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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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Most popular questions from this chapter

In a series RLC circuit, \(V=(12.0 \mathrm{~V})(\sin \omega t), R=10.0 \Omega\) \(L=2.00 \mathrm{H},\) and \(C=10.0 \mu \mathrm{F}\). At resonance, determine the voltage amplitude across the inductor. Is the result reasonable, considering that the voltage supplied to the entire circuit has an amplitude of \(12.0 \mathrm{~V} ?\)

A transformer contains a primary coil with 200 turns and a secondary coil with 120 turns. The secondary coil drives a current \(I\) through a \(1.00-\mathrm{k} \Omega\) resistor. If an input voltage \(V_{\mathrm{rms}}=75.0 \mathrm{~V}\) is applied across the primary coil, what is the power dissipated in the resistor?

Which statement about the phase relation between the electric and magnetic fields in an LC circuit is correct? a) When one field is at its maximum, the other is also, and the same for the minimum values. b) When one field is at maximum strength, the other is at minimum (zero) strength. c) The phase relation, in general, depends on the values of \(L\) and \(C\).

A series RLC circuit is in resonance when driven by a sinusoidal voltage at its resonant frequency, \(\omega_{0}=(L C)^{-1 / 2}\) But if the same circuit is driven by a square-wave voltage (which is alternately on and off for equal time intervals), it will exhibit resonance at its resonant frequency and at \(\frac{1}{3}, \frac{1}{5}\), \(\frac{1}{7}, \ldots,\) of this frequency. Explain why.

Show that the power dissipated in a resistor connected to an AC power source with frequency \(\omega\) oscillates with frequency \(2 \omega\).
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