For a radioactive meterial , its activity `A` and rate of charge of its activity `R` are defined as `A = - (dN)/(dt) and R = -(dA)/(dt) ` where `N(t)` is the number of nuclei at time `t` .Two radioactive source P (mean life `tau` ) and Q(mean life `2 tau` ) have the same activity at `t = 2 tau R_(p) and R_(Q)` respectively , if `(R_(p))/(R_(Q)) = (n)/(e)`

Answer

2

Explanation

`N=N_(0)e^(-lambdat)`
`A=lambdaN_(0)e^(-lambdat)`
`R=lambda^(2)N_(0)e^(-lambdat)`
`A_(P)=A_(Q)` at `t=0`
`lambda_(P)N_(P)e^(-lambda_(P)t)=lambda_(Q)N_(Q)e^(-lambda_(Q)t)` at `t=0`
`lambda_(P)N_(P)=lambda_(Q)N_(Q)` …(i)
`R_(P)/R_(Q)=(lambda_(P)/lambda_(Q))^(2) (N_(P)/N_(Q)) e^(-lambda_(P)2tau)/e^(-lambda_(Q)2tau)`
`R_(P)/R_(Q)=(lambda_(P)/lambda_(Q))^(2)(N_(P)/N_(Q))1/e` ...(ii)
`[lambda_(P)=1/("mean life")]`
from equation (i) and equation (ii)
`R_(P)/R_(Q)=2/e`
Ans. 2