A charged capacitor of capacitance C is discharged through a resistance R. A radioactive sample decays with an average-life `tau`.Find the value of R for which the ratio of the electrostatic field energy stored in the capacitor to the activity of the radioactive sample remains constant in time.

Explanation

`bar(T) = tau`
`lambda = (1)/(bar(T)) = (1)/(tau)`
Activity of radioactive smaple at time `t`
`A = A_(0)e^(-lambdat) = A_(0)e^(-t//tau) ….(i)`
In discharging `RC` circuit
`q = q_(0)e^(-t//RC)`
Energy stored in capacitor
`U = (q^(2))/(2C) = (q_(0)^(2))/(2C)e^(-(2t)/(RC))`
`(U)/(A) =((q^(2))/(2C)e^(-(2t)/(RC)))/(A_(0)e^(-t//tau))`
If `U//A` is independent of time,
`e^(-2t//RC) = e^(-t//tau)`
`(2)/(RC) = (1)/(tau)`
`R =(2tau)/(C )`