To solve the problem step by step, we will go through each part of the question systematically.
### Given Data:
- Inductance, \( L = 0.120 \, \text{H} \)
- Resistance, \( R = 240 \, \Omega \)
- Capacitance, \( C = 7.30 \, \mu\text{F} = 7.30 \times 10^{-6} \, \text{F} \)
- RMS Current, \( I_{\text{rms}} = 0.450 \, \text{A} \)
- Frequency, \( f = 400 \, \text{Hz} \)
### (a) Phase Angle and Power Factor
1. **Calculate Inductive Reactance \( X_L \)**:
\[
X_L = 2 \pi f L = 2 \pi (400) (0.120) = 301.59 \, \Omega \approx 301.6 \, \Omega
\]
2. **Calculate Capacitive Reactance \( X_C \)**:
\[
X_C = \frac{1}{2 \pi f C} = \frac{1}{2 \pi (400) (7.30 \times 10^{-6})} \approx 54.5 \, \Omega
\]
3. **Calculate Phase Angle \( \phi \)**:
\[
\tan \phi = \frac{X_L - X_C}{R} = \frac{301.6 - 54.5}{240} \approx 1.030
\]
\[
\phi = \tan^{-1}(1.030) \approx 45.8^\circ
\]
4. **Calculate Power Factor**:
\[
\text{Power Factor} = \cos \phi = \cos(45.8^\circ) \approx 0.697
\]
### (b) Impedance of the Circuit
1. **Calculate Impedance \( Z \)**:
\[
Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{240^2 + (301.6 - 54.5)^2}
\]
\[
Z = \sqrt{240^2 + 247.1^2} \approx 344 \, \Omega
\]
### (c) RMS Voltage of the Source
1. **Calculate RMS Voltage \( V_{\text{rms}} \)**:
\[
V_{\text{rms}} = I_{\text{rms}} \times Z = 0.450 \times 344 \approx 154.8 \, \text{V} \approx 155 \, \text{V}
\]
### (d) Average Power Delivered by the Source
1. **Calculate Average Power \( P_{\text{avg}} \)**:
\[
P_{\text{avg}} = V_{\text{rms}} \times I_{\text{rms}} \times \cos \phi
\]
\[
P_{\text{avg}} = 155 \times 0.450 \times 0.697 \approx 48.6 \, \text{W}
\]
### (e) Average Rate of Thermal Energy Conversion in the Resistor
1. **Calculate Average Power in Resistor**:
\[
P_R = I_{\text{rms}}^2 \times R = (0.450)^2 \times 240 \approx 48.6 \, \text{W}
\]
### (f) Average Rate of Energy Dissipated in the Capacitor
1. **Average Power in Capacitor**:
\[
P_C = 0 \, \text{W} \quad (\text{No power is consumed in the capacitor})
\]
### (g) Average Rate of Energy Dissipated in the Inductor
1. **Average Power in Inductor**:
\[
P_L = 0 \, \text{W} \quad (\text{No power is consumed in the inductor})
\]
### Summary of Results:
- (a) Phase Angle \( \phi \approx 45.8^\circ \), Power Factor \( \approx 0.697 \)
- (b) Impedance \( Z \approx 344 \, \Omega \)
- (c) RMS Voltage \( V_{\text{rms}} \approx 155 \, \text{V} \)
- (d) Average Power \( P_{\text{avg}} \approx 48.6 \, \text{W} \)
- (e) Power in Resistor \( P_R \approx 48.6 \, \text{W} \)
- (f) Power in Capacitor \( P_C = 0 \, \text{W} \)
- (g) Power in Inductor \( P_L = 0 \, \text{W} \)
