In the circuit shown in Fig. P30.60, switch S1 has been closed for a long enough time so that th

step 1 A circuit consists of a inductor and a capacitor called LC circuit. The total energy in the LC c

In the circuit shown in Fig. P30.60, switch S1 has been...

In the circuit shown in $\textbf{Fig. P30.60}$, switch $S_1$ has been closed for a long enough time so that the current reads a steady 3.50 A. Suddenly, switch $S_2$ is closed and $S_1$ is opened at the same instant. (a) What is the maximum charge that the capacitor will receive? (b) What is the current in the inductor at this time?

Key Concepts

Steady-State Behavior in DC Circuits

In DC circuits that have reached steady state, reactive components show unique behaviors. Inductors act like short circuits because, after a long time, the rate of change of current is zero and thus the voltage drop across an inductor is zero. Conversely, capacitors behave as open circuits under steady state conditions since no further charge flows through once they are fully charged.

Current Continuity in Inductors

A key property of inductors is that their current cannot change instantaneously. This current continuity means that at the moment when the circuit is reconfigured by the switching events, the inductor carries over its previous steady-state current without sudden change, which is critical in predicting transient behavior immediately following the switching.

Energy Conservation in LC Circuits

In an ideal LC circuit, energy is exchanged between the magnetic field of an inductor and the electric field of a capacitor. The total energy remains constant, so the energy initially stored in the inductor (given by ½ L I²) is fully transferable to the capacitor, where it is stored as ½ C V². This principle allows one to determine the maximum voltage—or equivalently the maximum charge—on the capacitor when the energy conversion is complete.

Capacitor Charging Dynamics in Transient Analysis

When a circuit is reconfigured to allow energy exchange between the inductor and capacitor, the capacitor begins charging. The maximum charge on the capacitor is reached when all the magnetic energy from the inductor has been transferred into electrical energy stored in the capacitor. Analyzing this process involves applying energy conservation principles to equate the inductor's initial energy with the capacitor's energy at the peak voltage.

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