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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

`alog((1)/(1-(Q)/(2piaA)))`

B

`(a)/(2)log((1)/(1-(Q)/(2piaA)))`

C

`(a)/(2)log(1-(Q)/(2piaA))`

D

`alog(1-(Q)/(2piaA))`

Text Solution

Verified by Experts

The correct Answer is:
B
`underset(0)overset(Q)intdq=4piunderset(0)overset(R)intr^(2)drrho(r)`
`Q=underset(0)overset(R)int4pir^(2)(A)/(r^(2))e^(-(2R)/(a))drimpliesQ=-4piA(a)/(2)[e^(-(2R)/(a))-1]`
`implies(Q)/(2piaA)=1-e^((-2R)/a)impliese^(-(2R)/(a))=1-(Q)/(2piaA)`
`implies-(2R)/(a)=ln[1-(Q)/(2piaA)]implies(2R)/(a)=ln"(1)/(1-(Q)/(2piaA))`
`implies R=(a)/(2)ln((1)/(1-(Q)/(2piaA)))`.
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