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Chapter 30: Problem 13

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

Expert verified
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

01

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.
02

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: \(V_{total} = V_R + V_L = 13.1 \mathrm{~V} + 30.3 \mathrm{~V} = 43.4 \mathrm{~V}\)
03

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: \(V_{total} = 43.4 \mathrm{~V} \ne 33.0 \mathrm{~V}\)
04

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.

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