In an oscillating LC circuit in which $\mathit{C}\mathbf{=}\mathbf{}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{}\mathit{\mu }\mathit{F}$, the maximum potential difference across the capacitor during the oscillations is $\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{}\mathbf{V}$ and the maximum current through the inductor is $\mathbf{50}\mathbf{.}\mathbf{0}\mathbf{}\mathit{m}\mathit{A}$. (a)What is the inductance L? (b)What is the frequency of the oscillations? (c) How much time is required for the charge on the capacitor to rise from zero to its maximum value?

An ac generator with emf amplitude ${\mathit{\epsilon }}_{\mathbf{m}}\mathbf{=}\mathbf{220}\mathbf{}\mathbf{\text{V}}$ and operating at frequency ${\mathit{f}}_{\mathbf{d}}\mathbf{=}\mathbf{400}\mathbf{}\mathbf{\text{Hz}}$ causes oscillations in a series $\mathit{R}\mathit{L}\mathit{C}$ circuit having $\mathit{R}\mathbf{}\mathbf{=}\mathbf{}\mathbf{220}\mathbf{}\mathbf{\text{Ω}}$, $\mathit{L}\mathbf{=}\mathbf{150}\mathbf{}\mathbf{\text{mH}}$ , and $\mathit{C}\mathbf{=24.0}\mathbf{\text{μF}}$ . Find (a) the capacitive reactance $\mathbf{}{\mathit{X}}_{\mathbf{C}}$, (b) the impedance Z, and (c) the current amplitude I. A second capacitor of the same capacitance is then connected in series with the other components. Determine whether the values of (d) ${\mathit{X}}_{\mathbf{C}}$ , (e) Z, and (f) I increase, decrease, or remain the same.

The values of the phase constant $\mathit{\varphi }$ for four sinusoidally driven series RLC circuits are (1) $\mathit{\varphi }\mathbf{=}\mathbf{-}\mathbf{15}\mathbf{°}$, (2) $\mathit{\varphi }\mathbf{=}\mathbf{+}\mathbf{35}\mathbf{°}$, (3) $\mathit{\varphi }\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{3}}\mathbf{}\mathbf{\text{rad}}$, and (4) $\mathit{\varphi }\mathbf{=}\mathbf{-}\frac{\mathbf{\pi }}{\mathbf{6}}\mathbf{}\mathbf{\text{rad}}$. (a) In which is the load primarily capacitive? (b) In which does the current lag the alternating emf?

In Figure, $\mathit{R}\mathbf{=}\mathbf{}\mathbf{14}\mathbf{.}\mathbf{0}\mathbf{\Omega }\mathbf{,}\mathbf{}\mathbf{C}\mathbf{=}\mathbf{}\mathbf{6}\mathbf{.}\mathbf{20}\mathbf{}\mathbf{\mu F}$, and $\mathit{L}\mathbf{=}\mathbf{}\mathbf{54}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{mH}$, and the ideal battery has emf $\mathit{\epsilon }\mathbf{=}\mathbf{34}\mathbf{.}\mathbf{0}\mathbf{V}$. The switch is kept at a for a long time and then thrown to position b.

(a)What is the frequency? (b)What is the current amplitude of the resulting oscillations?

A series RLC circuit is driven by an alternating source at a frequency of 400Hz and an emf amplitude of 90.0V. The resistance is $\mathbf{20}\mathbf{.}\mathbf{0}\mathit{\Omega }$, the capacitance is $\mathbf{12}\mathbf{.}\mathbf{1}\mathit{\mu }\mathit{F}$, and the inductance is $\mathbf{24}\mathbf{.}\mathbf{2}\mathit{m}\mathit{H}$. What is the RMS potential difference across (a) the resistor, (b) the capacitor, and (c) the inductor? (d) What is the average rate at which energy is dissipated?

In Fig. 31-38, a three-phase generator G produces electrical power that is transmitted by means of three wires. The electric potentials (each relative to a common reference level) are ${\mathit{V}}_{\mathbf{1}}\mathbf{=}\mathit{A}\mathbf{}\mathbf{\text{sin}}\mathbf{}{\mathit{\omega }}_{\mathbf{d}}\mathit{t}$ for wire 1, ${\mathit{V}}_{\mathbf{2}}\mathbf{=}\mathit{A}\mathbf{}\mathbf{\text{sin}}\mathbf{}\left({\omega }_{d}t-{\text{120}}^{\text{0}}\right)\mathbf{}$ for wire 2, and ${\mathit{V}}_{\mathbf{3}}\mathbf{=}\mathit{A}\mathbf{}\mathbf{\text{sin}}\mathbf{}\left({\omega }_{d}t-{\text{240}}^{\text{0}}\right)\mathbf{}\mathbf{}$for wire 3. Some types of industrial equipment (for example, motors) have three terminals and are designed to be connected directly to these three wires. To use a more conventional two-terminal device (for example, a lightbulb), one connects it to any two of the three wires. Show that the potential difference between any two of the wires (a) oscillates sinusoidally with angular frequency ${\mathit{\omega }}_{\mathbf{d}}$ and (b) has an amplitude of $\mathit{A}\sqrt{\mathbf{\text{3}}}$.