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ph1bA_08_q3s

# ph1bA_08_q3s - Ph1b Analytic Quiz 3 Solution Chefung Chan...

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Ph1b Analytic Quiz 3 Solution Chefung Chan, Winter 2008 Problem 1 (6 points total) A spherically symmetric charge distribution is given by ρ ( r ) = 0 ; 0 < r < a = K r ; a < r < b = 0 ; r > b where K is a constant. At the center, r = 0, there is a point charge q . (a) (2 points) Use Gauss’s Law to find the electric field E ( r ) in the region a < r < b in terms of q, K, a, and r . Solution: Consider Gauss law with a spherical gaussian surface, we have E (4 πr 2 ) = 4 πq enclosed In this case, charge enclosed has 2 components. One is the center point charge q while another one is the charge distribution. For the distribution one : q ( r ) = r a 4 πr 2 ρ ( r ) dr = r a 4 πKr dr = 2 πK ( r 2 a 2 ) E ( r ) = 1 r 2 q enclosed = 1 r 2 ( q + 2 πK ( r 2 a 2 ) ) = q r 2 + 2 πK 1 a 2 r 2 ( a < r < b ) (b) (1 point) Find the value of K in terms of q and a for which the electric field is a constant E 0 (independent of r ) in the region a < r < b Solution: From (a), E ( r ) = q r 2 + 2 πK 1 a 2 r 2 If E a<r<b = E 0 = constant, then we have dE ( r ) dr = 0 2 q r 3 + 2 πK 2 a 2 r 3 = 0 q = 2 πKa 2 K = Q 2 πa 1

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With K = q/ (2
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ph1bA_08_q3s - Ph1b Analytic Quiz 3 Solution Chefung Chan...

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