P24_041 - q enc = 4 Z r o dr r 2 = 4 s R r 4 o 4 ....

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41. (a) We integrate the volume charge density over the volume and require the result be equal to the total charge: Z dx Z dy Z dz ρ =4 π Z R 0 dr r 2 ρ = Q. Substituting the expression ρ = ρ s r/R and performing the integration leads to 4 π ³ ρ s R ´ µ R 4 4 = Q = Q = πρ s R 3 . (b) At a certain point within the sphere, at some distance r o from the center, the Feld (see Eq. 24-8 through Eq. 24-10) is given by Gauss’ law: E = 1 4 πε 0 q enc r 2 o where q enc is given by an integral similar to that worked in part (a):
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Unformatted text preview: q enc = 4 Z r o dr r 2 = 4 s R r 4 o 4 . Therefore, E = 1 4 s r 4 o R r 2 o which (using the relation between s and Q derived in part (a)) becomes E = 1 4 Q R 3 r 2 o R and simpliFes to the desired result (shown in the problem statement) if we change notation r o r ....
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