A radioactive sample has an average life of \(30\, {ms}\) and is decaying. A capacitor of capacitance \(200\, \mu\, {F}\) is first charged and later connected with resistor \(^{\prime}{R}^{\prime}\). If the ratio of charge on capacitor to the activity of radioactive sample is fixed with respect to time then the value of \(^{\prime}R^{\prime}\) should be \(....\,\Omega\)

where, \(R C\) is the time constant of \(R C\) - circuit.

At time \(t\), activity \(A\) is given by

\(A=A_{0} e ^{-\lambda t} \) where, \(A_{0}=\) initial activity and \(\quad \lambda=\) decay constant. On dividing Eqs. (i) and (ii), we get

\(\frac{q}{A} =\frac{q_{0} e^{-\frac{t}{R C}}}{A_{0} e^{-\lambda t}}\)

\(1=\frac{e^{-\frac{t}{R C}}}{e^{-\lambda t}}\)

\(e^{-\lambda t} =e^{-\frac{t}{R C}}\)

\(\Rightarrow\) Taking log on both sides of above equation, we get

\(\ln \left(e^{-\lambda t}\right) =\ln \left(e^{-\frac{t}{R C}}\right)\)

\(-\lambda t =-\frac{t}{R C}\)

\(\lambda =\frac{1}{R C}\)

\(R =\frac{1}{\lambda C}\)

\(R =\frac{30 \times 10^{-3}}{200 \times 10^{-6}}\)

\(R =150 \Omega\)

\(\Rightarrow \lambda=\frac{1}{R C}\)

\(\Rightarrow R=\frac{1}{\lambda C}\)

\(\Rightarrow R=\frac{30 \times 10^{-3}}{200 \times 10^{-6}}\)

\(\Rightarrow R=150 \Omega\)