A $\mathbf{50}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\Omega }$ resistor is connected, as in Figure to an ac generator with ${\mathbf{\in }}_{\mathbf{m}}\mathbf{=}\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{V}$.

(a) What is the amplitude of the resulting alternating current if the frequency of the emf is $\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{kHz}$? (b) What is the amplitude of the resulting alternating current if the frequency of the emf is $\mathbf{8}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{kHz}$?

A resistor is connected across an alternating-current generator.

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?

In an RLC circuit such as that of Fig. 31-7 assume that $\mathit{R}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{}\mathit{\Omega }\mathbf{,}\mathit{L}\mathbf{=}\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{}\mathit{m}\mathit{H}\mathbf{,}{\mathit{f}}_{\mathbf{d}}\mathbf{=}\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{}\mathit{H}\mathit{z}$ and ${\mathit{\epsilon }}_{\mathbf{m}}\mathbf{=}\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{}\mathit{V}$ . For what values of the capacitance would the average rate at which energy is dissipated in the resistance be (a) a maximum and (b) a minimum? What are (c) the maximum dissipation rate and the corresponding (d) phase angle and (e) power factor? What are (f) the minimum dissipation rate and the corresponding (g) phase angle and (h) power factor?

Using the loop rule, derive the differential equation for an LC circuit (Equation $\mathit{L}\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{q}}{\mathbf{d}{\mathbf{t}}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{C}}\mathit{q}\mathbf{=}\mathbf{0}\mathbf{}$).

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?

A series RLC circuit is driven by a generator at a frequency of $\mathbf{2000}\mathbf{}\mathbf{Hz}$ and an emf amplitude of $\mathbf{170}\mathbf{}\mathbf{V}$. The inductance is $\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{mH}$ , the capacitance is $\mathbf{0}\mathbf{.}\mathbf{400}\mathbf{}\mathbf{\mu F}$ , and the resistance is $\mathbf{200}\mathbf{}\mathbf{\Omega }$. (a) What is the phase constant in radians? (b) What is the current amplitude?