Physics 417G : Solutions for Problem set 3 Due : February 5, 2016 Please show all the details of your computations including intermediate steps.

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Problem 1

A thick spherical shell of inner radius a and outer radius b carries a polarization P~ (~r) = k~r, where k is a constant and ~r is the vector from the center. a) Calculate the surface bound charges σa at r = a and σb at r = b and ρ inside the thick spherical shell. b) What is the total charge? c) Find the electric fields in the three regions r < a, a < r < b and r > b. Sol: a) The bound charges are given by σ = P~ · n ˆ with the unit vector n ˆ outward normal to the surface ~ P~ (~r). Thus and ρ = −∇ σa = P~ · (−ˆ r) = −ka , σb = P~ · (ˆ r) = kb ,
~ · P~ (~r) = − 1 ∂r r2 kr = −3k . ρ = −∇ r2 b) The total charge should be 0. This can be checked by adding all the charges   4πa3 4πb3 2 2 − ρ=0. Qtotal = 4πa σa + 4πb σb + 3 3 H ~ · dS ~ = Qencl /0 and the charges, we can get c) Using the Gauss law E  ~ =0,  E r