Electromagnetic Field Theory: A Problem Solving Approach Part 67 pot

... 8-29 Thus, although Ak and k, are real, k can be either pure real or pure imaginary. A real value of k. represents power flow down the waveguide in the z direction. An imaginary ... direction. An imaginary value of k, means exponential decay with no time-average power flow. The transition from propagating waves (kh real) to evanes- cence (k,...

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... so that in later chapters most of our attention can be devoted to the applications of the mathematics rather than to its development. Additional mathematical material will ... ARRAYS 681 9.3.1 A Simple Two Element Array 681 (a) Broadside Array 683 (b) End-fire Array 685 (c) Arbitrary Current Phase 685 9.3.2 An N Dipole Array 685 9.4 LONG DIPOLE ANTEN...

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... Vector Algebra 9 scalar. If the scalar is negative, the direction of the vector is reversed: aA = aAi. + aA,i, + aA i, 1-2-3 Addition and Subtraction The sum of ... the area of the parallelogram formed with A and B as adjacent sides. Interchanging the order of A and B reverses the sign of the cross product: AxB= -BxA AxB BxA=-AxB Figure ......

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... the radial and 0 directions, but sym- metrically located charge elements at -0 have equal field magnitude components that add radially but cancel in the 0 direction. Realizing ... surface charge show that the electric field is purely radial. (b) Gauss's law applied to concentric cylindrical -surfaces shows that the field inside the surface...

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... their centers a distance D apart as in Figure 2-26. We place a line charge A a distance b, from the center of cylinder 1 and a line charge -A a distance b 2 from the ... 0 plane. If a conductor were placed along the x = 0 plane with a single line charge A at x = -a, the potential and electric field for x < 0 is the...

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... potential must approach that of an isolated point charge. Note that for small r the potential becomes very large and the small poten- tial approximation is violated. (c) ... generalize the results of Section 2.4.6 to include dielec- tric media by again choosing a small Gaussian surface whose upper and lower surfaces of area dS are parallel .....

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... Vo (33) (iwsE 2 + 2 ) (iaEl +1) [b(al +jcee) +a( o'2+jWae 2 )] which gives the interfacial surface charge amplitude as °' = e 2 • 2 I1I = (34) [b (a 1 +jOai1) +a( a-2 +jW6 2 )] As the frequency ... that the dielectric can be modeled as a parallel resistor-capacitor combination, edAz 1 s Cz =z s GAz odAz C is the capacitance per unit length and...

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... charge density on the upper electrode is then ev or(f a) = -eE4.(0 = a) =- ra with total charge per unit length E ev b A( ,b=ea)= ofrT( 4a= a)dr=-Eln Ja a a so that the capacitance ... r 0 a b Figure 4-7 Two conducting planes at angle a stressed by a voltage v have a 4-directed electric field. · 270 Electric Field Boundary Value Problems...

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... Problems 305 These relations are known as the Cauchy-Riemann equations and u and v are called conjugate functions. (c) Show that both u and v obey Laplace's equation. (d) ... surface charge at (c) What is the resistance and capacitance? 10. The potential on an infinitely long cylinder is constrained to be V(r = a) = Vo sin n6 -Vo/2 Vo/2...

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... variable in (44), (48) integrates to M=- mN -auedu 2a sinh aa -mN N Se"(u - 1) •, 2a sinh a -mN [e-"( -a - 1)- e" (a - 1)] 2a sinh a -mN - [ -a cosh a+ sinh a] a sinh ... the magnetization saturates at M = mN as all the dipoles have their moments aligned with the field. At room temperature, a is typically very small. Usin...

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