If the switch in the circuit shown in Figure P5.34 is closed...

If the switch in the circuit shown in Figure P5.34 is closed at $t=0$ and: $$ \begin{array}{ll} V_S=12 \mathrm{~V} & C=130 \mu \mathrm{~F} \\ R_1=2.3 \mathrm{k} \Omega & R_2=7 \mathrm{k} \Omega \\ L=30 \mathrm{mH} & \end{array} $$ Determine the current through the inductor and the voltage across the capacitor and across $R_1$ after the circuit has returned to a steady state.

If the switch in the circuit shown in Figure P5.34 is closed at $t=0$ and:

$$
\begin{array}{ll}
V_S=12 \mathrm{~V} & C=130 \mu \mathrm{~F} \\
R_1=2.3 \mathrm{k} \Omega & R_2=7 \mathrm{k} \Omega \\
L=30 \mathrm{mH} &
\end{array}
$$
Determine the current through the inductor and the voltage across the capacitor and across $R_1$ after the circuit has returned to a steady state.

Key Concepts

DC Steady-State Analysis

In circuits with capacitors and inductors, DC steady state is achieved when all transient effects have died out. At steady state for a DC source, the capacitor reaches a constant voltage (and draws no further current) while the inductor carries a constant current (with zero voltage drop across it). This simplifies the analysis because it allows replacing the capacitor with an open circuit and the inductor with a short circuit.

Capacitor Behavior in DC Circuits

A capacitor stores energy in its electric field and its current is related to the rate of change of voltage across it. In DC steady state, the voltage becomes constant (dV/dt = 0), so the capacitor current becomes zero. This means that the capacitor effectively behaves as an open circuit, isolating itself from further current flow.

Inductor Behavior in DC Circuits

An inductor stores energy in its magnetic field and its voltage is related to the rate of change of current flowing through it. In DC steady state, since the current is constant (dI/dt = 0), the voltage across the inductor drops to zero. This allows the inductor to be modeled as a short circuit, essentially acting as a wire with negligible resistance.

Kirchhoff's Circuit Laws

Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) are essential tools in circuit analysis. KVL states that the sum of voltage drops around any closed loop equals the total applied voltage, while KCL ensures that the total current entering any node equals the total current leaving it. These laws, when combined with the steady state behavior of capacitors and inductors, allow for determining unknown currents and voltages in the circuit including across individual elements like resistors.

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