3209-2009-Solutions7

3209-2009-Solutions7 - ECE 3209 — Electromagnetic Fields...

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Unformatted text preview: ECE 3209 — Electromagnetic Fields University of Virginia Fall 2009 Homework # 7 Solutions 1. Average potential over a sphere with point charge q outside. Consider the geometry of the problem shown below: The average potential over the spherical surface is, Φ avg ≡ 1 4 πR 2 I sphere Φ da, where Φ( r ) = 1 4 π² q r is the potential at position ~ r from a point charge. From the Law of Cosines, r 2 = s 2 + R 2- 2 sR cos θ , so Φ avg = 1 4 πR 2 q 4 π² Z 2 π dφ Z π R 2 sin θ dθ √ s 2 + R 2- 2 sR cos θ = q 4 π² 1 2 sR p s 2 + R 2- 2 sR cos θ fl fl fl π = q 4 π² 1 2 sR ( p ( s + R ) 2- p ( s- R ) 2 ) Φ avg = 1 4 π² q s which is the potential due to the point charge evaluated at the center of the sphere. 2. Conductive medium between concentric spheres. (a) For a conducting medium, ~ J = σ c ~ E , where σ c is the conductivity. For a uniform conductor and steady current, ~ ∇ · ~ E = 0 , within the medium, and thus, ∇ 2 Φ = 0 . The potential between the conductors satisfies Laplace’s equation which, in spherical coordinates, is written, ∇ 2 Φ = 1 r 2 d dr ‡ r 2 d Φ dr · = 0 Since the problem is spherically symmetric and independent of the θ and φ coordinates. The region of interest does not contain the point r = 0 , so we can recast Laplace’s equation as, ∂ ∂r ‡ r 2 Φ · = 0 ⇒ r 2 d Φ dr =- C ⇒ d Φ dr =- C r 2 where C is an integration constant. Integrating this equation, we obtain Φ( r ) = C r + B where B is the second integration constant. Setting the boundary conditions to be V ( a ) = 0 and V ( b ) = V we get C a + B = 0 ⇒ B =- C a C b + B = C b- C a = C a- b ab = V ⇒ C = ab a- b V and thus the potential between the conductors is, Φ( r ) = ab a- b V ‡ 1 r- 1 a · and the electric field is, ~ E ( r ) =- ~ ∇ Φ = ˆ r d Φ dr = ab b- a V r 2 ˆ r The current density is, ~ J = σ c ~ E = σ c ‡ ab b- a · V r 2 ˆ r We find the total current by integrating over the cross-sectional area (a spherical surface)...
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This note was uploaded on 10/16/2010 for the course ECE 309 taught by Professor Weikle during the Fall '08 term at UVA.

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3209-2009-Solutions7 - ECE 3209 — Electromagnetic Fields...

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