A charged capacitor with capacitance C undergoes discharge through...

The charge on the capacitor during discharge at time $t$ is given by $q(t) = q_0 e^{-t/RC}$.

The electrostatic field energy stored in the capacitor is $U(t) = \dfrac{q(t)^2}{2C} = \dfrac{q_0^2}{2C} e^{-2t/RC}$.

The activity of a radioactive sample with average life $\tau$ at time $t$ is $A(t) = A_0 e^{-t/\tau}$.

The ratio of the electrostatic field energy to the activity is given by:

$\dfrac{U(t)}{A(t)} = \dfrac{q_0^2}{2C A_0} e^{-t\left(\dfrac{2}{RC} - \dfrac{1}{\tau}\right)}$

For this ratio to remain constant over time, the power of the exponential term must be zero. Thus:

$\dfrac{2}{RC} - \dfrac{1}{\tau} = 0$

$\dfrac{2}{RC} = \dfrac{1}{\tau}$

$R = \dfrac{2\tau}{C}$