Resnick & Halliday Solutions for Chapter: Electromagnetic Oscillations and Alternating Current, Exercise 1: Problems

 The current amplitude $I$ versus driving angular frequency, ${\omega }_{\mathrm{d}}$  for a driven $RLC$ circuit is given in the figure where the vertical axis scale is set by, $I_{\mathrm{s}}=4.00 \mathrm{A}$ . The inductance is $450 \mu \mathrm{H},$ and the emf amplitude is $6.0 \mathrm{V}.$ What are $\left(\mathrm{a}\right) C$ and $\left(\mathrm{b}\right) R$? 

The energy in an oscillating  $LC$ circuit containing a $2.50 \mathrm{H}$ Inductor is $5.70 \mu \mathrm{J}$. The maximum charge on the capacitor is $175 \mu \mathrm{C}$. For a mechanical system with the same period, find the (a) mass, (b) spring constant, (c) maximum displacement, and (d) maximum speed.

 An alternating source with a variable frequency, an inductor with inductance  $L,$ and a resistor with resistance $R$ are connected in series. Below figure gives the impedance $Z$of the circuit versus the driving angular frequency  ${\omega }_{\mathrm{d}},$ with the horizontal axis scale set by ${\omega }_{\mathrm{ds}}=1600 \mathrm{rad} {\mathrm{s}}^{-1}.$ The figure also gives the reactance $X_{\mathrm{L}}$ for the inductor versus ${\omega }_{\mathrm{d}}.$ What are $\left(\mathrm{a}\right) R$and $\left(\mathrm{b}\right) L?$ 

 What resistance$R$ should be connected in series with an inductance  $L=220 \mathrm{mH}$  and capacitance $C=12.0 \mu \mathrm{F}$  for the maximum charge on the capacitor to decay to $99.0\%$ of its initial value in $50.0$ cycles? (Assume ${\omega }^{\text{'}}\approx \omega$)

 In an oscillating $LC$ circuit, when $75.0\%$ of the total energy is stored in the inductor's magnetic field, (a) what multiple of the maximum charge is on the capacitor and (b) what multiple of the maximum current is in the inductor?

Using the loop rule, derive the differential equation for $LC$ circuit as $L\frac{d^{2}q}{d{t}^2}+\frac{1}{C}q=0$

The frequency of oscillation of a certain $LC$  circuit is $200 \mathrm{kHz}.$ At time $t=0,$ plate $A$ of the capacitor has maximum positive charge. At what earliest  time  $t>0$ will

(a) plate $A$ again have maximum positive charge, (b) the other plate of the capacitor have maximum positive charge, and (c) the inductor have maximum magnetic field?

 (a) At what frequency would a $12 \mathrm{mH}$  inductor and a $10 \mu \mathrm{F}$ capacitor have the same reactance? (b) What would the reactance be? (c) Show that this frequency would be the natural frequency of an oscillating circuit with the same $L$and $C$.