Week 10 Solns

Week 10 Solns - Electromagnetics Workshop Week 10 Solutions...

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Electromagnetics Workshop – Week 10 Solutions Boundary Conditions & Review 1 Problem 1: Given the potential, 20 cos ρ φ Φ= for 2 < and ( ) 50 cos 120 cos φρ for 2 > : a) Find E in both regions. b) Show that the boundary conditions are satisfied at the surface 2 = for all if 0 ε = for 2 > and 0 r εε = for 2 < . Find r . Problem 1 Solution: For cylindrical coordinates: 1 ˆˆ ˆ z ρρ ⎛⎞ ∂Φ ∂Φ ∂Φ =−∇Φ=− + + ⎜⎟ ∂∂ ⎝⎠ E ρφ z () () () () 1 ˆ ˆ 20cos 20 sin 20sin , 2 =− + < E ρ φ ( ) 22 cos 50 120 sin 50 120 , 2 + + > E At 2 = , the tangential components of the electric field should be equal: () () 2 2 20sin sin 50 120 φφ = The tangential electric field components are equal for all at the surface 2 = . At 2 = , the normal components of the electric flux density should be equal: 2 2 cos 50 120 r φε = −= + The normal electric flux density components are equal for all at the surface 2 = if 4 r = .

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Electromagnetics Workshop – Week 10 Solutions Boundary Conditions & Review 2 Problem 2: The uniform fields shown in the figure below are near a dielectric-dielectric boundary but on opposite sides of it. Which configurations are correct? Assume that the boundary is charge free and that 21 ε > .
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Week 10 Solns - Electromagnetics Workshop Week 10 Solutions...

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