Inductive reactance:

$\overline{){{\mathbf{X}}}_{{\mathbf{L}}}{\mathbf{=}}{\mathbf{2}}{\mathbf{\pi}}{\mathbf{f}}{\mathbf{L}}}$, where f is the frequency and L is the inductance.

Capacitive reactance:

$\overline{){{\mathbf{X}}}_{{\mathbf{C}}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}\mathbf{\pi}\mathbf{f}\mathbf{C}}}$, where C is the capacitance.

Impedance, resistors, capacitors, and inductors are related by:

$\overline{){\mathbf{Z}}{\mathbf{=}}\sqrt{{\mathbf{R}}^{\mathbf{2}}\mathbf{+}{\mathbf{(}\mathbf{X}}^{}}}$_{L}_{C}

**(a)**

The impedance, Z is obtained by the power factor:

$\begin{array}{rcl}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\varphi}& \mathbf{=}& \frac{\mathbf{R}}{\mathbf{Z}}\\ \mathbf{Z}& \mathbf{=}& \frac{\mathbf{R}}{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\varphi}}\\ & \mathbf{=}& \frac{\mathbf{200}}{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{(}\mathbf{45}\mathbf{.}\mathbf{0}\mathbf{\xb0}\mathbf{)}}\end{array}$

An RLC series circuit has a 200 Ω resistor and a 25.0 mH inductor. At 8000 Hz, the phase angle is 45.0°.

(a) What is the impedance?

(b) Find the circuit’s capacitance.

(c) If Vrms = 408 V is applied, what is the average power supplied?

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