# FinalExam_EM_B_Jul_2007_Answers - Θ µα 1(α σa2 4π 0...

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Θ μα 1 ( α ) Φ( r, θ, φ ) = σa 2 4 π 0 2 π φ =0 π/ 2 θ =0 sin θ R ( r, θ, φ, a, θ , φ ) dθ dφ R ( r, θ, φ, a, θ , φ ) = [ r 2 + a 2 - 2 ra (cos θ cos θ + sin θ sin θ cos( φ 0 - φ ))] 1 / 2 ( β ) Φ( z ) = σa 2 2 0 z 2 + a 2 - | z - a | az z > 0 σa 2 2 0 z 2 + a 2 - | z + a | - az z < 0 ( γ ) Φ( z = a ) - Φ( z = 0) = σa 2 0 [ 2 - 1] Θ μα 2 ( α ) Φ 1 = 2 V π 1 1 + σ 1 2 φ + V - 1 + σ 1 2 1 + σ 1 2 , π 2 < φ < π Φ 2 = 2 V π σ 1 2 1 + σ 1 2 φ, 0 < φ < π 2 ( β ) R = π (1 + σ 1 2 ) 2 σ 1 d ln( a/b ) Θ μα 3 ( α ) K = r T σω ˆ i φ ( β ) H = σω 4 π 2 π φ =0 a r T =0 r 2 T R 2 ( ˆ i φ × ˆ i R ) dr T dφ , R = [ r 2 + r 2 T - 2 rr T sin θ cos( φ - φ )] 1 / 2 , ˆ i R = 1 R [( r sin θ cos φ - r T cos φ ) ˆ i x + ( r sin θ sin φ - r T sin φ ) ˆ i y + r cos θ ˆ i z ] , ˆ i φ = - sin φ ˆ i x + cos φ ˆ i y

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( γ ) H (0 , 0 , z ) = ˆ i z σω 2 a 2 + 2 z 2 z 2 + a 2 - 2 | z | ( δ ) H (0 , 0 , z ) = ˆ i z ρaω 2 π θ =0
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