Consider the circuittery has emf 72.0 V and negligible internal resistance. R2=2.00 Ω, C1=3.00

Consider the circuittery has emf 72.0 V and negligible...

Consider the circuittery has emf $72.0 \mathrm{~V}$ and negligible internal resistance. $R_{2}=2.00 \Omega$, $C_{1}=3.00 \mu \mathrm{F},$ and $C_{2}=6.00 \mu \mathrm{F}$ After the capacitors have attained their final charges, the charge on $C_{1}$ is $Q_{1}=18.0 \mu \mathrm{C} .$ What is (a) the final charge on $C_{2} ;$ (b) the resistance $R_{1} ?$

Consider the circuittery has emf $72.0 \mathrm{~V}$ and negligible internal resistance. $R_{2}=2.00 \Omega$, $C_{1}=3.00 \mu \mathrm{F},$ and $C_{2}=6.00 \mu \mathrm{F}$
After the capacitors have attained their final charges, the charge on $C_{1}$ is $Q_{1}=18.0 \mu \mathrm{C} .$ What is (a) the final charge on $C_{2} ;$ (b) the resistance $R_{1} ?$

Key Concepts

Capacitor Charge-Voltage Relationship

A fundamental property of capacitors is expressed by the equation Q = C × V, which indicates that the charge stored on a capacitor is equal to the product of its capacitance and the voltage across it. This relationship is vital for determining charge distribution in circuits where capacitors are present, regardless of their specific arrangement.

Capacitors in Series and Parallel Configurations

The configuration of capacitors—whether in series or parallel—significantly influences how voltage and charge are distributed among them. In series, each capacitor carries the same charge while the voltages add up to the total source voltage. In parallel, all capacitors share the same voltage across their terminals, although the stored charge may differ based on individual capacitance.

Role of Resistors in Charging Circuits

Resistors in circuits containing capacitors determine the rate at which capacitors charge or discharge, described by the RC time constant. Although resistors dictate the transient behavior, in the long-term steady state of a DC circuit, their effect becomes negligible as the capacitors essentially block direct current.

Kirchhoff’s Voltage Law (KVL) in Circuit Analysis

Kirchhoff’s Voltage Law is employed to analyze closed loops in circuits, ensuring that the sum of voltage drops equals the applied electromotive force. In capacitor circuits, KVL is used to calculate voltage distribution across various circuit elements, particularly in establishing the conditions needed for reaching steady state.

Steady?State Behavior of Capacitors

In DC circuits, once capacitors are fully charged, they eventually reach a steady state where the current ceases to flow through them. At this point, capacitors behave like open circuits, meaning that no further charging or discharging occurs. This concept is critical for analyzing circuits after a long time has elapsed since the application of the voltage source.

Transcript

0:00 Hi.

00:01 In the given problem, this is the circuit diagram having the resistance r1 here, then the two capacitors in parallel.

00:20 Here this is c1 and here this is c2.

00:28 And then another resistance r2 that is also in parallel with these two capacitors.

00:38 And finally, the battery having emf epsilon.

00:45 The value of the emf is 72 .0 volt.

00:51 Resistance r2 is 2 .00 oom.

00:57 R1 is missing here.

01:00 Capacitor.

01:03 The first capacitor is having capacity and c1 is equal to 3 .00 microferret.

01:09 And the second one is having c2 is equal to 6 .00 microfarrant.

01:17 It is said that the final charge stored by the capacitor c1 is q1 is equal to 18 .0 microculum.

01:30 And here final charge means the circuit is in steady state.

01:48 So no current will pass through the branches having capacitors.

02:02 So whole of the current will pass through these two resistors in series.

02:09 Now for the first capacitor, first of all, we consider here these two terminals as a and b.

02:19 Now, in the first part of the problem, vab, first of all, the potential drop taking place across these two terminals will be given by q1 by c1 because all these three means the two capacitors and one resistor, r2, all these three are in parallel and we know in parallel potential remains the same...

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