How to find Qmax of a capacitor in parallel RC circuit?

[woaname]

Homework Statement

i tried using the usual Q=VC to answer this question for a multiloop RC circuit, "What is Q(∞), the charge on the capacitor after the switch has been closed for a very long time?" and my answer was wrong. clearly, i can't find Qmax because the resistors contribute to the voltage drop.
i can't seep to wrap my head around finding the proper equations for finding Q using kirchhoffs laws and end up having multiples unknowns. the image of the schematic is attached. any help? thnx.

Homework Equations

kirchoff's loop rule and junction rules
I) I1=I2+I3
II) -I2R2+Q/C+I3R3=0
III) -I3R3-I4R4+V-I1R1=0

The Attempt at a Solution

based on a previous question, i was able to find I4= V ((Req for R2andR3)+R4+R1)

 

[gneill]

Homework Statement

i tried using the usual Q=VC to answer this question for a multiloop RC circuit, "What is Q(∞), the charge on the capacitor after the switch has been closed for a very long time?" and my answer was wrong. clearly, i can't find Qmax because the resistors contribute to the voltage drop.
i can't seep to wrap my head around finding the proper equations for finding Q using kirchhoffs laws and end up having multiples unknowns. the image of the schematic is attached. any help? thnx.

Homework Equations

kirchoff's loop rule and junction rules
I) I1=I2+I3
II) -I2R2+Q/C+I3R3=0
III) -I3R3-I4R4+V-I1R1=0

The Attempt at a Solution

based on a previous question, i was able to find I4= V ((Req for R2andR3)+R4+R1)

Suppose you were to remove the capacitor from the circuit. What voltage would be presented at the terminals where it was connected?

 

[woaname]

removing the capacitor, it would be a regular combination circuit where V3=V2, R3 and R2 would have an equivalent Req= 1/((1/r2)+(1/r3)). but would the current be additive, or equivalent (based on kirchhoffs rule i1=i2+i3) ?
the potential would be same at both junctions.

 

[gneill]

By remove I mean cut it out of the circuit and leave its connection points open. Like this:

 

[woaname]

oh! then only one loop remains. that's what establishes one of the loop equations:
-i3r3-i4r4+v-i1r1=0.
here, we'd have known value for V and i4r4, but unknowns for i3 and i1

 

[gneill]

oh! then only one loop remains. that's what establishes one of the loop equations:
-i3r3-i4r4+v-i1r1=0.
here, we'd have known value for V and i4r4, but unknowns for i3 and i1

If it's a single loop, and all the components are connected in series, what can you say about the current in all the components?

 

[woaname]

If it's a single loop, and all the components are connected in series, what can you say about the current in all the components?

same throughout. so the equation would simplify to :
-I(R3+R4+R1)= -V
the I would be the one i found for R4...?

 

[gneill]

same throughout. so the equation would simplify to :
-I(R3+R4+R1)= -V
the I would be the one i found for R4...?

Perhaps. I never saw your work for finding that.

So what's the potential at the output terminals?

 

[woaname]

Perhaps. I never saw your work for finding that.

So what's the potential at the output terminals?

the answer i found for r4 was marked correct, therefore my confident guarantee of it :D.
just to get clear on the terminology, would the output terminals be the junctions where i1 splits into i3 and i2, or the void created by removing the capacitor?
i would presume the potential would be the same at either end.

 

[gneill]

the answer i found for r4 was marked correct, therefore my confident guarantee of it :D.
just to get clear on the terminology, would the output terminals be the junctions where i1 splits into i3 and i2, or the void created by removing the capacitor?
i would presume the potential would be the same at either end.

I intended the "void" as you call it, and as I drew it in the diagram that I provided in a post above. I'm looking for Vc as indicated.

 

[woaname]

So maybe in wrong here, but then there are no unknowns?All the R values for the resistors and the V from the battery are given, and current for R4 I found was based on this question,"The switch has been open for a long time when at time t = 0, the switch is closed. What is I4(0), the magnitude of the current through the resistor R4 just after the switch is closed?"

I'm lost now. What am I solving for?

 

[woaname]

Perhaps. I never saw your work for finding that.

So what's the potential at the output terminals?

the answer i found for r4 was marked correct, therefore my confident guarantee of it :D.
just to get clear on the terminology, would the output terminals be the junctions where i1 splits into i3 and i2, or the void created by removing the capacitor?
i would presume the potential would be the same at either end.

 

[gneill]

I have no idea what things you solved for before, or what the component values might be since you haven't posted them in this thread.

I am approaching the problem that you stated in the first post of this thread: "What is Q(∞), the charge on the capacitor after the switch has been closed for a very long time?" So we are looking for Q(∞), right?

So as far as I know, we are dealing with the problem symbolically (without numerical values to plug in), and the switch has been closed for a long time so that the circuit has reached a steady state with the capacitor charged to its maximum amount. To accomplish this, it would be expedient to find the steady-state potential across the capacitor. That is why the potential across the output terminals that I drew is important; that voltage will be the same as that across the capacitor at steady state.

 

[woaname]

you're right. should i post the whole question with all the known values? it might be better approachable then

 

[woaname]

so here is the question:

A circuit is constructed with four resistors, one capacitor, one battery and a switch as shown. The values for the resistors are: R1 = R2 = 31 Ω, R3 = 108 Ω and R4 = 141 Ω. The capacitance is C = 44 μF and the battery voltage is V = 12 V. The positive terminal of the battery is indicated with a + sign.

The switch has been open for a long time when at time t = 0, the switch is closed. What is I4(0), the magnitude of the current through the resistor R4 just after the switch is closed?
What is Q(∞), the charge on the capacitor after the switch has been closed for a very long time?

After the switch has been closed for a very long time, it is then opened. What is Q(topen), the charge on the capacitor at a time topen = 599 μs after the switch was opened?

What is IC,max(closed), the current that flows through the capacitor whose magnitude is maximum during the time when the switch is closed? A positive value for the current is defined to be in the direction of the arrow shown.

What is IC,max(open), the current that flows through the capacitor whose magnitude is maximum during the time when the switch is open? A positive value for the current is defined to be in the direction of the arrow shown.

 

[gneill]

so here is the question:

A circuit is constructed with four resistors, one capacitor, one battery and a switch as shown. The values for the resistors are: R1 = R2 = 31 Ω, R3 = 108 Ω and R4 = 141 Ω. The capacitance is C = 44 μF and the battery voltage is V = 12 V. The positive terminal of the battery is indicated with a + sign.

The switch has been open for a long time when at time t = 0, the switch is closed. What is I4(0), the magnitude of the current through the resistor R4 just after the switch is closed?
What is Q(∞), the charge on the capacitor after the switch has been closed for a very long time?

After the switch has been closed for a very long time, it is then opened. What is Q(topen), the charge on the capacitor at a time topen = 599 μs after the switch was opened?

What is IC,max(closed), the current that flows through the capacitor whose magnitude is maximum during the time when the switch is closed? A positive value for the current is defined to be in the direction of the arrow shown.

What is IC,max(open), the current that flows through the capacitor whose magnitude is maximum during the time when the switch is open? A positive value for the current is defined to be in the direction of the arrow shown.

Okay, that makes things nice and clear

Am I correct in assuming that we are working on this part:

The switch has been open for a long time when at time t = 0, the switch is closed. What is I4(0), the magnitude of the current through the resistor R4 just after the switch is closed?
What is Q(∞), the charge on the capacitor after the switch has been closed for a very long time?

 

[woaname]

yes sir :D.

 

[gneill]

yes sir :D.

Okay, so I come back to the question, what is the voltage across the open terminals where the capacitor would sit?

 

[woaname]

Okay, so I come back to the question, what is the voltage across the open terminals where the capacitor would sit?

using the equation we reached at before, -I(R3+R4+R1)= -V, i used the current for I4 from part 1 and multiplied it by the sum of the three resitors in the left loop and got 17.135 V ... :S which is greater than V(battery) !

 

[gneill]

using the equation we reached at before, -I(R3+R4+R1)= -V, i used the current for I4 from part 1 and multiplied it by the sum of the three resitors in the left loop and got 17.135 V ... :S which is greater than V(battery) !

What's your current value? Remember, this is steady state when there's no current through R2 and the capacitor. This will not be the same current as when the capacitor is first starting to charge up.

 

[woaname]

i guess i didnt pay attention before, but the I4 from part 1 (which was 0.06119 A) pertains to t=0 (thanks for pointing it out). so at present, i don't have a value for the current. would i find the sum of the resistors and divide it from the battery voltage? if so, v=0.042857143

 

[gneill]

i guess i didnt pay attention before, but the I4 from part 1 (which was 0.06119 A) pertains to t=0 (thanks for pointing it out). so at present, i don't have a value for the current. would i find the sum of the resistors and divide it from the battery voltage? if so, v=0.042857143

Okay, so I = 42.86 mA .

What then is the potential drop across R3?

 

[woaname]

Okay, so I = 42.86 mA .

What then is the potential drop across R3?

multiplied the current by each of the three resistors: R1= 1.328 V, R3=4.628 V, R4=6.042 V. so from here, would i be safe to presume that V3 is equivalent to the branch with R2 and the capacitor (reconnected)? I'm just trying to actively be a part of the solution :D, but i may be wrong.

 

[gneill]

Yes, the potential across R3 will be the potential presented at the open terminals where the capacitor will connect. It will thus be the steady-state potential across the capacitor...

 

[woaname]

oh finally! thankyou. but could you summarize the theory behind the process we went through? it could help clarify the big picture for me. thnx

 

[gneill]

The idea is that at steady state there will be no current flowing through the capacitor branch. This is the case when the potential across the capacitor is the same as that provided at its branch's connection points to the rest of the network. No potential difference means no current.

So, to find out what that potential difference is, remove the capacitor and determine the potential at the open terminals thus created.

 

[woaname]

thankyou! one last thing. if i have further questions for this same scenario, do i just add to this thread or begin another one with all the information? thanks once again!

 

[gneill]

thankyou! one last thing. if i have further questions for this same scenario, do i just add to this thread or begin another one with all the information? thanks once again!

Now that the full question statement is in the thread, might as well keep everything on one place

 

[woaname]

so here is the question:

A circuit is constructed with four resistors, one capacitor, one battery and a switch as shown. The values for the resistors are: R1 = R2 = 31 Ω, R3 = 108 Ω and R4 = 141 Ω. The capacitance is C = 44 μF and the battery voltage is V = 12 V. The positive terminal of the battery is indicated with a + sign.

The switch has been open for a long time when at time t = 0, the switch is closed. What is I4(0), the magnitude of the current through the resistor R4 just after the switch is closed?
What is Q(∞), the charge on the capacitor after the switch has been closed for a very long time?

After the switch has been closed for a very long time, it is then opened. What is Q(topen), the charge on the capacitor at a time topen = 599 μs after the switch was opened?

regarding the bolded question: i gathered that the answer i got for the previous (first) question, 203.65 microseconds, that i would use the equation
Q(t)=Q(1-e^(-t/RC)
with the time constant RC = R(2and3)*C. doing the calculation, the answer is definitely wrong. am i mistaken in using the answer from the previous question as Qmax ? or is my time constant wrong?

 

[gneill]

The equation you have used represents the charge on a capacitor increasing from zero to a maximum of Q over time. You want instead an equation that represents a decay from Q down towards zero. Your time constant contains the right components.

 

[woaname]

oh, charging vs discharging. right. thanks.

 

[woaname]

What is IC,max(closed), the current that flows through the capacitor whose magnitude is maximum during the time when the switch is closed? A positive value for the current is defined to be in the direction of the arrow shown.

What is IC,max(open), the current that flows through the capacitor whose magnitude is maximum during the time when the switch is open? A positive value for the current is defined to be in the direction of the arrow shown.
[/I]

for these questions, the first one would use the charging equation for current:
I(t)=(V(battery)/Rtotal)*EXP(-t/RC)
and the second would the discharging equation:
I(t)=(Qmax)/RC)*EXP(-t/RC)

is this correct?

 

[gneill]

for these questions, the first one would use the charging equation for current:
I(t)=(V(battery)/Rtotal)*EXP(-t/RC)
and the second would the discharging equation:
I(t)=(Qmax)/RC)*EXP(-t/RC)

is this correct?

In the first case where the capacitor is charging, the resistor network will complicate things a bit since there's a voltage, and hence current, division taking place. You'll have to find the current through R2 when the capacitor is replaced by a short circuit (i.e. what the capacitor "looks like" when it's initially uncharged) to determine the leading coefficient for the exponential.

For the second equation, again the correct choice for R makes all the difference, but essentially that's the correct form for the equation.

 

[woaname]

no luck trying on my own... the equations aren't working out. i haven't been given time values to substitute for t. and for the value of RC, this is what i have in understanding:
1) for Ic,max(closed), i can't simply add all the resistances, but i can't find how to include R3 in there.
2) only the smaller loop will be in action, so the RC value would include only the two live resistors in circuit. but then, the time variable stands unsolved.

 

[gneill]

no luck trying on my own... the equations aren't working out. i haven't been given time values to substitute for t. and for the value of RC, this is what i have in understanding:
1) for Ic,max(closed), i can't simply add all the resistances, but i can't find how to include R3 in there.

If you're looking for Ic's maximum value, that occurs the instant the switch is closed and the capacitor is completely uncharged. How does an uncharged capacitor behave (what does it "look like" to the rest of the circuit)?

2) only the smaller loop will be in action, so the RC value would include only the two live resistors in circuit. but then, the time variable stands unsolved.

What are you looking to determine? Why is a time value required?