Electromagnetic Field Theory: A Problem Solving Approach Part 30. Electromagnetic field theory is often the least popular course in the electrical engineering curriculum. Heavy reliance on vector and integral calculus can obscure physical phenomena so that the student becomes bogged down in the mathematics and loses sight of the applications. This book instills problem solving confidence by teaching through the use of a large number of worked problems. To keep the subject exciting, many of these problems are based on physical processes, devices, and models. This text is an introductory treatment on the junior level for a two-semester electrical engineering. | Boundary Value Problems in Cartesian Geometries 265 Figure 4-3 The exponential and hyperbolic functions for positive and negative arguments. solution because Laplace s equation is linear. The values of the coefficients and of k are determined by boundary conditions. When regions of space are of infinite extent in the x direction it is often convenient to use the exponential solutions in 21 as it is obvious which solutions decay as x approaches oo. For regions of finite extent it is usually more convenient to use the hyperbolic expressions of 20 . A general property of Laplace solutions are that they are oscillatory in one direction and decay in the perpendicular direction. 4-2-4 Spatially Periodic Excitation A sheet in the x 0 plane has the imposed periodic potential V Vo sin ay shown in Figure 4-4. In order to meet this boundary condition we use the solution of 21 with k a. The potential must remain finite far away from the source so 266 Electric Fứld Boundary Value Problems Figure 4-4 The potential and electric field decay away from an infinite sheet with imposed spatially periodic voltage. The field lines emanate from positive surface charge on the sheet and terminate on negative surface charge. we write the solution separately for positive and negative x as V I x fl 22 I Vo sin aye x 0 where we picked the amplitude coefficients to be continuous and match the excitation at x 0. The electric field is then E -W -V ae X COSa3 i _Sina ix 1 X 23 I Voa cos ayi sin ayix x 0 The surface charge density on the sheet is given by the discontinuity in normal component of D across the sheet Tf x 0 e x x 0 Ex x 0_ 2eVoasinay 24 Boundary Value Problems in Cartesian Geometries 267 The field lines drawn in Figure 4-4 obey the equation dy E x 0 T cot ay cos ay e const dx Ex I x 0 25 4 2-5 Rectangular Harmonics When excitations are not sinusoidally periodic in space they can be made so by expressing them in terms of a trigonometric Fourier series. Any periodic function of y can be
