Question

In Solved Problem 30.1 , the voltage supplied by the source of time-varying emf is \(33.0 \mathrm{~V}\), the voltage across the resistor is \(V_{R}=I R=13.1 \mathrm{~V}\), and the voltage across the inductor is \(V_{L}=I X_{L}=30.3 \mathrm{~V}\). Does this circuit obey Kirchhoff's rules?

Short answer

Answer: No, the circuit does not obey Kirchhoff's rules, as the total voltage drop (43.4 V) does not equal the supplied voltage (33.0 V).

Step-by-step solution

Recall Kirchhoff's Voltage Law

Kirchhoff's Voltage Law (KVL) states that the sum of the voltages around any closed loop in a circuit must be equal to zero. In other words, the supplied voltage must be equal to the sum of the voltage drops across the components in the circuit.

Calculate the sum of the voltage drops across the resistor and inductor

We are given the voltage across the resistor, \(V_R = 13.1 \mathrm{~V}\), and the voltage across the inductor, \(V_L = 30.3 \mathrm{~V}\). To find the sum of the voltage drops, simply add these two values together: \(V_{total} = V_R + V_L = 13.1 \mathrm{~V} + 30.3 \mathrm{~V} = 43.4 \mathrm{~V}\)

Compare the sum of the voltage drops with the supplied voltage

Now, we need to compare the total voltage drop \(V_{total}\) with the supplied voltage to see if the circuit obeys Kirchhoff's rules. The supplied voltage is given as \(33.0 \mathrm{~V}\). Comparing the values, we see that: \(V_{total} = 43.4 \mathrm{~V} \ne 33.0 \mathrm{~V}\)

Conclusion

Since the total voltage drop, \(V_{total}\), does not equal the supplied voltage, the circuit does not obey Kirchhoff's rules. Remember that Kirchhoff's rules apply to ideal circuits where the wires have no resistance and there is no mutual inductance between the components. In a real circuit, there might be some additional factors causing the discrepancy, such as resistance in the wires or mutual inductance.

Key concepts

Voltage Drops

Voltage drops occur when electrical energy is converted into another form of energy as it moves through a circuit component. In a typical circuit, components like resistors, inductors, and capacitors facilitate this energy conversion.

Here's how it works: the supplied voltage from a power source is distributed through the components of the circuit. As the current flows, energy is dissipated primarily in the form of heat or magnetic energy, depending on the component.

• In a resistor, the voltage drop is determined by the product of the current and the resistance (\(V_R = IR\)). The energy is mainly dissipated as heat.
• For an inductor, the voltage drop (\(V_L = IX_L\)) is linked to the inductive reactance and current, with energy stored temporarily in the magnetic field.

According to Kirchhoff's Voltage Law, the algebraic sum of these voltage drops in a closed circuit loop should equal zero, ensuring that all supplied energy is accounted for.

Resistor and Inductor Circuits

Resistor and inductor circuits are commonly found in electrical engineering applications, and they play distinct roles in a circuit's behavior. A resistor opposes the flow of current, dissipating energy as heat, while an inductor stores energy in a magnetic field when electric current passes through it.

Why is it important to analyze these components together?

• Resistors help manage the flow of current, maintaining a steady operation of the circuit despite fluctuations in voltage.
• Inductors, due to their ability to store and release energy, can influence the phase and frequency response of a circuit. They are crucial in applications such as filters or oscillators.

When analyzing RL (Resistor-Inductor) circuits, you consider the impedance, which is the combination of real resistance and reactance from the inductor. This affects how the circuit dynamically responds to AC signals, with factors like phase shift and resonance. Understanding these components' interaction is key to mastering circuit analysis.

Circuit Analysis

Circuit analysis involves using set rules and techniques to deduce unknown values within an electric circuit. Effective circuit analysis allows one to predict how electrical circuits will perform. A foundational principle in this process is Kirchhoff's Voltage Law, which asserts that the sum of all voltages in a closed-loop circuit is zero. This law helps ensure that energy conservation principles are adhered to.

Here are some steps often involved in circuit analysis under KVL:

• Identify closed loops in the circuit: The loop refers to a complete path through which current can flow.
• Assign polarities to voltage sources and component voltage drops: Properly identifying positive and negative sides assists in using KVL correctly.
• Sum all voltage gains and losses: Ensure this sum equals zero for the loop, indicating that energy is neither lost nor gained—only transformed.

These analyses are vital for diagnosing issues in real-world circuits, designing reliable systems, and understanding the theoretical and practical limitations of circuits. Whether dealing with simple configurations or complex network designs, a firm grasp of circuit analysis is essential for any budding engineer or technician.