RC Circuits

Physics 102: Lecture 7

Exam I: Monday February 18PRACTICE EXAM IS AVAILABLE-CLICK ON “COURSE INFO” ON WEB PAGE

Exam I

Review, Sunday Feb. 17 3 PM, 141 Loomis

• I will go over a past exam

• On Monday, you get to vote for which exama) Fall 2007

b) Spring 2007

c) Fall 2006

d) Spring 2006

e) Fall 2005

Recall ….

• First we covered circuits with batteries and capacitors

– series, parallel

• Then we covered circuits with batteries and resistors

– series, parallel– Kirchhoff’s Loop Law– Kirchhoff’s Junction Law

• Today: circuits with batteries, resistors, and capacitors

RC CircuitsRC Circuits

RC Circuits

Charging Capacitors

Discharging Capacitors

Intermediate Behavior

RC CircuitsRC 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).

V

R2

R1C

SS

CapacitorsCapacitors

Charge on Capacitors cannot change instantly. 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.

Current through a Capacitor eventually goes to zero. Long term behavior of Capacitor:

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.

Charging CapacitorsCharging Capacitors

Example: an RC circuit with a switch

When switch is first closed … (short term behavior): No charge on the capacitor since charge can not change

instantly – the capacitor acts as a wire or short circuit

After switch has been closed for a while… (long term behavior): No current flows through the capacitor – the capacitor

acts as a break in the circuit or open circuit

RC

S

Charging CapacitorsCharging Capacitors

Capacitor is initially uncharged and switch is open. Switch is then closed. What is current in circuit immediately thereafter?

What is current in circuit a long time later?

RC

S

ACT/Preflight 7.1

2R

C R

S2

Both switches are initially open, and the capacitor is uncharged. What is the current through the battery just after switch S1 is closed?

1) Ib = 0 2) Ib = E /(3R)

3) Ib = E /(2R) 4) Ib = E /R

S1

KVL: -E + I(2R) + q/C = 0

q = 0

I = E /(2R)

Ib

+

-

+

+

-

-

34%

44%

6%

17%

20

ACT/Preflight 7.3 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?

1) Ib = 0 2) Ib = E/(3R)

3) Ib = E/(2R) 4) Ib = E/R

2R

C R

S2

S1

Ib

+

-

+

+

-

-

• 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

56%

28%

5%

11%

24

Discharging CapacitorsDischarging Capacitors

Example: a charged RC circuit with a switch

When switch is first closed… (short term behavior): Full charge is on the capacitor since charge can not

change instantly – the capacitor acts as a battery

After switch has been closed for a while… (long term behavior): No current flows through the capacitor – the

capacitor is fully discharged

RC

S

ACT/Preflight 7.5After 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?

1) IR = 0 2) IR = V/(3R)

3) IR = V/(2R) 4) IR = V/R

KVL: -q0/C+IR = 0

Recall q is charge on capacitor after charging:q0=VC (since charged w/ switch 2 open!)-V + IR = 0 I = V/R

2R

C R

S2S1

IR

+

-

+

+

-

-

+

-

32%

16%

15%

37%

42

ACT: RC CircuitsBoth switches are closed. What is the final charge on

the capacitor after the switches have been closed a long time?

1) Q = 0 2) Q = C E /3

3) Q = C E /2 4) Q = C E

KVL (right loop): -Q/C+IR = 0

KVL (outside loop), -E + I(2R) + IR = 0I = E /(3R)

KVL: -Q/C + E /(3R) R = 0Q = C E /3

R

2R

C

S2S1

IR

+

-

+

+

-

-

+

-

27

RC Circuits: Charging

• KVL: - + I(t)R + q(t) / C = 0

• Just after…: q =q0– Capacitor is uncharged

– - + I0R = 0 I0 = / R

• Long time after: Ic= 0– Capacitor is fully charged

– - + q/C =0 q = C

• Intermediate (more complex)q(t) = q(1-e-t/RC)

I(t) = I0e-t/RC

C

R

S1

S2

+

+

+

I-

-

-

The switches are originally open and the capacitor is uncharged. Then switch S1 is closed.

t

q

RC 2RC

0

q

17

RC Circuits: Discharging• KVL: - 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

C

R

S1

-+

+

I-

+

-

S2

q

RC 2RC

t40

What is the time constant?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.

Time Constant Demo

• Which system will be brightest?• Which lights will stay on longest?• Which lights consumes more energy?

2

Each circuit has a 1 F capacitor charged to 100 Volts.When the switch is closed:

1

= 2RC = RC/2

2 I=2V/R1

Same U=1/2 CV2

50

Summary of ConceptsSummary of Concepts Charging Capacitors

Short Term: capacitor has no charge, no potential difference, acts as a wire

Long Term: capacitor fully charged, no current flows through capacitor, acts as an open circuit

Discharging Capacitors Short Term: capacitor is fully charged, potential difference is a

maximum, acts as a battery Long Term: capacitor is fully discharged, potential difference is

zero, no current flows

Intermediate Behavior Charge & Current exponentially approach their long-term values = RC

Charging: Intermediate Times

2R

C R

S2

S1

Ib

+

-

+

+

-

-

Calculate the charge on the capacitor 310-3

seconds after switch 1 is closed.

q(t) = q(1-e-t/RC)

= q(1-e-310-3 /(2010010-6)))

= q (0.78)

Recall q = C

= (50)(100x10-6) (0.78)

= 3.9 x10-3 Coulombs

R = 10

V = 50 Volts

C = 100F

38

Practice!

Calculate current immediately after switch is closed:

Calculate current after switch has been closed for 0.5 seconds:

Calculate current after switch has been closed for a long time:

Calculate charge on capacitor after switch has been closed for a long time:

R

CE

S1

R=10C=30 mF

E =20 Volts

-E + I0R + q0/C = 0I

+

+

+

-

-

--E + I0R + 0 = 0I0 = E/R

After a long time current through capacitor is zero!

-E + IR + q∞/C = 0

-E + 0 + q∞ /C = 0q∞ = E C 32

0

t

RCI I e

0.5

RCeR

0.5

10 0.0320

10e

0.5

10 0.0320

10e

0.38 Amps

ACT: RC Challenge

1) 0.368 q0 2) 0.632 q0

3) 0.135 q0 4) 0.865 q0

R

CE 2R

S1

E = 24 Volts

R = 2

C = 15 mFAfter 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?

q(t) = q0 e-t/RC

= q0 (e-0.06 /(4(1510-3)))

= q0 (0.368)45

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/RCI(t) = I0e-t/RC

Short term: Charge doesn’t change (often zero or max)

Long term: Current through capacitor is zero.

Time Constant = RC Large means long time to charge/discharge

45