1. Physics 102: Lecture 7
RC Circuits 1

2. Recall …. First we covered circuits with batteries and capacitors
series, parallel
Then we covered circuits with batteries and resistors
Kirchhoff’s Loop and Junction Relations
Today: circuits with batteries, resistors, and capacitors

3. RC Circuits RC Circuits Charging Capacitors Discharging Capacitors
Intermediate Behavior

4. RC Circuits Circuits that have both resistors and capacitors:
With resistance in the circuits, capacitors do not charge and discharge instantaneously – it takes time (even if only fractions of a second). RK C S +
RNa
RCl εK
εNa
εCl

5. Capacitors
Charge (and therefore voltage) on Capacitors cannot change instantly: remember VC = Q/C
Short term behavior of Capacitor:
If the capacitor starts with no charge, it has no potential difference across it and acts as a wire
If the capacitor starts with charge, it has a potential difference across it and acts as a battery.
Long term behavior of Capacitor: Current through a Capacitor eventually goes to zero.
If the capacitor is charging, when fully charged no current flows and capacitor acts as an open circuit.
If capacitor is discharging, potential difference goes to zero and no current flows.

6. Charging Capacitors
Capacitor is initially uncharged and switch is open. Switch is then closed. What is current I0 in circuit immediately thereafter?
What is current I in circuit a long time later? C R S 
Initial current through battery.
Final current; final voltage across C.

7. Charging Capacitors: t=0
Capacitor is initially uncharged and switch is open. Switch is then closed. What is current I0 in circuit immediately thereafter?
Capacitor initially uncharged
Therefore VC is initially 0
Therefore C behaves as a wire (short circuit)
 – I0 R = 0
I0 = /R C R S  R 
Initial current through battery.
Final current; final voltage across C.
Opposite sign convention compared to your handouts!

8. Charging Capacitors: t>0
I0 = /R
Positive charge flows
Onto bottom plate (+Q)
Away from top plate (-Q)
As charge builds up, VC rises (VC=Q/C)
Loop:  – VC – I R = 0
I = (-VC)/R
Therefore I falls as Q rises
When t is very large ()
I = 0: no current flow into/out of capacitor for t large
VC =  - C R +  R 
Initial current through battery.
Final current; final voltage across C.
Demo

9. ACT/Preflight 7.1 Loop: E – I(2R) – q/C = 0 q = 0  I = E /(2R) e
Both switches are initially open, and the capacitor is uncharged. What is the current through the battery just after switch S1 is closed? + 2R -
1) Ib = 0 2) Ib = E /(3R)
3) Ib = E /(2R) 4) Ib = E /R Ib + + e C R - -
Loop: E – I(2R) – q/C = 0
q = 0
 I = E /(2R) S2 S1 20

10. ACT/Preflight 7.3 KVL: E – 0 – q/C = 0  q = E C
Both switches are initially open, and the capacitor is uncharged. What is the current through the battery after switch 1 has been closed a long time? 2R C e R S2 S1 Ib + -
1) Ib = 0 2) Ib = E/(3R)
3) Ib = E/(2R) 4) Ib = E/R
Long time  current through capacitor is zero
Ib=0 because the battery and capacitor are in series.
KVL: E – 0 – q/C = 0  q = E C 24

11. Discharging Capacitors
Capacitor is initially charged (Q) and switch is open. Switch is then closed. What is current I0 in circuit immediately thereafter?
What is current I in circuit a long time later? R C S

12. Discharging Capacitors+
Capacitor is initially charged (Q) and switch is open. Switch is then closed. What is current I0 in circuit immediately thereafter?
Q/C – I0R = 0
I0 = Q/RC
What is current I in circuit a long time later?
I = 0 C - R
Demo

13. ACT/Preflight 7.5 e Loop: q0/C – IR = 0
After switch 1 has been closed for a long time, it is opened and switch 2 is closed. What is the current through the right resistor just after switch 2 is closed? 2R C e R S2 S1 IR + -
1) IR = 0 2) IR = e /(3R)
3) IR = e /(2R) 4) IR = e /R
Loop: q0/C – IR = 0
Recall q is charge on capacitor after charging:
q0= e C (since charged w/ switch 2 open!)
e – IR = 0
 I = e /R
Followup…what is current a long time later? What is charge on capacitor a long time later? 42

14. ACT: RC Circuits right loop: Q/C – IR = 0
Both switches are closed. What is the final charge on the capacitor after the switches have been closed a long time? + 2R -
1) Q = 0 2) Q = C E /3
3) Q = C E /2 4) Q = C E IR + + + e C - R - -
right loop: Q/C – IR = 0
outside loop: E – I(2R) – IR = 0
I = E /(3R)
Q/C – E /(3R) R = 0
Q = C E /3 S1 S2 27

15. RC Circuits: Charging Loop: e – I(t)R – q(t) / C = 0
The switches are originally open and the capacitor is uncharged. Then switch S1 is closed.
Loop: e – I(t)R – q(t) / C = 0
Just after…: q =q0
Capacitor is uncharged
e – I0R = 0  I0 = e / R
Long time after: Ic= 0
Capacitor is fully charged
e – q/C =0  q = e C
Intermediate (more complex)
q(t) = q(1-e-t/RC)
I(t) = I0e-t/RC + R + e - I - C + - S2 S1 t q RC
2RC q 17

16. RC Circuits: Discharging
Loop: q(t) / C + I(t) R = 0
Just after…: q=q0
Capacitor is still fully charged
q0 / C + I0 R = 0  I0 = -q0 / (RC)
Long time after: Ic=0
Capacitor is discharged (like a wire)
q / C = 0  q = 0
Intermediate (more complex)
q(t) = q0 e-t/RC
Ic(t) = I0 e-t/RC + R + e - I - C + - S1 S2 q RC
2RC t 40

17. What is the time constant?
The time constant  = RC.
Given a capacitor starting with no charge, the time constant is the amount of time an RC circuit takes to charge a capacitor to about 63.2% of its final value.
The time constant is the amount of time an RC circuit takes to discharge a capacitor by about 63.2% of its original value.

18. Time Constant Demo Example
Each circuit has a 1 F capacitor charged to 100 Volts.
When the switch is closed:
2 I=2V/R
Which system will be brightest?
Which lights will stay on longest?
Which lights consumes more energy? 1
Same U=1/2 CV2 2 1
t = 2RC
t = RC/2 50

19. Summary of Concepts
Charge (and therefore voltage) on Capacitors cannot change instantly: remember VC = Q/C
Short term behavior of Capacitor:
If the capacitor starts with no charge, it has no potential difference across it and acts as a wire
If the capacitor starts with charge, it has a potential difference across it and acts as a battery.
Long term behavior of Capacitor: Current through a Capacitor eventually goes to zero.
If the capacitor is charging, when fully charged no current flows and capacitor acts as an open circuit.
If capacitor is discharging, potential difference goes to zero and no current flows.
Intermediate behavior: Charge and current exponentially approach their long-term values  = RC

20. Practice! Example I E – I0R – q0/C = 0 E – I0R – 0 = 0 I0 = E/Rε S1
R=10W
C=30 mF
ε =20 Volts + -
Calculate current immediately after switch is closed: I
E – I0R – q0/C = 0 + + -
E – I0R – 0 = 0 -
I0 = E/R
Calculate current after switch has been closed for 0.5 seconds:
Calculate current after switch has been closed for a long time:
After a long time current through capacitor is zero!
Calculate charge on capacitor after switch has been closed for a long time:
E – IR – q∞/C = 0
E – 0 – q∞ /C = 0
q∞ = E C 32

21. ACT: RC Challenge q(t) = q0 e-t/RC = q0 (e-0.06 /(4(1510-3)))
ε = 24 Volts
R = 2 W
C = 15 mF
After being closed for a long time, the switch is opened. What is the charge Q on the capacitor 0.06 seconds after the switch is opened? R C 2R ε
1) q0 2) q0
3) q0 4) q0 S1
q(t) = q0 e-t/RC
= q0 (e-0.06 /(4(1510-3)))
= q0 (0.368) 45

22. Charging: Intermediate Times
Example
Calculate the charge on the capacitor 310-3 seconds after switch 1 is closed.
R = 10 W
V = 50 Volts
C = 100mF
q(t) = q(1-e-t/RC)
= q(1-e-310-3 /(2010010-6)))
= q (0.78)
Recall q =  C
= (50)(100x10-6) (0.78)
= 3.9 x10-3 Coulombs 2R C R S2 S1 Ib + - e 38

23. RC Summary Charging Discharging q(t) = q(1-e-t/RC) q(t) = q0e-t/RC
V(t) = V(1-e-t/RC)
V(t) = V0e-t/RC
I(t) = I0e-t/RC
I(t) = I0e-t/RC
Time Constant t = RC
Large t means long time to charge/discharge
Short term: Charge doesn’t change (often zero or max)
Long term: Current through capacitor is zero. 45