Question #42168

This question exploits the conceptual boundary between state functions (stored reactive energy $\Delta U$) and path functions (work $W$ and dissipated energy $E_d$) in an LCR circuit.

Students often fall into the trap of attempting to calculate $W$ and $\Delta U$ independently.

Calculating $\Delta U$ seems impossible at first glance because $L$ and $C$ are not individually given, leading to algebraic paralysis.

Even if they realize that $\Delta U$ collapses using the reactance condition $\omega L - 1/(\omega C) = R \tan \phi$, the path integrals for $W$ involving the power factor are highly prone to sign errors.

The elegant 'lethal collision' is bypassed by recognizing the circuit analogue of the Work-Energy Theorem: $W - \Delta U = E_d$.

By shifting focus from the elusive state variables to the purely path-dependent Joule heating $E_d = \int I^2 R dt$, the problem simplifies to a single, straightforward integral, rewarding deep conceptual understanding over brute-force ODE manipulation.