Consider the circuit shown with R1=25 Ω, R2=33 ΩC1=12 μF, C2=23 μF, C3=46 μF, and V=6.0 V. (a) D

Consider the circuit shown with R1=25 Ω, R2=33 ΩC1=12 μF,...

Consider the circuit shown with $R_{1}=25 \Omega, R_{2}=33 \Omega$ $C_{1}=12 \mu \mathrm{F}, C_{2}=23 \mu \mathrm{F}, C_{3}=46 \mu \mathrm{F},$ and $V=6.0 \mathrm{V}$. (a) Draw an equivalent circuit with one resistor and one capacitor and label it with the values of the equivalent resistor and capacitor. (b) $\mathrm{A}$ long time after switch $S$ is closed, what are the charge on capacitor $C_{1}$ and the current in resistor $R_{1} ?$ (c) What is the time constant of the circuit?

Consider the circuit shown with $R_{1}=25 \Omega, R_{2}=33 \Omega$ $C_{1}=12 \mu \mathrm{F}, C_{2}=23 \mu \mathrm{F}, C_{3}=46 \mu \mathrm{F},$ and $V=6.0 \mathrm{V}$.
(a) Draw an equivalent circuit with one resistor and one capacitor and label it with the values of the
equivalent resistor and capacitor.
(b) $\mathrm{A}$ long time after switch $S$ is
closed, what are the charge on capacitor $C_{1}$ and the current in resistor $R_{1} ?$ (c) What is the time constant of the circuit?

Key Concepts

Equivalent Circuit Representation

This concept involves simplifying a more complex circuit into a basic form that has the same electrical behavior, typically by combining multiple resistors into a single equivalent resistor and multiple capacitors into a single equivalent capacitor. This simplification makes it easier to analyze and understand the overall behavior of the circuit, particularly when studying transient and steady state responses.

Series and Parallel Combinations

Understanding how components like resistors and capacitors combine in series and parallel is fundamental. Resistors in series add directly, while capacitors in series combine inversely, and vice versa for parallel arrangements. These rules are essential for calculating the equivalent values needed for circuit simplification.

DC Steady State Behavior in RC Circuits

In a DC circuit, after a long period, the transient effects disappear and the circuit reaches a steady state. For RC circuits, this means that the capacitor becomes fully charged and acts as an open circuit, leading to zero current through the resistor. The voltage across the capacitor equals the supply voltage or is determined by the voltage divider if other elements are present.

RC Time Constant

The time constant, represented as ? = RC, is crucial in predicting how quickly a capacitor charges or discharges in an RC circuit. It defines the rate at which transient behavior decays, with the voltage across the capacitor reaching about 63.2% of its final value after one time constant.

Transcript

00:02 Hi, in the given problem, a combination of resistors and capacitors is shown in this figure.

00:11 Two resistors in series combination, then two capacitors in parallel combination, and then one more.

00:20 Capacitor is again in series combination with all of them.

00:28 A source of emf and a switch.

00:34 Here this is r1, r2, capacetan c1, capacetton c2, and capacetin c3.

00:45 The values are r1 is 25 om, r2 is 33 oom.

00:55 Capacetan c1 is 12 micro -farrat, c2 is 23 micro -farrate, and c -3.

01:05 Is 6 .0 sorry c3 is 46 micro ferret the potential applied is 6 .0 volt in the first part of the problem we have to change this circuit diagram this replace this circuit diagram with one having a single equivalent resistance and a single equivalent capacitance so equivalent resistance as the two resistors are in series combination so this is r1 plus r2 means this is 25 plus 33 om which comes out to be 58 om and now for the equivalent capacitance initially the two capacitors are in parallel these two so their parallel combination is taken as cp is equal to c1 plus c2 means this is 12 plus 23 micro ferret which comes out to be 35 micro ferret now this parallel combination is in series with c3 so final equivalent capacitance comes out to be the parallel sorry series combination of c3 with cp and this is given as product of the two in numerator and their addition in denominator so this may be given as for c3 this is 46 micro -farrad multiplied by cp which is 35 micro -farrad we have just found it divided by 46 plus 35 micro -faird so canceling this one micro -faird it it comes out to be nearly equal to 20 micro ferret.

03:13 So now the equivalent circuit diagram here is, this is the single resistor representing the equivalent resistance and having a value of 58 -oom and this is a single capacitor representing the equivalent capacitance having capacitance of 20 micro -faird.

03:35 Then a source of potential 6 .0 volt and a switch here so this is the answer for the first part of the problem now in the second part of the problem a long time after switch s is closed the capacitors will be fully charged all the capacitors so there will be no current in the the circuit...

Please give Ace some feedback