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Chapter.5.fall09.ver2

Chapter.5.fall09.ver2 - Chapter 5 Cylindrical Cavities and...

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Chapter 5 Cylindrical Cavities and Waveguides We shall consider an electromagnetic field propagating inside a hollow (in the present case cylindrical) conductor. There are no sources inside the conductor, but we shall assume the material is isotropic with electric permittivity , and magnetic permeability, . The speed of the propagating wave is 1 /  The direction of propagation will be along the cylindrical axis which is the z ̂ direction We shall assume that E ( r , t ) = E ( r ) e i t and B ( r , t ) = B ( r ) e i t . Maxwell’s equations give: [∇ 2  2 t 2 ] E ( r . t ) = 0 [∇ 2  2 t 2 ] B ( r . t ) = 0 [∇ 2 +  2 ] B ( r ) = 0 [∇ 2 +  2 ] E ( r ) = 0 5.1 5.2 5.3 5.4 Since the wave is propagating along the z ̂ direction we shall further assume that: ∇ × E = i B ( r ) ; ∇ × B = − i  E ( r ) . 5.5 5.6 Since the wave is propagating along the z ̂ direction we shall further assume that: E ( r ) = E ( x , y ) e ± ikz B ( r ) = B ( x , y ) e ± ikz 5.7a 5.7b Thus Eq. (5.3 and 5.4) become [∇ t 2 +  2 k 2 ] E ( x , y ) = 0 [∇ t 2 +  2 k 2 ] B ( x , y ) = 0 5.8b 5.8a where
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