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Chapter 28: Problem 58

A series \(R L C\) circuit has power factor 0.80 and impedance \(100 \Omega\) at \(60 \mathrm{Hz}\). (a) What's the resistance? (b) If the inductance is \(0.10 \mathrm{H},\) what's the resonant frequency?

Short Answer

Expert verified
The resistance in the circuit is \(80 \Omega\). The exact resonant frequency cannot be calculated without knowing the capacitance.

Step by step solution

01

Calculation of Resistance

By definition, the power factor in a circuit is the cosine of the phase angle (\(\theta\)) between current and voltage (cos(\(\theta\))) and is equal to Resistance (\(R\)) divided by Impedance (\(Z\)). So, rearranging for \(R\), we get: \(R = Z \times cos(\(\theta\))\). Plug in the given values: \(R = 100 \Omega \times 0.80\).
02

Calculation of Resonant Frequency

The resonant frequency (\(f\)) in a series \(R L C\) circuit is given by the formula \(f = \frac{1}{2\pi \sqrt{L C}}\). However, since we are not given the capacitance (\(C\)) in this case, we cannot calculate the resonant frequency directly. But since the power factor is less than 1, we know the circuit is inductive and the resonant frequency will be less than the given frequency of 60Hz. Thus, we cannot calculate the exact resonant frequency in this case.

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