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Solution to quiz8

# Solution to quiz8 - 1&3 ⇒ A l = 0,B l = 0 2 ⇒ E l =...

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ENEE 380 Fall 2011 October 21, 2011 Discussion Quiz 8 1. A grounded spherical conducting shell of radius R 1 is enclosed by a charged thin spherical shell of radius R 2 with surface charge density cos o . The spherical shells are concentric. a) Write the general solution of Laplace’s equation for the electric potential in all regions. b) Determine the boundary conditions. c) Using the boundary conditions in part (b), determine the potential in all regions. d) Determine the induced surface charge density on the grounded conducting shell.

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a) General solutions V 1 ( r, θ ) = X l =0 ( A l r l + B l r l +1 ) P l (cos θ ) V 2 ( r, θ ) = X l =0 ( C l r l + D l r l +1 ) P l (cos θ ) V 3 ( r, θ ) = X l =0 ( E l r l + F l r l +1 ) P l (cos θ ) b) Boundary conditions 1. V 1 (0) finite 2. V 3 ( ) = 0 3. V 1 ( R 1 ) = 0 4. V 2 ( R 1 ) = 0 5. V 2 ( R 2 ) = V 3 ( R 2 ) 6. ∂r V 2 ( R 2 ) - ∂r V 3 ( R 2 ) = σ 0 cos θ/ε
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Unformatted text preview: 1&3 ⇒ A l = 0 ,B l = 0 2 ⇒ E l = 0 4 ⇒ for all l C l R l 1 + D l R l +1 1 = 0 5 ⇒ for all l C l R l 2 + D l R l +1 2 = F l R l +1 2 6 ⇒ for all l 6 = 1 lC l R l-1 2-( l + 1) D l R l +2 2 + ( l + 1) F l R l +2 2 = 0 for l = 1 C 1-2 D 1 R 3 2 + 2 F 1 R 3 2 = σ ε We have for all l 6 = 1, C l = D l = F l = 0 and C 1 = σ 3 ε 1 D 1 =-σ R 3 1 3 ε F 1 = σ ( R 3 2-R 3 1 ) 3 ε So that the potential in all three regions are V 1 = 0 V 2 = σ 3 ε ± r-R 3 1 r 2 ² cos θ V 2 = σ 3 ε ± R 3 2-R 3 1 r 2 ² cos θ d) ∂ ∂r V 1 ( R 1 )-∂ ∂r V 2 ( R 1 ) = σ ind ε ⇒ σ ind =-σ cos θ . 2...
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Solution to quiz8 - 1&3 ⇒ A l = 0,B l = 0 2 ⇒ E l =...

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