In a nuclear physics experiment, radioactive isotopes are produced at a constant rate dN/dt = R . A series circuit is formed by connecting an inductor of inductance 100 mH , a resistor of resistance 100 , and a battery. The circuit is switched on at the exact instant the production of the radioactive isotope begins. It is observed that the ratio i/N remains constant over time, where i represents the current in the ci

Step-by-step solution

The rate of change of the number of active nuclei is given by dN dt = R - N , where R is the constant production rate and is the decay constant. Integrating with the initial condition N(0) = 0 , we get N(t) = R (1 - e^ - t ) . For the series LR circuit, the current at time t is given by i(t) = i₀(1 - e^ - R_ res L t ) , where R_ res is the resistance and L is the inductance. Given that the ratio i(t) N(t) is constant over time, the exponential terms must be identical. Thus, = R_ res L . Substituting the given value