Chapter 31: Electromagnetic Oscillations and Alternating Current

Does the phasor diagram of Fig. 31-26 correspond to an alternating emf source connected to a resistor, a capacitor, or an inductor? (b) If the angular speed of the phasors is increased, does the length of the current phasor increase or decrease when the scale of the diagram is maintained?

To construct an oscillating LCsystem, you can choose from a 10mH inductor, a 5.0 µF capacitor, and a 2.0 µF capacitor. (a)What is the smallest largest oscillation frequency that can be set up by these elements in various combinations? (b)What is the second smallest largest oscillation frequency that can be set up by these elements in various combinations? (c)What is the second largest oscillation frequency that can be set up by these elements in various combinations? (d)What is the largest oscillation frequency that can be set up by these elements in various combinations?

An oscillating LC circuit consisting of a1µFcapacitor and a$\mathit{L}\mathbf{=}\mathbf{3}\mathbf{}\mathbf{\text{mH}}$coil has a maximum voltage of 3.0 V. What are (a) the maximum charge on the capacitor, (b) the maximum current through the circuit, and (c) the maximum energy stored in the magnetic field of the coil?

An inductor is connected across a capacitor whose capacitance can be varied by turning a knob. We wish to make the frequency of oscillation of this LC circuit vary linearly with the angle of rotation of the knob, going from$\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{}\mathbf{Hzto}\mathbf{}\mathbf{}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{}\mathbf{Hz}$as the knob turns through 180°. If $\mathit{L}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{mH}$, plot the required capacitance C as a function of the angle of rotation of the knob.

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?

An oscillating LCcircuit has current amplitude of $\mathbf{7}\mathbf{.}\mathbf{50}\mathbf{}\mathit{m}\mathit{A}$, potential amplitude of$\mathbf{250}\mathbf{}\mathit{m}\mathit{V}$, and a capacitance of$\mathbf{220}\mathbf{}\mathit{n}\mathit{F}$. (a) What is the period of oscillation? (b) What is the maximum energy stored in the capacitor? (c) What is the maximum energy stored in the inductor? (d) What is the maximum rate at which the current changes? (e) What is the maximum rate at which the inductor gains energy?

Using the loop rule, derive the differential equation for an LCcircuit (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) Does the phasor diagram of Fig. 31-26 correspond to an alternating emf source connected to a resistor, a capacitor, or an inductor? (b) If the angular speed of the phasors is increased, does the length of the current phasor increase or decrease when the scale of the diagram is maintained?

Figure 31-19 shows three oscillating LC circuits with identical inductors and capacitors. At a particular time, the charges on the capacitor plates (and thus the electric fields between the plates) are all at their maximum values. Rank the circuits according to the time taken to fully discharge the capacitors during the oscillations greatest first.

In an oscillating LCcircuit 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?