ABSTRACT
Maxwell’s equations are vector PDEs coupling the electric and magnetic fields (read Chapter 1 for a quick review of Maxwell’s equations). The independent variables are the three spatial coordinates and the time variable. For time-harmonic fields where we assume the time variation as harmonic (cosinusoidal), one can use the phasor concepts and replace ∂/∂t in the time domain with multiplication (by jω) in the phasor domain. Thus, one can reduce the dimensionality of the problem from four independent variables to three, thus reducing the mathematical complexity of the problem. Further reduction in the dimensionality of the problem is possible for certain applications. Let us review the waveguide problem (see Chapter 3) from this viewpoint. If we consider the TM modes of a rectangular waveguide E ˜ z ≠ 0 , H ˜ z = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/eqi15_1.jpg"/> , the longitudinal component satisfies the scalar Helmholtz equation ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 E ˜ z + k 2 E ˜ z = 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/eqn15_1.jpg"/> k 2 = ω 2 μ ε . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/eqn15_2.jpg"/>
