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Question #43694

Difficulty: 8/10 | 🟢 Free

The Question

A specialized radioisotope thermoelectric generator (RTG) is powered by a successive radioactive decay chain $A \xrightarrow{\lambda_1} B \xrightarrow{\lambda_2} C$ (stable). At time $t=0$ , the RTG contains $N_0$ nuclei of $A$ and zero nuclei of $B$ and $C$ . The decay constants are $\lambda_1 = \lambda$ and $\lambda_2 = 2\lambda$ .

The RTG is modeled as a time-varying ideal voltage source $E(t)$ in series with a time-varying internal resistance $r(t)$ . The electromotive force is directly proportional to the instantaneous number of $B$ nuclei: $E(t) = E_0 \frac{N_B(t)}{N_0}$ . The internal resistance is directly proportional to the instantaneous number of $A$ nuclei: $r(t) = R_0 \frac{N_A(t)}{N_0}$ .

An external load resistance $R$ is connected to the RTG. An onboard controller continuously adjusts $R$ to extract the maximum possible electrical power at every instant $t$ . What is the absolute maximum power $P_{max}$ extracted by the load during the entire operational lifetime of the RTG? (Assume $E_0$ and $R_0$ are positive constants with standard SI units).

Options

A.
$\frac{E_0^2}{27 R_0}$
B.
$\frac{E_0^2}{32 R_0}$
C.
$\frac{E_0^2}{4 R_0}$
D.
$\frac{4 E_0^2}{27 R_0}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Statistical Nature of Decay|Successive Decay Equilibrium|Maximum Power Transfer

Why This Question Is Tricky

Common Mistakes

The core trap in this problem is the assumption that the maximum power output of a dynamic system occurs when its driving force (the EMF, which is proportional to the daughter nuclei population) is at its maximum.

Students often calculate the time of maximum activity for the daughter nucleus (transient equilibrium, where $e^{-\lambda t} = 1/2$ ) and plug this into the power formula, yielding the trap answer $E_0^2 / 32 R_0$ .

However, because the internal resistance (proportional to the parent nuclei) is also decaying exponentially, the true maximum power transfer requires optimizing the ratio $E(t)^2 / 4r(t)$ .

This tests the synthesis of successive radioactive decay kinetics with the Maximum Power Transfer theorem, penalizing those who treat the dynamic internal resistance as a static parameter or decouple the circuit optimization from the time-evolution of the decay chain.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Statistical Nature of Decay
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free