At time t = 0, the switch in the circuit shown in Fig. 30–30 is closed. After a sufficiently long time, steady currents I₁, I₂, and I₃ flow through resistors R₁, R₂, and R₃, respectively. Determine these three currents.


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Verified step by step guidance

1. Identify the configuration of the circuit (series, parallel, or a combination) and note the values of the resistors R₁, R₂, and R₃, and the voltage source if provided.
2. Apply Kirchhoff's Voltage Law (KVL) to the loops in the circuit. KVL states that the sum of the electromotive forces and the product of currents and resistances around any closed loop in a circuit is zero.
3. Use Ohm's Law, V = IR, where V is the voltage across the resistor, I is the current through the resistor, and R is the resistance, to relate the currents and voltages in the circuit.
4. Set up equations based on the KVL and Ohm's Law for each loop and solve the simultaneous equations to find the currents I₁, I₂, and I₃.
5. Check the consistency of your answers by substituting the values of I₁, I₂, and I₃ back into the original circuit equations to ensure they satisfy both KVL and Ohm's Law.

Key Concepts

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed mathematically as V = IR. Understanding this law is essential for analyzing circuits, as it allows us to calculate the current through each resistor when the voltage and resistance values are known.

Kirchhoff's Laws

Kirchhoff's Laws consist of two fundamental principles for circuit analysis: Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL). KCL states that the total current entering a junction must equal the total current leaving it, while KVL states that the sum of the electrical potential differences (voltage) around any closed circuit loop must equal zero. These laws are crucial for determining the currents in complex circuits with multiple branches.

Steady State in Circuits

The steady state in electrical circuits refers to the condition where all circuit variables (currents and voltages) remain constant over time after transient effects have dissipated. In this state, the circuit has reached equilibrium, and the currents through the resistors can be calculated using Ohm's Law and Kirchhoff's Laws. Understanding the steady state is important for analyzing circuits that have been powered for a sufficient duration, as it simplifies the calculations needed to find the currents.