– Damped exillatios is a circuit
whem
\[
4=4=e^{-A} \cos (e t+a)
\]
– Legarithmic damping decrement i and quality lacter $Q$ of a eircuit am defined by Eqs. (4.1d). When damping is los:
\[
\lambda=\pi R \sqrt{\frac{C}{L}}, \quad Q=\frac{1}{h} \sqrt{\frac{T}{C}} .
\]
– Steady-atate forced escillatios in seireuit with a voltag $V=V_{m} \cos$ ot coanected is series:
\[
t=I_{m} \cos (\omega t-9) \text {. }
\]
where
\[
\begin{array}{l}
t_{m}=\frac{V_{m}}{\sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}}}, \\
\tan \varphi=\frac{\omega L-\frac{1}{\omega C}}{R} \text {. } \\
\end{array}
\]

Fie. 4.26 .
The eorraposding vector diacram for veltages is shows is Fie. 4.26.
– Power goaerated in an ac eircait.
\[
P=V I \cos \% \text {. }
\]
where $V$ aad $I$ are the ellective valoes of voltage aad curnent:
4.94. Due to a certain cause the free electrons in a plane copper plate shifted over a small distance $x$ at right angles to its surface. As a result, a surface charge and a corresponding restering foree emerged, giving rise to so-called plasma oscillations. Find the angular frequency of these oscillations if the free electron concentration in copper is $n=0.85 \cdot 10^{20} \mathrm{~m}^{-1}$.
4.95. An oscillating circuit consisting of a capacitor with capacitance $C$ and a coil of inductance $L$ maintains free undamped oscillations with voltage amplitude across the capacitor equal to $V_{m}$. For an arbitrary moment of time find the relation between the current $I$ in the circuit and the veltage $V$ across the capacitor. Solve this problem using Ohm’s law and then the energy conservation law.
4.96. An oscillating circuit consists of a capacitor with capacitance $C$, a coil of induetance $L$ with negligible resistance, and a switch. With the switch disconnected, the capacitor was charged to a voltage $V_{m}$ and then at the moment $t=0$ the switch was closed. Find:
(a) the current $I$ ( $f$ ) in the circuit as a function of time;
(b) the emf of seli-inductance in the coil at the moments when the electric energy of the capacitor is equal to that of the current in the coil.
4.97. In an oscillating circuit consisting of a parallel-plate capaeitor and an inductance coil with negligible active resistance the oscillations with energy $W$ are sustained. The capaciter plates were slowly drawn apart to increase the oscillation frequeney $\eta$-fold. What work was done in the process?
4.98. In an oseillating eireuit shown in Fig. 4.27 the ceil inductance is equal to $L=2.5 \mathrm{mH}$ and the capacitor have capacitances $C_{1}=$ $=2.0 \mu \mathrm{F}$ and $c_{1}=3.0 \mu \mathrm{F}$. The capacitors were charged to a voltage $\bar{V}=180 \mathrm{~V}$, and then the switch $S w$ was closed. Find:
(a) the aatural oscillation frequency;
(b) the peak value of the current llowing through the coil.
Fie. 4.27.
Fig. 4.28.
4.99. An electric circuit shown in Fig. 4.28 has a segligibly small active resistance. The left-hand capacitor was charged to a voliage $V_{\text {o }}$ and then at the moment $t=0$ the switch $S \mathrm{w}$ was elosed. Find the time dependence of the voltages in left and right capacitors.
4.100. An oscillating circuit consists of an inductance coil $L$ and a capacitor with capacitance $C$. The resistance of the coil and the lead
wires is aegligible. The coil is placed in a permanent magnetic field so that the total Aux passing through all the turns of the coil is equal to $\Phi$. At the moment $t=0$ the magnetie field was switched off. Assuming the switching of time to be aegligible compared to the natural oscillation period of the circuit, find the circuit current as a function of time $t$.
4.101. The free damped oscillations are maintained in a eireuit, Find the moments of time when the modulus of the voltage across the capacitor reaches
(a) peak values;
(b) maximum (extremum) values.
4.102. A certain oscillating circuit consists of a capacitor with capacitance $C$, a coil with inductance $L$ and active resistance $R$, and a switch. When the switch was disconnected, the capacitor was eharged; then the switeh was elosed and oscillations set in. Find the ratio of the voltage across the capacitor to its peak value at the moment immediately after closing the switch.
4.103. A circuit with capacitance $C$ and inductance $L$ generates free damped oscillations with current varying with time as $I=$ $=I_{\mathrm{m}} \mathrm{e}^{-7} \mathrm{sin}$ et. Find the voltage across the capacitor as a function of time, and in particular, at the moment $t=0$.
4.104. An oscillating circuit consists of a capacitor with capacitance $C=4.0 \mu \mathrm{F}$ and a coil with inductance $L=2.0 \mathrm{mH}$ and active resistance $R=10 \Omega$. Find the ratio of the energy of the coil’s magnetic field to that of the capacitor’s electric field at the moment when the current has the maximum valae.
4.105. An oscillating eireait consists of two ceils connected in series whose inductances are $L_{1}$ and $L_{2}$, active resistances are $R_{1}$ and $\boldsymbol{R}_{2}$, and mutual indactance is negligible. These coils are to be replaced by one, keeping the frequency and the quality factor of the eireuit constant. Find the inductance and the active resistance of such a coil.
4.106. How soon does the current amplitude in an oscillating eireait with quality factor $Q=5000$ decrease $\eta=2.0$ times if the oscillation frequency is $\mathrm{v}=2.2 \mathrm{MHz}$ ?
4.107. An oscillating eircuit consists of capacitance $C=10 \mu \mathrm{F}$, inductance $L=25 \mathrm{mH}$, and active resistance $H$ oscillation periods does it take for the carrent amplitude to decrease e-fold?
4.108. How much (in per cent) does the free oscillation frequency $\theta$ of a circuit with quality factor $Q=5.0$ differ from the natural oscillation trequency $\omega_{0}$ of that circuit?
4.109. In a circuit shown in Fig. 4.29 the battery emf is equal to $\mathrm{z}-2.0 \mathrm{~V}$, its internal resistance is $r-9.0 \Omega$. the capacitance of the capacitor is $C=10 \mu \mathrm{F}$, the coil inductance is $L=100 \mathrm{mH}$, and the resistance
is $R=1.0 \Omega$. At a certain moment the switch $S w$ was disconnected. Find the energy of oscillations in the circuit
(a) immediately after the switch was disconnected;
(b) $t=0.30$ s after the switeh was disconnected.
4.110. Damped oscillations are induced in a circuit whose quality factor is $Q=50$ and natural oscillation frequency is $v_{4}=5.5 \mathrm{kHz}$. How soon will the energy stored in the eircuit decrease $\eta=2.0$ times?
4.111. An oscillating circuit incorporates a leaking capacitor. Its eapacitance is equal to $C$ and active resistance to $R$. The coil inductance is $L$. The resistance of the coil and the wires is negligible. Find:
(a) the damped oscillation frequency of such a circuit;
(b) its quality factor.
4.112. Find the quality factor of a circuit with eapacitance $C=$ $=2.0 \mu \mathrm{F}$ and inductance $L=5.0 \mathrm{mH}$ if the maintenance of undamped oscillations in the eireuit with the voltage amplitude across the capacitor being equal to $V_{N}=1.0 \mathrm{~V}$ requires a power $(P)=$ $=0.10 \mathrm{~mW}$. The damping of oscillations is sufficiently low.
4.113. What mean power should be fed to an oscillating circuit with active resistance $R=0.45 \Omega$ to maintain undamped harmonic oscillations with current amplitude $I_{m}=30 \mathrm{~mA}$ ?
4.114. An oseillating eireuit consists of a capacitor with capacitance $C=1.2 \mathrm{nF}$ and a coil with induetance $L=6.0 \mu \mathrm{H}$ and active resistance $R=0.50 \Omega$. What mean power should be fed to the circuit to maintain undamped harmonic oscillations with voltage amplitude across the capacitor being equal to $V_{m}=10 \mathrm{~V}$ ?
4.115. Find the damped oscillation frequency of the circuit shown in Fig. 4.30. The capacitance $C$, inductance $L$, and active resistance $R$ are supposed to be known. Find how must $\boldsymbol{C}, \boldsymbol{L}$, and $\mathrm{B}$ be interrelated to make oscillations possible.
rie. 4.30 ,
ทie. 4.31.
4.116. There are two oscillating eircuits (Fig. 4.31) with capacitors of equal capacitances. How must inductances and active resistances of the coils be interrelated for the frequencies and damping of free oscillations in both eircuits to be equal? The mutual induetance of coils in the left circuit is negligible.
4.117. A circuit consists of a capacitor with capacitance $C$ and a coil of inductance $L$ connected in series, as well as a switch and a resistance equal to the critical value for this circait. With the switeh disconnected, the capacitor was charged to a voltage $V_{4}$, and at the moment $t=0$ the switch was closed. Find the current $t$ in the circuit as a function of time $t$. What is $t_{\operatorname{mex}}$ equal to?
4.118. A coil with active resistance $h$ and inductance $L$ was connected at the moment $t=0$ to a source of voltage $V=V_{\mathrm{m}} \cos \omega t$. Find the curreat in the coil as a function of time $t$.
4.119. A circuit consisting of a capacitor with capacitance $C$ and a resistance $\boldsymbol{R}$ connected in series was connected at the moment $t=0$ to a source of ac voltage $V=V_{\mathrm{m}}$ cos at. Find the current in the circuit as a function of time $t$.
4.120. A long one-layer solenoid tightly wound of wire with resistivity $p$ has $n$ terns per unit length. The thickness of the wire insulation is negligible. The cross-sectional radius of the solenoid is equal to a. Find the phase diflerence between current and alternating voltage fed to the solenoid with frequency v.
4.121. A circuit consisting of a capacitor and an active resistance $R=110 \Omega$ connected in series is fed an alternating voltage with amplitude $V_{m}=110 \mathrm{~V}$. In this case the amplitude of steady-state current is equal to $I_{m}=0.50 \mathrm{~A}$. Find the phase difference between the current and the voltage fed.
4.122. Fig. 4.32 illustrates the simplest ripple filter. A voltage $V=V_{0}(1+\cos \omega t)$ is fed to the left input. Find:
(a) the output voltage $\boldsymbol{V}^{* \prime}(t)$;
(b) the magraitude of the product $R C$ at which the output amplitude of alternating voltage component is $\eta=7.0$ times less than the direct voltage component, if $\omega=314,-1$.
Fie. 4.32 .
Fig. 4.33.
4.123. Draw the approximate voltage vector diagrams in the electric circuits shown in Fig. $4.33 \mathrm{a}, \mathrm{b}$. The external voltage $\boldsymbol{V}$ is assumed to be alternating harmonically with frequeney $\Leftrightarrow$.
4.124. A series circuit consisting of a capacitor with capacitance $C=22 \mu \mathrm{F}$ and a coil with active resistance $\boldsymbol{R}-20 \Omega$ and induetance $L=0.35 \mathrm{H}$ is connected to a source of alternating voltage with amplitude $V_{\mathrm{m}}=180 \mathrm{~V}$ and frequency $\omega=314 \mathrm{~s}-1$. Find:
(a) the current amplitude in the circuit;
(b) the phase difference between the current and the external voltage:
(c) the amplitudes of voltage across the capacitor and the coil.
4.125. A series circuit consisting of a capacitor with capacitance $C$, a resistance $R$, and a coil with inductance $L$ and negligible active
resistance is connected to an oseillator whose frequency can be varied without changing the voltage amplitude. Find the frequency at which the voltage amplitude is maximum
(a) across the capacitor;
(b) across the coil.
4.126. An alternating voltage with frequency $\omega=314 \mathrm{~s}-1$ and amplitude $V_{\mathrm{m}}=180 \mathrm{~V}$ is fed to a series circuit consisting of a capacitor and a coil with active resistance $R=40 \Omega$ and inductance $L=0.36 \mathrm{H}$. At what value of the capacitor’s capacitance will the voltage amplitude scross the coil be maximum? What is this amplitude equal to? What is the corresponding voltage amplitude across the condenser?
4.127. A capacitor with capacitance $C$ whose interelectrode space is filled up with poorly conducting medium with active resistance $\boldsymbol{R}$ is connected to source of alternating voltage $V=V_{m} \cos \omega t$. Find the time dependence of the steady-state current fowing in lead wires. The resistance of the wires is to be neglected.
4.128. An oscillating circuit consists of a capacitor of capacitance $C$ and a solenoid with inductance $L_{4}$. The solenoid is inductively connected with a short-circuited coil having an inductance $L_{y}$ and a negligible active resistance. Their mutual inductance coefficient is equal to $L_{n 2}$. Find the natural frequency of the given oscillating circuit.
4.129. Find the quality factor of an oscillating circuit consected in series to a source of alternating emf if at resonance the voltage across the capacitor is $n$ times that of the source.
4.130. An oscillating circuit consisting of a coil and a capacitor connected in series is fed an alternating emf, with coil inductance being chosen to provide the maximum current in the circuit. Find the quality factor of the system, provided an $n$-fold increase of inductance results in an 7 -fold decrease of the current in the circuit.
4.131. A series circuit consisting of a capacitor and a coil with active resistance is connected to a source of harmonic voltage whose frequency can be varied, keeping the voltage amplitudeconstant. At frequencies $\omega_{1}$ and $\omega_{2}$ the current amplitudes are $n$ times less then the resonance amplitude. Find:
(a) the resonance frequency;
(b) the quality factor of the circuit.
4.132. Demonstrate that at low damping the quality factor $Q$ of a circuit maintaining forced oscillations is approximately equal to $\omega_{4} / \Delta \omega$, where $\omega_{4}$ is the natural oscillation frequency, $\Delta \omega$ is the width of the resonance curve $I(\omega)$ at the \”height\” which is $V \overline{2}$ times less than the resonance current amplitude.
4.133. A eireuit consisting of a capacitor and a coll consected in series is fed two alternating voltages of equal amplitudes but different frequencies. The frequency of one voltage is equal to the natural oscillation frequency $\left(\omega_{0}\right)$ of the circuit, the frequency of the other voltage is $\eta$ times higher. Find the ratio of the current amplitedes $(I, I)$ generated by the two voltages if the quality factor of the syster is equal to $Q$. Calculate this ratio for $Q=10$ and 100 , if $\boldsymbol{\eta}=\mathbf{1 . 1 0}$.
4.134. It takes $t_{0}$ hours for a direct current $I_{0}$ to charge a storage battery. How long will it take to charge such a battery from the mains using a half-wave rectifier, if the effective current value is also equal to $I_{0}$ ?
4.135. Find the effective value of eurrent if its mean value is $I_{6}$ and its time dependence is
(a) shown in Fig. 4.34;
(b) $I \sim \mid$ sin et $\mid$.
Fie. 4.34.
4.136. A solenoid with inductance $L-7 \mathrm{mH}$ and active resistance $R=\mathbf{4} \Omega$ is first consected to a source of direct voltage $V_{\text {, and }}$, then to a source of sinusoidal voltage with effective value $V^{0}=V_{4}$. At what frequency of the oscillator will the power consumed by the solenoid be $\eta=5.0$ times less than in the former case?
4.137. A coil with inductive resistance $X_{L}=30 \Omega$ and impedance $\mathbf{Z}=50 \mathrm{Q}$ is connected to the mains with eflective voltage value $V=100 \mathrm{~V}$. Find the phase dillerence between the current and the voltage, as well as the heat power generated in the coil.
4.138. A coil with inductance $L=0.70 \mathrm{H}$ and active resistance $r=20 \Omega$ is connected in series with an inductance-Iree resistance $\boldsymbol{R}$. An alternating voltage with eflective value $\boldsymbol{V}=220 \mathrm{~V}$ and frequency $\omega=314 \mathrm{~s}^{-1}$ is applied across the terminals of this eircuit. At what value of the resistance $\boldsymbol{R}$ will the maximum heat power be generated in the circuit? What is it equal to?
4.139. A circuit consisting of a capacitor and a coil in series is connected to the mains. Varying the capacitance of the capacitor, the heat power generated in the coll was increased $n=1.7$ times. How nuch (in per cent) was the value of cos $\$$ changed in the process?
4.140. A source of sinusoidal emt with constant voltage is connected in series with an oseillating circuit with quality factor $Q=$ -100 . At a certain frequency of the external voltage the heat power generated in the circuit reaches the maximum value. How much (in per cent) should this frequency be shifted to decrease the power generated $n=2.0$ times?
4.141. A series cireuit consisting of an indwetance-free resistance $\boldsymbol{R}=0.16 \mathrm{k} \Omega$ and a coil with active resistance is corinected to the mains with effective voltage $V=220 \mathrm{~V}$. Find the heat power generated in the coil if the eflective voltage values across the resistance $\boldsymbol{A}$ and the coil are equal to $V_{1}=80 \mathrm{~V}$ and $V_{i}=180 \mathrm{~V}$ respectively.
186
4.142. A coil and an inductance-free resistance $R=25 \Omega$ are connected in parallel to the ac mains. Find the heat power generated in the coil provided a current $t=0.90 \mathrm{~A}$ is drawn from the mains. The coil and the resistance $R$ carry currents $I_{1}=0.50 \mathrm{~A}$ and $I_{1}=$ $=0.60 \mathrm{~A}$ respectively.
4.143. An alternating current of frequency $e-314 \mathrm{~s}^{-1}$ is fed to a circuit consisting of a capacitor of capacitance $C=73 \mu \mathrm{F}$ and an active resistance $\boldsymbol{R}-100 \Omega$ connected in parallel. Find the impedance of the circuit.
4.144. Draw the approximate vector diagrams of currents in the eireuits shown in Fig. 4.35. The voltage applied across the points A and $B$ is assumed to be sinusoidal; the parameters of each circuit are so chosen that the total current $I_{\text {, lags in }}$ voltage by an angle $\uparrow$.
nie. 4.35.
4.145. A capacitor with capacitance $C=1.0 \mu \mathrm{F}$ and a coil with active resistance $R=0.10 \Omega$ and inductance $L=1.0 \mathrm{mH}$ are conneeted in parallel to a source of sinasoidal voltage $\boldsymbol{V}=31 \mathrm{~V}$. Find:
(a) the frequency o at which the resonance sets in;
(b) the eflective value of the fed current in resonance, as well as the corresponding currents lowing through the coil and through the capacitor.
4.146. A capacitor with capacitance $C$ and a coll with active resistance $R$ and inductance $L$ are connected in parallel to a source of sinusoidal voltage of frequency e. Find the phase difference between the carrent fed to the circuit and the source voltage.
4.147. A circuit consists of a capacitor with capacitunce $C$ and a coil with active resistance $A$ and inductance $L$ connected in parallel. Find the impedance of the circuit at frequency of alternating voltage.
4.148. A ring of thin wire with active resistance $R$ and inductance $L$ rotates with constant angular velocity $\omega$ in the external uniform magnetie field perpendicular to the rotation axis. In the process, the flux of magnetic induction of external field across the ring varies with time as $\Phi=\Phi_{0} \cos$ ot. Demonstrate that
(a) the inductive current in the ring varies with time as $I=$ $=I_{7} \sin (\omega t-\vartheta)$, where $I_{\mathrm{s}}=\omega \Phi_{v} / \sqrt{R^{2}+\omega^{2} L}$ with $\tan \varphi=$ $=\omega L / R$
(b) the mean mechanical power developed by external forces to maintain rotation is defined by the formula $P=1 / 2^{2} \Phi ! R /\left(R^{n}+\right.$ $+\omega^{2} L^{2}$ ).
4.149. A wooden core (Fig. 4.36) supports two coils: coil $I$ with inductance $L_{1}$ and short-eircuited coil 2 with active resistance $R$ and inductance $L_{2}$. The mutual inductance of the cofls depends on
Fig. 4.36 ,
the distance $x$ between them according to the $1 n w L_{n}(x)$. Find the mean (averaged over time) value of the interaction force between the coils when coil $I$ carries an alternating cerrent $I_{1}-I_{0}$ eos $0 t$.