with nulls at intervals of λ/2. 2.8 Uniqueness theorem Throughout this text we often implicitly assume uniqueness when we first guess the solution to Maxwell’s equations for a given set of boundary conditions and then test that solution against those equations. This process does not guarantee that the resulting solution is unique, and often there are an infinite number of possible solutions, of which we might guess only one. The - 62 - uniqueness theorem is quite useful for it sets forth constraints on the boundary conditions that guarantee there is only one solution to Maxwell’s equations, which we can find as usual. To prove the uniqueness theorem we begin by considering a volume V enclosed by surface S and governed by Maxwell’s equations: ∇ • Di = ρ ∇ • Bi = 0 (2.8.1) ∇ × Ei = − ∂Bi ∂t ∇ × Hi = J + ∂Di ∂t (2.8.2) where i = 1, 2 correspond to two possible solutions consistent with the given source distributions ρ and J . We can now show that the difference Ad = A1 − A 2 between these two solutions must be zero under certain conditions, and therefore there can then be no more than one solution: A represents D, B, E, H, or J . If we subtract (2.8.1) for i = 2 from (2.8.1) for i = 1 we obtain: ∇ • ( D1 − D 2 ) = ∇ • Dd = 0 (2.8.3) Similar subtraction of corresponding equations for (2.8.2) yield three more Maxwell’s equations that the difference fields Bd and Dd must satisfy: ∇ • Bd = 0 ∇ × Ed = − ∂Bd ∂t ∇ × Hd = ∂Dd ∂t (2.8.4) where we note that the source terms ρ and J have vanished from (2.8.3) and (2.8.4) because they are given and fixed. The boundary constraints that ensure uniqueness are: (1) At some time t = 0 the fields are known everywhere so that at that instant Ed = Dd = Hd = Bd = 0 . (2) At all times and at each point on the surface S either the tangential E or the tangential H is known. Applying Poynting’s theorem (2.7.10) to the difference fields at time t proves uniqueness subject to these constraints: ⎡ ∫∫∫V ⎢⎣Hd • ∂Bd + E • ∂Dd ⎤ dv + d ∂t ∂t ⎥⎦ ∫∫S ( Ed × Hd ) • da = 0 - 63 - (2.8.5) Boundary constraint (2) ensures that the tangential component of either E d or Hd is always zero, thus forcing the cross product in the second integral of (2.8.5) to zero everywhere on the enclosing surface S. The first integral can be simplified if D = εE and B = μH , where both ε and μ can be functions of position. Because this volume integral then involves only the time ( 2 2 ) derivative of the squares of the difference fields H d and E d , and because these fields are zero at t = 0 by virtue of constraint (1), the difference fields can never depart from zero while satisfying (2.8.5). Since (2.8.5) holds for all time, the difference fields must therefore always be zero everywhere, meaning there can be no more than one solution to Maxwell’s equations subject to the two constraints listed above. - 64 -