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HW8_Solution

# HW8_Solution - ECE 201A Homework 8 solution 1(a Maxwells...

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ECE 201A Homework 8 solution 1. (a) Maxwell’s equations in a uniform, source free and isotropic dielectric medium are 0 E j H ωμ ∇× = − G G H j E ωε ∇× = G G 0 H = G 0 E ∇ • = G N ( ) ( ) 2 0 0 0 E E E j H j j E ωμ ωμ ωε ∇×∇× = ∇ ∇ • − ∇ = − ∇× = − G G G G G 2 2 0 0 E E ω μ ε + = G G (b) ˆ T z a z ∇ = ∇ + , where ˆ ˆ T x y a a x y = + . Hence 2 2 2 2 T z = ∇ •∇ = ∇ + . If ( ) , j z E E x y e β = G G , ( ) ( ) 2 2 2 2 2 2 2 2 2 , , T T T j z j z E E x y e E E x y e E E z z β β β = + = ∇ + = ∇ G G G G G G Substituting this result in to the wave equation 2 2 2 2 2 0 0 0 T E E E E E ω μ ε β ω μ ε + = ∇ + = G G G G G , or ( ) 2 2 2 0 0 T E E ω μ ε β + = G G . (c) For a TEM wave T E E = G G and T H H = G G , i.e., fields have only x and y components. Then 0 E j H ωμ ∇× = − G G becomes 0 ˆ T z T T a E j H z ωμ + × = − G G 0 z directed x and y directed x and y directed ˆ T T T z T E E a j H z ωμ × + × = − G G G ±²³²´ ±²³²´ ±²³²´ Hence for the equality to hold the z component of this vector equation should vanish. Therefore z directed 0 T T E × = G ±²³²´ . Hence T E G can be taken as the gradient of a scalar function since ( ) 0 T T φ × ∇ . Then

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