Other Single-Loop Circuits — Question 9

Question 9

Nearly balanced sources can leave current direction unresolved by tolerances

Two opposing sources have hard emf bounds $E_1=12.0\pm.2$ V and $E_2=11.9\pm.3$ V. Total loop resistance is exactly 2.00 ohm. Define positive current in the E1-driven direction.

1. Find nominal current and its exact interval.

2. Find the allowed resistor-heating interval and explain why the minimum cannot be obtained simply by squaring the two current endpoints.

Question 9 - Solution

Nearly balanced sources can leave current direction unresolved by tolerances

(a) Method and calculation

$$I=(E_1-E_2)/R,\quad I_0=(12.0-11.9)/2=.0500\,\mathrm A.$$$$I\in[(11.8-12.2)/2,(12.2-11.6)/2]=[-.200,.300]\,\mathrm A.$$

(b) Interpretation and further calculation

$$0\in[I_{\min},I_{\max}]\Rightarrow P_{\min}=0.$$$$P_{\max}=R\max(I_{\min}^2,I_{\max}^2)=2(.300)^2=.180\,\mathrm W.$$

Physical check

The uncertainty interval permits either current direction and includes exact cancellation.