1. a. The instantaneous emf value is the projection down onto the horizontal axis. The emf is negative but increasing in magnitude as the phasor, which rotates ccw, approaches the horizontal axis.

2. c. Voltage and current are measured using different scales and units. You can’t compare the length of a voltage phasor to the length of a current phasor.

3. a. There is “no capacitor” when the separation between the two capacitor plates becomes zero and the plates touch. Capacitance $C$ is inversely proportional to the plate spacing $d$, hence $C\to \infty \text{ as }d\to 0$. The capacitive reactance is inversely proportional to $C$, so $X_{\text{C}}\to 0\text{as C}\to \infty$.

4. $(\omega {}_{\text{C}}){}_{\text{D}}>{\left({\text{ω}}_{\text{C}}\right)}_{\text{C}}={\left({\omega }_{\text{C}}\right)}_{\text{A}}>{\left({\omega }_{\text{C}}\right)}_{\text{B}}$. The crossover frequency is 1/RC.

5. Above. $V_{\text{L}}>V_{\text{C}}$ tells us that $X_{\text{L}}>X_{\text{C}}$. This is the condition above resonance, where $X_{\text{L}}$ is increasing with $\omega$ while $X_{\text{C}}$ is decreasing.

6. a, b, and c. You can always increase power by turning up the voltage. Power is maximum at resonance when the power factor is zero. The current leads the emf, telling us that the resonance frequency is higher than the emf frequency. We need to lower the resonance frequency to match the emf frequency, which can be done by increasing either $L$ or $C$.

1.

1. $2.0\times {10}^3\text{rad}\text{/}\text{s}$

2. 71 V

3.

5.

1. 50 mA

2. 50 mA

7.

1. 1.9 mA

2. 1.9 A

9.

1. 50 Hz

2. $4.8\mu \text{F}$

11.

1. 80 kHz

2. 0 V

13.

• b.

  

  Full Alternative Text

15. $1.6\mu \text{F}$

17.

1. 10.0 Hz

2. $f$	$V{}_c(\text{V})$
   ${\displaystyle \frac{1}{2}}{f}_{\text{c}}$	4.47
   $f_{\text{c}}$	3.53
   $2f_{\text{c}}$	2.24

19.

1. 3.0 V, 9.5 V

2. 10 V, 3.2 V

21.

1. 0.9770

2. 0.9998

23.

1. 0.80 A

2. 0.80 mA

25.

1. $24\mu \text{H}$

2. $165\mu \text{A}$

27.

1. 200 kHz

2. 141 kHz

29.

1. $Z=70\text{Ω}\text{,}I=0.072\text{A, }\varphi =-44\text{°}$

2. $Z=50\text{Ω}\text{,}I=0.10\text{A, }\varphi =0.0\text{°}$

3. $Z=62\text{Ω}\text{,}I=0.080\text{A, }\varphi =37\text{°}$

31. $1.0\text{Ω}$

33.

1. $5\times {10}^3\text{Hz}$

2. 10 V, 32 V

35. 30°

37. $44\text{ Ω}$

39. 0.75

43.

1. $\omega ={\displaystyle \frac{1}{\sqrt{3}RC}}$

2. ${\displaystyle \frac{\sqrt{3}}{2}}{\epsilon }_0$

45. 0.50 mm

47.

1. $I_{\text{R}}={\displaystyle \frac{{\epsilon }_0}{R}}\text{,}{I}_{\text{C}}={\displaystyle \frac{{\epsilon }_0}{{\left(\omega C\right)}^{-1}}}$

2. ${\epsilon }_0\sqrt{{\left(\omega C\right)}^2+{\displaystyle \frac{1}{R^2}}}$

49.

1. ${\epsilon }_0/\sqrt{R^2+{\omega }^2{L}^2}\text{,}{\epsilon }_0\text{R}/\sqrt{R^2+{\omega }^2{L}^2}\text{,}{\epsilon }_0\omega L/\sqrt{R^2+{\omega }^2{L}^2}$

2. $V_{\text{R}}\to {\epsilon }_0\text{,}{V}_{\text{R}}\to 0$

3. Low pass

4. R/L

51.

1. 2.0 A

2. $-30\text{°}$

3. 150 W

53. $10\mu \text{T}$

55.

1. 64 mA

2. 48 mA

57.

1. 3.6 V

2. 3.5 V

3. $-3.6\text{V}$

61.

1. $0.49\mu \text{H}$

2. $10.3\text{Ω}$

63.

1. $1.25\times {10}^6\text{A}$

2. 300 A

65.

1. 0.44 kA

2. $1.8\times {10}^{-4}\text{F}$

3. 7.4 MW

67.

1. 0.83

2. 100 V

3. $13\text{Ω}$

4. $3.2\times {10}^{-4}\text{F}$

71.

2. $I={\epsilon }_0/R$ in both cases

3. 0