Question

A charged capacitor of capacitance through a resistance R. A radioactive sample decays with an average life $\tau$. Find the value of R for which the ratio of the activity of the radioactive sample is independent of time,

• A. $\frac{\tau }{C}$
• B. $\frac{2\tau }{C}$
• C. $\frac{\tau }{2C}$
• D. $\frac{3\tau }{2C}$

Answer 1

The correct option is B $\frac{2\tau }{C}$

$Q=q{e}^{-t/CR}$

$A=A_0{e}^{-\lambda t}$

$\frac{Energy}{activity}=\frac{q^2\times e^{-\frac{2t}{CR}}}{2C{A}_0{e}^{-\lambda t}}$

Since the term is independent of time, so the coefficients can be equated

So,

$\frac{2t}{CR}=\lambda t$ or $\lambda =\frac{2}{CR}$

or,

$\frac{1}{T}=\frac{2}{CR}$

$R=\frac{2T}{C}$

hence option (B) is the correct answer.