Week 7 Solns

Week 7 Solns - Electromagnetics Workshop Week 7 Solutions...

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Electromagnetics Workshop – Week 7 Solutions Grounding and Divergence Theorem 1 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 () abcd <<< . Find the surface charge density at rb = , rc = , and at , rdd −+ = (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. The total charge for ra is given by 2 2 4 000 sin a a r Qr d d d r a ππ θ θφ ⎛⎞ = ⎜⎟ ⎝⎠ ∫∫∫ [] 4 4 0 4 1C 4 a a r Q a == ⎢⎥ The field needs to be zero inside the conducting shell ( ) brc . Hence, 2 2 1 C/m 4 b s b =− , 2 2 1 C/m 4 c s c = . The field is also zero outside of rd = due to the grounding of the outer shell. 2 2 1 C/m 4 d s d , and 0 d s + = . The fields can be solved for using Gauss’ law: 2 4 0 ˆ V/m 4 r a πε = Er , 2 0 1 ˆ V/m 4 r = , arb ≤< 0 = E , << 2 0 1 ˆ V/m 4 r = , crd 0 = E , > a b c d v
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Electromagnetics Workshop – Week 7 Solutions Grounding and Divergence Theorem 2 Problem 2: A vector field 3 ˆ ρ = D ρ exists in the region between two concentric cylindrical surfaces defined by 1 = and 2 =
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Week 7 Solns - Electromagnetics Workshop Week 7 Solutions...

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