Question #43336

This question tests the propagation of errors in AC circuits, specifically targeting the trap of treating dependent variables as independent.

Students often calculate the wattless current as $I_w = I_{rms} \sin\phi$.

Finding the relative error in $I_{rms}$ ($2.4\%$) and $\sin\phi$ ($0.6\%$) separately, they mistakenly add them to get $3.0\%$.

This linear error propagation fails because both $I_{rms}$ and $\sin\phi$ depend on the same physical quantity (inductance $L$).

An increase in $L$ decreases $I_{rms}$ but increases $\sin\phi$, meaning their errors are anti-correlated and partially cancel.

The correct approach requires expressing $I_w$ entirely in terms of $L$ before applying calculus, revealing a sensitivity factor $\cos(2\phi)$ that scales the relative error.