Winter 2008 Lecture 5 Page 1 of 5 Lecture 5Multiple Charge Distributions In this lecture we will expand on the previous discussion of Gauss’ law and investigate the analysis of electric fields due to multiple charge distributions and how Gauss’ law may be applied to very small structures. The first geometry we will investigate is the co-axial structure. The coaxial structure consists of two concentric circular based cylinders. The inner cylinder has an outer radius “a”. the outer cylinder is hollow with inner radius “b” and outer radius “c”. Both cylinders have length “L”. Example: Given the following co-axial structure: The inner cylinder (29a≤≤r0has a surface charge SArand the outer cylinder (29cb≤≤rhas a volume charge density ofVBr. The cylinders will be treated as infinite length structures. The total electric field outside the outer conductor is zero. Compute the electric field as a function of radial distance from the center of the inner conductor. Solution:In this case we will treat the inner conductor as hollow (only charge on the surface) in terms of charge distribution. The outer cylinder will have thickness that contains charge so the outer cylinder has a charge distribution. Region 1: a≤≤r0In this region there is no enclosed charge so that the flux density is zero. 0=⋅=Ψ=∫∫SeenclSdDQRegion 2: ba≤≤rIn this region the enclosed charge is SAenclaLQrp2=. The flux is computed using LDdzdDSdDLSeprfrrpr2020==⋅=Ψ∫ ∫∫∫. This will allow us to determine the electric field as: errrSAaaEˆ=. Region 3 cb≤≤rThe enclosed charge includes the first surface charge and a portion of the second (volume) charge. (29VBSALbVBSAenclLbaLdzddaLQrprrpfrrrrpp r22020''22-+=+=∫ ∫ ∫
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