A charged capacitor of capacitance C is discharged through a resistance R. A radioactive sample decays with an average-life τ. 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.

A sample contains a mixture of ¹¹⁰Ag and ¹⁰⁸Ag isotopes each having an activity of 8.0 × 10⁸ disintegrations per second. ¹⁰⁸Ag is known to have larger half-life than ¹¹⁰Ag. The activity A is measured as a function of time and the following data are obtained.

(a) Plot ln(A/A₀) versus time.

(b) See that for large values of time, the plot is nearly linear. Deduce the half-life of ¹⁰⁸Ag from this portion of the plot.

(c) Use the half-life of ¹⁰⁸Ag to calculate the activity corresponding to ¹¹⁰Ag in the first 50 s.

(d) Plot ln(A/A₀) versus time for ¹¹⁰Ag for the first 50 s.

(e) Find the half-life of ¹⁰⁸Ag.

A human body excretes (removes by waste discharge, sweating, etc.) certain materials by a law similar to radioactivity. If technetium is injected in some form in a human body, the body excretes half the amount in 24 hours. A patient is given an injection containing ⁹⁹Tc. This isotope is radioactive with a half-life of 6 hours. The activity from the body just after the injection is 6 μCi. How much time will elapse before the activity falls to 3 μCi?

Radioactive isotopes are produced in a nuclear physics experiment at a constant rate dN/dt = R. An inductor of inductance 100 mH, a resistor of resistance 100 Ω and a battery are connected to form a series circuit. The circuit is switched on at the instant the production of radioactive isotope starts. It is found that i/N remains constant in time where i is the current in the circuit at time t and N is the number of active nuclei at time t. Find the half-life of the isotope.

Calculate the energy released by 1g of natural uranium assuming 200 MeV is released in each fission event and that the fissionable isotope ²³⁵U has an abundance of 0.7% by weight in natural uranium.