Charge is distributed inside a sphere of radius R with volume charge density ρ(r) = A exp(-2r/a)/r², where A and a are constants. If Q is the total charge of the distribution, find R.
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Question

Charge is distributed inside a sphere of radius R with volume charge density ρ(r) = A exp(−2r/a)/r², where A and a are constants. If Q is the total charge of the distribution, find R.

G1 Physics · Accepted Answer

Q = 4π∫₀ᴿρ(r)r²dr = 4πA∫₀ᴿexp(−2r/a)dr = 2πaA[1 − exp(−2R/a)]. Hence exp(−2R/a) = 1 − Q/(2πaA). Therefore R = −(a/2)ln[1 − Q/(2πaA)] = (a/2)ln{1/[1 − Q/(2πaA)]}. Correct answer: D.