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Week 6 Solns

# Week 6 Solns - Electromagnetics Workshop Week 6 Solutions...

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Electromagnetics Workshop – Week 6 Solutions Grounding and Divergence Theorem Problem 1: Concentric spheres are placed in the configuration below. A solid ball (radius a ) of volume charge 3 4 C/m v r a ρ π = is surrounded by a neutral spherical metal shell of finite thickness ( c-b ) and an infinitesimally thin metallic shell of radius d which is grounded ( 29 a b c d < < < . Find the surface charge density at r b = , r c = , and at , r d d - + = (just inside of and just outside of the outermost shell, respectively). Once the charge distribution is known throughout the geometry solve for the electric field in all regions of space. Error: Reference source not found The total charge for r a is given by ( 29 2 2 4 0 0 0 sin a a r Q r d d dr a π π θ θ φ π = ÷ ∫ ∫ ∫ [ ] 4 4 0 4 1 C 4 a a r Q a π π = = The field needs to be zero inside the conducting shell ( 29 b r c . Hence, 2 2 1 C/m 4 b s b ρ π = - , 2 2 1 C/m 4 c s c ρ π = . The field is also zero outside of r d = due to the grounding of the outer shell. 2 2 1 C/m 4 d s d ρ π - = - , and 0 d s ρ + = . The fields can be solved for using Gauss’ law: [ ] 2 4 0 ˆ V/m 4 r a π ε = E r , r a [ ] 2 0 1 ˆ V/m 4 r π ε = E r , a r b < 0 = E , b r c < < [ ] 2 0 1 ˆ V/m 4 r π ε = E r , c r d < < 0 = E , r d 1 a b c d v ρ

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Electromagnetics Workshop – Week 6 Solutions Grounding and Divergence Theorem Problem 2:
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Week 6 Solns - Electromagnetics Workshop Week 6 Solutions...

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