Advanced Engineering Electromagnetics (Second Edition) by C A Balanis PDF
By C A Balanis
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Additional info for Advanced Engineering Electromagnetics (Second Edition)
Similar equations can be derived for time-harmonic electromagnetic ﬁelds using the complex spatial forms of the ﬁeld vectors. Before we attempt this, let us ﬁrst rewrite the instantaneous Poynting vector in terms of the complex ﬁeld vectors. M. TIME-VARYING AND TIME-HARMONIC ELECTROMAGNETIC FIELDS where the asterisk (*) indicates complex conjugate. Substituting (1-67a) and (1-67b) into (1-66), we have that = Ᏹ × Ᏼ = 12 (Ee j ωt + E∗ e −j ωt ) × 12 (He j ωt + H∗ e −j ωt ) = 1 2 1 [E 2 × H∗ + E∗ × H] + 12 [E × He j 2ωt + E∗ × H∗ e −j 2ωt ] = 1 2 1 [E 2 × H∗ + (E × H∗ )∗ ] + 12 [E × He j 2ωt + (E × He j 2ωt )∗ ] (1-68) Using the equalities (1-67a) or (1-67b) in reverse order, we can write (1-68) as = 12 [Re(E × H∗ ) + Re(E × He j 2ωt )] (1-69) Since both E and H are not functions of time and the time variations of the second term are twice the frequency of the ﬁeld vectors, the time-average Poynting vector (average power density) over one period is equal to av = S = 12 Re[E × H∗ ] (1-70) Since E × H∗ is, in general, complex and the real part of E × H∗ represents the real part of the power density, what does the imaginary part represent?
G. Balmain, Electromagnetic Waves and Radiating Systems, Second Edition, Prentice-Hall, Englewood Cliffs, NJ, 1968. 33. R. F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, New York, 1961. 34. J. R. Wait, Electromagnetic Wave Theory, Harper & Row, New York, 1985. 35. J. A. Kong, Theory of Electromagnetic Waves, John Wiley & Sons, New York, 1975. 36. C. C. Johnson, Field and Wave Electrodynamics, McGraw-Hill, New York, 1965. 37. J. D. Jackson, Classical Electrodynamics, Third Edition, John Wiley & Sons, New York, 1999.
Each side of the cube is 1 m long. 4. 56) to the right of the interface. 2 C/m2 exists along the entire interface. tex V3 - 11/22/2011 PROBLEMS Determine the corresponding vector: (a) Electric ﬁeld intensity inside the polystyrene. (b) Electric polarization vector inside the polystyrene. (c) Electric ﬂux density inside the air medium. (d) Electric ﬁeld intensity inside the air medium. (e) Electric polarization vector inside the air medium. Leave your answers in terms of εo , μo . 5. 6. A static electric ﬁeld of intensity/strength Eo is established inside a free-space medium as shown below.
Advanced Engineering Electromagnetics (Second Edition) by C A Balanis