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Charge is distributed within a sphere of...

Charge is distributed within a sphere of radius R with a volume charge density `p(r)=(A)/(r^(2))e^(-2r//a),` where A and a are constants. If Q is the total charge of this charge distribution, the radius R is :

A

`a/2 log (1 - Q/(2pi a A))`

B

`a/2 log(1/(1- Q/(2pi a A))`

C

`a log (1 - Q/(2pi a A))`

D

`a log(1/(1- Q/(2pi a A)))`

Text Solution

Verified by Experts

The correct Answer is:
B
`Q = int_(0)^R rho 4 pi r^2 dr`
`=int_(0)^(R ) A/(r^2) e^(-(2r)/(a)) 4 pi r^2 dr = 4 pi A int_(0)^(k) e^(-2r//a) dr = (4 pi A)/((-2//a)) [e^(-2r//a)]_(0)^(R )`
`Q = -2 pi A a(e^(-(2R)/a - 1))`
`Q = - 2pi A a e^(-(2R)/a) + 2 pi a A`
`implies e^(-2R//a) = (2 pi a A -Q)/(2 pi a A) implies R = a/2 log(1/(1 - Q/(2pi a A)))`.
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