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

Step-by-step solution

ρ r = A r 2   e - 2 r a Charge enclosed between r and r + d r is d q = ρ r   4 π r 2 d r To get total charge Q Q = ∫ 0 R d q = ∫ 0 R A r 2 e - 2 r / a   4 π r 2 d r = - 4 πA a 2 [ e - 2 r / a ] 0 R Q = - 4 π   A a 2 e - 2 R / a - 1 R = - ln ⁡ 1 - Q 2 4 πAa a 2 = a 2 ln ⁡ 1 1 - Q 2 πaA