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Charge is distributed inside a sphere of radius R with volume charge density rho(r) = (A/r²) * e^(-2r/a), where A and a are constants. If Q is the total charge contained in the sphere, find R.

1. (a/2) log(1 - Q/(2 pi a A))
2. a log(1 - Q/(2 pi a A))
3. a log(1/(1 - Q/(2 pi a A)))
4. (a/2) log(1/(1 - Q/(2 pi a A)))

Correct answer: (a/2) log(1/(1 - Q/(2 pi a A)))

Solution

Q = integral₀^R rho * 4 pi r² dr = integral₀^R (A/r²) e^(-2r/a) * 4 pi r² dr = 4 pi A integral₀^R e^(-2r/a) dr = 4 pi A * (a/2)[1 - e^(-2R/a)] = 2 pi a A [1 - e^(-2R/a)]. Solving: 1 - e^(-2R/a) = Q/(2 pi a A), so e^(-2R/a) = 1 - Q/(2 pi a A), giving 2R/a = log(1/(1 - Q/(2 pi a A))) and R = (a/2) log(1/(1 - Q/(2 pi a A))).

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