# Two Loop RC Circuit

1. Feb 16, 2017

### marchcha

1. The problem statement, all variables and given/known data
A circuit is constructed with four resistors, one capacitor, one battery and a switch as shown. The values for the resistors are: R1 = R2 = 41 Ω, R3 = 68 Ω and R4 = 154 Ω. The capacitance is C = 49 μF and the battery voltage is V = 12 V. The positive terminal of the battery is indicated with a + sign.

What is Q(∞), the charge on the capacitor after the switch has been closed for a very long time?
2. Relevant equations
Kirchhoff's Laws

3. The attempt at a solution
I attempted to find Q(inf) by setting up kirchoff's voltage and loop equations as shown:
1. I1 = I4
2. 14 = I2 + I3
3. V - I1R1 - I3R3 - I4R4 = 0
4. V - I1R1 - I2R2 - (Q/C) - I4R4 = 0

I used the equalities in equation 1 to solve equation 3 in terms of I3. I then used I3 from equation 3 and plugged it into equation 2 to get an equation for I2. I then plugged in what I got for I2 into equation 4 as well as plugging in I4 for I1. The final equation I got for the value of Q is: Q = C(V - I4R4 - (I4 - (V - I4(R1 + R4))/R3) * R2 - I4R4). I had already found the value for I4 in a previous problem using V = IR when the switch is closed for a very long time. I4's value is 0.54A.

Thank you so much in advance!

2. Feb 16, 2017

### Staff: Mentor

Hi marchcha,

You may be putting too much effort into this one

You're looking for the charge on the capacitor at steady state, Q(inf). What do you know about the value of the capacitor current $I_c$ at steady state? What does that tell you about the loop current for that loop?

3. Feb 16, 2017

### marchcha

The capacitors current will be zero at steady state. I'm a little confused as to what this tells us about the loop current for that loop.

4. Feb 16, 2017

### Staff: Mentor

Yes, $I_c = 0$ at steady state.

Is there any way to distinguish $I_c$ from the loop current? Aren't they identical? Just as the loop current for the first loop is identical to the currents flowing through the voltage source, R1, and R4.

5. Feb 16, 2017

### marchcha

So would I2 also be 0?

6. Feb 16, 2017

### Staff: Mentor

Yes. $I2 = I_c = 0$ at steady state.

7. Feb 16, 2017

### marchcha

So then Q = C (V - I1R1 - I4R4)?

8. Feb 16, 2017

### Staff: Mentor

Sure. Or Q = C*I3R3, to use your notation. You really don't need to invent so many different currents if there's just one loop current flowing....

9. Feb 16, 2017

### marchcha

Ok, then how do you find I3?

10. Feb 16, 2017

### Staff: Mentor

Draw in the current flowing through loop 1. It's the only current flowing at steady state.

11. Feb 16, 2017

### marchcha

Ok so I3 is .0456A I dont understand how the capacitors charge has anything to do with the first loop though?

12. Feb 16, 2017

### Staff: Mentor

If $I_c = 0$, what's the current through R2? Hence, what's the potential drop across R2?

13. Feb 16, 2017

### marchcha

Zero for both.

14. Feb 16, 2017

### Staff: Mentor

So if the potential drop across R2 is zero, what's the difference between the potential at the top of the capacitor and the potential at the top of R3?

15. Feb 16, 2017

### marchcha

Would it be just V?

16. Feb 16, 2017

### Staff: Mentor

er, no. If the potential at the top of R3 is, say V3, and you "walk" from there across R2 which you say has a potential drop of zero, what's the potential at the other end of R2 (also the top of the capacitor)?

17. Feb 16, 2017

### marchcha

Ah so they'd be the same and you can say Q/C = I3R3 the equation above. Thanks!