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  • A radioactive sample has an average life of 30 ms and is decaying. A capacitor of capacitance 200  is first charged and later connected with resistor If

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 'R'. If the ratio of charge on capacitor to the activity of radioactive sample is fixed with respect to time then the value of 'R' should be ______\Omega.
 

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\begin{gathered} t_{\text {mean }}=30 \times 10^{-3} \mathrm{~S}=\frac{1}{\lambda} \\ C=200 \mu \mathrm{F} \end{gathered}

Let q_{0} be the initial change on the capacitor

\rightarrow q= q_{0}e^{-t/\tau }

Discharging of capacitor condition.

Activity after time  t

A_{t}=A_{0} e^{-\lambda t}

\begin{aligned} &\frac{q_{t}}{A_{t}}=\frac{q_{0}}{A_{0}} \frac{e^{-t / \tau}}{e^{-\lambda t}} \\ &\frac{q_{t}}{A_{t}}=\frac{q_{0}}{A_{0}} e^{(-t / \tau+\lambda t)}=K \end{aligned}

\begin{aligned} \therefore-\frac{t}{\tau} &+\lambda t=0 \\ \frac{1}{\tau} &=\lambda \\ \frac{1}{R C} &=\lambda \end{aligned}

\begin{aligned} t_{\text {mean }}=\frac{1}{\lambda} &=R C \\ \end{aligned}

\begin{aligned} 30 \times 10^{-3}=R \times 200 \times 10^{-6} \\ \end{aligned}

\begin{aligned} R &=\frac{30}{200} \times 10^{+3} \\ \\R &=150 \Omega \end{aligned}

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vishal kumar

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