HW-571-5 - R one conducting one having a uniform charge...

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1 Phys 571 , Fall 2010 Assignment #5 Due: Nov 3rd, 2010 1. Prove Green’s reciprocation theorem: If φ is the potential due to a volume-charge density ρ within volume V and a surface-charge density σ on the conducting surface S bounding the volume V , while φ 0 is the potential due to another distributions ρ 0 and σ 0 , then Z V ρφ 0 dV + φ 0 Q = Z V ρ 0 φdV + φQ 0 , where Q = R S σda . 2. The time-averaged potential of a neutral hydrogen atom is given by φ = e e - αr r ± 1 + αr 2 ² where e is the magnitude of the electronic charge, and α = 2 /a 0 , a 0 being the Bohr radius. Find the distribution of charge (both continuous and discrete) that will give this potential and interpret your result physically. 3. Each of three charged spheres of radius
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Unformatted text preview: R , one conducting, one having a uniform charge density within its volume, and one having a spherically symmetric charge density that varies radially as r n ( n >-3), has a total charge Q . Use Gauss’s theorem to obtain the electric fields both inside and outside each sphere. Sketch the behavior of the fields as a function of radius for the first two spheres, and for the third with n =-2 , +2. 4. Prove that the following charge distribution ρ = q n Y i =1 ( a i · ∇ ) δ ( r ) creates the potential φ ( r ) = q n Y i =1 ( a i · ∇ ) 1 r...
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