13 - Electromagnetics Workshop – Week 13 Solutions...

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Unformatted text preview: Electromagnetics Workshop – Week 13 Solutions Boundary Conditions & Review Problem 1: Given the potential, 20 cos ρ φ Φ = for 2 ρ < and ( 29 50 cos 120 cos ρ φ ρ φ Φ =- for 2 ρ : a) Find E in both regions. b) Show that the boundary conditions are satisfied at the surface 2 ρ = for all φ if ε ε = for 2 ρ and r ε ε ε = for 2 ρ < . Find r ε . Problem 1 Solution: For cylindrical coordinates: 1 ˆ ˆ ˆ z ρ ρ φ ∂Φ ∂Φ ∂Φ = -∇Φ = - + + ÷ ∂ ∂ ∂ Eρ φ z ( 29 ( 29 ( 29 ( 29 1 ˆ ˆ ˆ ˆ 20cos 20 sin 20cos 20sin , 2 φ ρ φ φ φ ρ ρ = -- = - + < ÷ Eρ φ ρ φ ( 29 ( 29 ( 29 ( 29 2 2 ˆ ˆ cos 50 120 sin 50 120 , 2 φ ρ φ ρ ρ = - + +- Eρ φ At 2 ρ = , the tangential components of the electric field should be equal: ( 29 ( 29 ( 29 2 2 20sin sin 50 120 ρ φ φ ρ = =- The tangential electric field components are equal for all φ at the surface 2 ρ = ....
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13 - Electromagnetics Workshop – Week 13 Solutions...

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