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8 - Electromagnetics Workshop Week 8 Solutions...

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Electromagnetics Workshop – Week 8 Solutions Electrostatic Potential and Divergence Theorem Problem 1: Given the following electric field, 10 ˆ [V/m], in [m] ρ ρ = , find the potential difference between the points a ( 29 5 [m], 4 ρ φ π = = and b ( 29 5 [m], 0 ρ φ = = . Use point b as the reference. Solution: [ ] 4 0 10 5 ˆ ˆ V 2 a ab b d d π π ρ φ ρ Φ = - × = - × = - E lφ φ 1
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Electromagnetics Workshop – Week 8 Solutions Electrostatic Potential and Divergence Theorem Problem 2: Spherical charge distribution: A spherical charge distribution exists in free space in the region 0 r a given by ( 29 2 0 2 1 r r a ρ ρ = - ÷ a) Find the total charge. b) Determine E everywhere. c) Determine Φ everywhere (assume Φ (0)=0). Solution: a) 2 3 2 4 2 2 0 0 0 2 2 0 0 0 0 8 1 sin 4 15 a a v V a r r Q dV r dr d d r dr a a π π π ρ ρ ρ θ θ φ π ρ = = - = - = ÷ ÷ ∫ ∫∫ b) ˆ encl S dS Q × = ∫∫ D n Ò 3 5 2 0 2 4 4 3 5 r r r D r a π π ρ = - ÷ 2 0 2 0 1 3 5 r r r
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