In the circuit shown in Fig. P 30.74 , neither the battery nor the inductors have any appreciabl

In the circuit shown in Fig. P 30.74 , neither the battery...

In the circuit shown in Fig. $\mathrm{P} 30.74$ , neither the battery nor the inductors have any appreciable resistance, the capacitors are initially uncharged, and the switch $S$ has been in position 1 for a very long time. (a) What is the current in the circuit? (b) The switith is now suddenly flipped to position $2 .$ Find the maximum charge that each capacitor will receive, and how much time after the switch is flipped it will take them to acquire this charge.

In the circuit shown in Fig. $\mathrm{P} 30.74$ , neither the battery nor the inductors have any
appreciable resistance, the capacitors are initially uncharged, and the switch $S$ has been in position 1 for a very long time. (a) What is the current in the circuit? (b) The switith is now suddenly flipped to position $2 .$ Find the maximum charge that each capacitor will receive, and how much time after the switch is flipped it will take them to acquire this charge.

Key Concepts

Steady-State Analysis in DC Circuits

In a DC circuit that has reached steady state, reactive components behave in characteristic ways; inductors act as short circuits because the voltage drop across them is zero once the current is constant, while capacitors act as open circuits since no further current flows once they are fully charged. This basic understanding is essential for determining initial conditions before a switch alters the circuit configuration.

Transient Response in Switching Circuits

When a switch in a circuit is flipped suddenly, the circuit undergoes a transient response as it adjusts from one set of initial conditions to new operating conditions. This transient period is governed by the reactive elements (inductors and capacitors) that momentarily store and release energy, causing time-dependent changes in current and voltage throughout the circuit.

LC Oscillations and Resonance

An LC circuit, consisting of an inductor and a capacitor, can exhibit oscillatory behavior where energy shuttles back and forth between the magnetic field of the inductor and the electric field of the capacitor. These oscillations occur at the circuit's natural (resonant) frequency, and the maximum charge on the capacitor is reached when all the energy initially stored in the inductor is transferred to the capacitor.

Energy Conservation in Reactive Circuits

In ideal reactive circuits, energy is not lost but rather dynamically transferred between components. When a circuit is reconfigured through switching, the energy stored in an inductor can be completely converted into electrostatic energy in a capacitor. By equating the initial magnetic energy to the final electric energy, one can determine the maximum charge that the capacitor can accumulate during the transient response.

Transcript

00:01 Okay, so the current in the circuits just simply equal to the voltage from the source over the total resistance.

00:06 We know the source voltage is 75 volts and the total resistance 125 omega.

00:12 So therefore we have the currents is just simply equal to 75 volts over 125 omega, which is equal to 0 .6 ampere.

00:31 So for the next question, while we know the total charge q here can be equal to i times square root lc, i is the current, l is the inductance.

00:38 C is the equivalent capacitance, okay? so we know the current is 0 .6 ampere, and the inductance should be equal to 15 million henry plus 5 million herons, since two inductors is connected in series.

00:54 Okay? and this will give us the total inductance is equal to 20 million heres, which is 2 times 1022 henries.

01:02 And the equivalent capacitance is equal to 1 over 25 micro phara.

01:06 Plus 1 over 35 micro phara, so the power of negative 1.

01:10 Since both capacitance are connected in a series, circuits, okay? and this will give us the equivalent capacitance, is 14 .5a micro phara, which is 14 .5a times 10 to 0 .96 phara.

01:26 So therefore can determine the charge, which is equal to 0 .6 ampere times square root, lc, which is 2 times 10 to the power of negative 2 henry, and then times 14 .5 times 102 power of 96, farrah.

02:08 And this will give us the total charge is about 3 .27 times 10 to the power of negative 4 cullums.

02:23 Okay, so therefore the maximum charge that each capacitor will receive is 3 .27 times 10 to power negative 4...

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