Curve a in Fig. 31-21 gives the impedance Z of a driven RC circuit versus the driving angular frequency ${\mathit{\omega }}_{\mathbf{d}}$. The other two curves are similar but for different values of resistance R and capacitance C. Rank the three curves according to the corresponding value of R, greatest first.

Figure 31-25 shows the current and driving emf $\mathit{\epsilon }$ for a series RLC circuit. (a) Does the current lead or lag the emf? (b) Is the circuit’s load mainly capacitive or mainly inductive? (c) Is the angular frequency $\mathbf{}{\mathit{\omega }}_{\mathbf{d}}$ of the emf greater than or less than the natural angular frequency $\mathit{\omega }$ ?

What values of phase constant $\mathit{\varphi }$ in Eq. 31-12 allow situations (a), (c), (e), and (g) of Fig. 31-1 to occur at $\mathit{t}\mathbf{=}\mathbf{0}$?

A variable capacitor with a range from 10 to 365pF is used with a coil to form a variable-frequency LC circuit to tune the input to a radio. (a) What is the ratio of maximum frequency to minimum frequency that can be obtained with such a capacitor? If this circuit is to obtain frequencies from 0.54 MHz to 1.6 0MHz, the ratio computed in (a) is too large. By adding a capacitor in parallel to the variable capacitor, this range can be adjusted. To obtain the desired frequency range,(b) What capacitance should be added ? (c) What inductance should the coil have?

An alternating emf source with a variable frequency ${\mathit{f}}_{\mathbf{d}}$ is connected in series with a $\mathit{R}\mathbf{=}\mathbf{50}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\text{Ω}}$ resistor and a $\mathit{C}\mathbf{=}\mathbf{20}\mathbf{}\mathbf{\text{μF}}$ capacitor. The emf amplitude is ${\mathbf{\in }}_{\mathbf{m}}\mathbf{=}\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\text{V}}$ . (a) Draw a phasor diagram for phasor ${\mathit{V}}_{\mathbf{R}}$ (the potential across the resistor) and phasor ${\mathit{V}}_{\mathbf{C}}$ (the potential across the capacitor). (b) At what driving frequency ${\mathit{f}}_{\mathbf{d}}$ do the two phasors have the same length? At that driving frequency, what are (c) the phase angle in degrees, (d) the angular speed at which the phasors rotate, and (e) the current amplitude?

A single-loop circuit consists of a $\mathbf{7}\mathbf{.}\mathbf{20}\mathit{\Omega }$resistor, an $\mathbf{12}\mathbf{.}\mathbf{0}\mathit{H}$ inductor, and a capacitor. Initially, the capacitor has a charge of $\mathbf{6}\mathbf{.}\mathbf{20}\mathbf{}\mathit{\mu }\mathit{C}$ and the current is zero. (a) Calculate the charge on the capacitor N complete cycles later for N=5. (b) Calculate the charge on the capacitor N complete cycles later for N=10. (c) Calculate the charge on the capacitor N complete cycles later for N=100.