au10_312ps3_sol

au10_312ps3_sol - ECE312 Autumn 2010 Problem Set 3...

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Unformatted text preview: ECE312 Autumn 2010: Problem Set 3 Solutions Problem 1 : Maxwell’s equations in free space with J = ρ V = 0: (i) ∇× E = − μ ∂ H ∂t , (ii) ∇· E = 0 , (iii) ∇× H = ǫ ∂ E ∂t , (iv) ∇· H = 0 Substitute given solutions into these equations to see if they are satisfied. (a) E = ˆ x cos (2 t + 2 z/c ) (i) ∇ × E = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ˆ x ˆ y ˆ z ∂ ∂x ∂ ∂y ∂ ∂z cos (2 t + 2 z/c ) vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = +ˆ y ∂ ∂z cos parenleftbigg 2 t + 2 z c parenrightbigg = − ˆ y 2 c sin parenleftbigg 2 t + 2 z c parenrightbigg ⇒ − μ ∂ H ∂t H = ˆ y 2 μ c integraldisplay t-∞ dτ sin parenleftbigg 2 τ + 2 z c parenrightbigg = − ˆ y 1 η cos parenleftbigg 2 t + 2 z c parenrightbigg where η = radicalbig μ /ǫ . (ii) ∇ · E = ∂E x ∂x + ∂E y ∂y + ∂E z ∂z = ∂ ∂x cos parenleftbigg 5 t + 2 z c parenrightbigg = 0 (OK) (iii) ∇ × H = 1 η vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ˆ x ˆ y ˆ z ∂ ∂x ∂ ∂y ∂ ∂z − cos (2 t + 2 z/c ) vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = 1 η ˆ x ∂ ∂z cos parenleftbigg 2 t + 2 z c parenrightbigg = − ˆ x 2 ǫ sin parenleftbigg 2 t + 2 z c parenrightbigg = ǫ ∂ E ∂t (OK) (iv) ∇ · H = ∂H y /∂y = 0 (OK) The field (a) satisfies Maxwell’s equations.Maxwell’s equations....
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This note was uploaded on 01/11/2012 for the course ECE 312 taught by Professor Johnson,j during the Fall '08 term at Ohio State.

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au10_312ps3_sol - ECE312 Autumn 2010 Problem Set 3...

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