problem24_78

University Physics with Modern Physics with Mastering Physics (11th Edition)

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24.78: a) For a normal spherical capacitor: ( 29 . 4 0 0 a b b a r r r r πε C - = Here we have, in effect, two parallel capacitors, L C and U C . - = = a b b a L r r r r K KC C 0 0 2 2 ε π and . 2 2 0 0 - = = a b b a U r r r r C C πε b) Using a hemispherical Gaussian surface for each respective half: 2 0 0 2 2 2 4 r πKε Q E Q πr E L L L L = = and . 2 2 4 2 0 0 2 r πε Q E ε Q πr E U U U U = = But L L VC Q = and . , Q Q Q VC Q U L U U = + = So: K Q Q Q K Q KQ K VC Q U U U L + = = + = = 1 ) 1 ( 2 0 and + = K KQ Q L 1 . 2 0 2 0 4 1 2 2 1 1 r πε Q K r πKε K KQ E L + = + = and . 4 1 2 2 1 1 2 0 2 0 r πKε Q K r πKε K Q E U + = + = c) The free charge density on upper and lower hemispheres are: ) 1 ( 4 4 ) ( 2 2 K πr Q πr Q σ a a U U f a r + = = and . ) 1 ( 4 4 ) ( 2 2 K πr Q πr Q σ b b u U f a r + = = ) 1 ( 4 4 ) ( 2 2 K πr KQ πr Q σ a a L L f s r + = = and . ) 1 ( 4 4 ) ( 2 2 K πr KQ πr Q σ b b L L f a r + = = d) . 4 1 1 1 4 ) 1 ( ) 1 1 ( 2 2 a a f i πr Q K
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