mt1_solution

# mt1_solution - Physics 501-Fall 2011 Midterm Name...

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Physics 501—Fall 2011 Midterm Name: Instructions: This is an in-class closed book exam. You will find a set of potentially useful formulas at the back of the exam. There are three problems, whose weights are indicated. Partial credit will be given provided your work is shown and can be understood. Undergrads: Be sure to write and sign the pledge in the exam booklet: “I pledge my honor that I have not violated the Honor Code during this examination.” 1

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c a κ b 1) (30 pts) The gap between two concentric con- ducting spherical shells having radius a and c is par- tially filled by a concentric spherical shell dielectric of inner radius b and outer radius c . The dielectric constant of the material between b and c is κ . The region between a and b is vacuum. A charge q resides on the surface of the inner conducting sphere. a) Find the electric field ~ E ( ~ r ) in the region a < r < b . Solution: ~ E = q 4 π 0 r 2 ˆ r b) Find the electric field ~ E ( ~ r ) and the polarization ~ P ( ~ r ) in the region b < r < c . Solution: ~ D = q 4 πr 2 ˆ r = ~ E = 0 ~ E + ~ P = ~ P = ~ D - 0 ~ E = 1 - 1 κ ~ D = 1 - 1 κ q 4 πr 2 ˆ r c) Find the capacitance between the inner and outer conducting shells. Solution: C = q V V = Z b a ~ E · d ~ dr + Z c b ~ E · d ~ dr = q 4 π " 1 0 Z b a dr r 2 + q Z c b dr r 2 # = q 4 π 0 1 a - 1 b + 1 κ 1 b - 1 c = C - 1 = 1 4 π 0 1 a - 1 b + 1 κ 1 b - 1 c 2) (30 pts) A azimuthally symmetric charge distribution has charge density in spherical
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