Physics 212 Lecture 11, Slide 1 Physics 212 Lecture 11 RC Circuits Change in schedule Exam 2 will be...

Physics 212 Lecture 11, Slide Physics 212 Lecture 11, Slide 11

Physics 212Physics 212Lecture 11Lecture 11

RC CircuitsRC Circuits

Change in scheduleChange in scheduleExam 2 will be on Thursday, July 12 from 8 – 9:30 AM.Exam 2 will be on Thursday, July 12 from 8 – 9:30 AM.


RC Circuit Charging

• Kirchoff’s Voltage Rule

• Initially (q = q0 = 0)

• Long Term (Ic =0)

RVbattery

Ca

bbattery

q-V + +IR = 0

C

00 0batteryV I R

battery0

VI =

R

0 0battery

qV R

C

batteryq CV

In general:

0battery

q dqV R

C dt

/( ) (1 )t RCq t q e /

0( ) t RCI t I e

11

a

RVbattery

C

b

• Capacitor uncharged, switch is moved to position “a”

II


Checkpoint 1aCheckpoint 1a

Checkpoint 1bCheckpoint 1b&&

Close S1, Close S1, VV11 = voltage across C immediately after = voltage across C immediately afterVV22 = voltage across C a long time after = voltage across C a long time after

A)A) VV11 = V V = V V22 = V= VB) VB) V11 = 0 V = 0 V22 = V= VC) VC) V11 = 0 V = 0 V22 = 0= 0D) VD) V11 = V V = V V22 = 0= 0

Immediately after Immediately after the switch Sthe switch S11 is is closed:closed:

Q = 0Q = 0 VV11 = 0 = 0V = Q/CV = Q/C

After the switch SAfter the switch S11 has has been closed for a long been closed for a long timetime

I = 0I = 0 VV22 = V = VVVRR = 0 = 0

A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open.


V 2RC

R

S1 S2

Close SClose S11 at t=0 at t=0(leave S(leave S22 open) open)

For t For t

V C

R

I=0

VVCC = V = V

At t = 0At t = 0

V C

R

I

VVCC = Q/C = Q/C = 0= 0

15


RC Circuit (Discharging)

• Kirchoff’s Voltage Rule

• Initially (q=q0)

• Long Term (Ic =0)

RVbattery

Ca

b 0q

IRC

00 0

qI R

C

00

qI

RC

0 0q

RC

0q

In general:

0q dq

RC dt

/0( ) t RCq t q e

/0( ) t RCdq qI t e

dt RC

V

19

• Capacitor has q0 = CV, switch is moved to position “b”

RVbattery

Ca

bII++ --

-I-I


ABCD

22

Checkpoint 1cCheckpoint 1c

++--

IIRR


After being closed a long time, switch 1 is opened and switch 2 is closed. What is the current through the right resistor immediately after switch 2 is closed?A. IR = 0 B. IR = V/3R C. IR = V/2R D. IR = V/R


V

2RCV

I

ABCD

22

Checkpoint 1cCheckpoint 1c

++--

IIRR


After being closed a long time, switch 1 is opened and switch 2 is closed. What is the current through the right resistor immediately after switch 2 is closed?A. IR = 0 B. IR = V/3R C. IR = V/2R D. IR = V/R


ABC

26

Checkpoint 1dCheckpoint 1d


Now suppose both switches are closed. What is the voltage across the capacitor after a very long time? A. VC = 0 B. VC = V C. VC = 2V/3


ABC

26

Checkpoint 1dCheckpoint 1d

• After both switches have been closed for a After both switches have been closed for a long timelong time

• The current through the capacitor is zeroThe current through the capacitor is zero

• The current through R = current through 2RThe current through R = current through 2R

• VVcapacitorcapacitor = V = V2R2R

• VV2R2R = 2/3 V = 2/3 V


Now suppose both switches are closed. What is the voltage across the capacitor after a very long time? A. VC = 0 B. VC = V C. VC = 2V/3


2RC

R

S1 S2

Close both S1 and S2 andClose both S1 and S2 andwait a long time…wait a long time…

V

2RC

R

V

No current flows No current flows through the through the capacitor after a long capacitor after a long time. time. This will always This will always be the case if the be the case if the sources of EMFsources of EMFdon’t change with don’t change with time.time.

VVCC

II

I = V/(3R)I = V/(3R)

27

VV2R2R = I(2R) = (2/3)V = V = I(2R) = (2/3)V = VCC VVCC = (2/3)V = (2/3)V


DEMO – ACT 1DEMO – ACT 1

VC

S

Bulb 1Bulb 1

Bulb 2Bulb 2

What will happen after I close the switch?What will happen after I close the switch?

A)A) Both bulbs come on and stay on.Both bulbs come on and stay on.B)B) Both bulbs come on but then bulb 2 fades out.Both bulbs come on but then bulb 2 fades out.C)C) Both bulbs come on but then bulb 1 fades out.Both bulbs come on but then bulb 1 fades out.D)D) Both bulbs come on and then both fade out.Both bulbs come on and then both fade out.

R

R

30

No initial charge No initial charge on capacitoron capacitor V(bulb 1) = V(bulb 2) = VV(bulb 1) = V(bulb 2) = V Both bulbs lightBoth bulbs light

No final current No final current through capacitorthrough capacitor

V(bulb 2) = 0V(bulb 2) = 0


DEMO – ACT 2DEMO – ACT 2

VC

S

Bulb 1Bulb 1

Bulb 2Bulb 2

Suppose the switch has been closed a long time.Suppose the switch has been closed a long time.Now what will happen after open the switch?Now what will happen after open the switch?

A)A) Both bulbs come on and stay on.Both bulbs come on and stay on.B)B) Both bulbs come on but then bulb 2 fades out.Both bulbs come on but then bulb 2 fades out.C)C) Both bulbs come on but then bulb 1 fades out.Both bulbs come on but then bulb 1 fades out.D)D) Both bulbs come on and then both fade out.Both bulbs come on and then both fade out.

R

R

32

Capacitor has charge (=CV)Capacitor has charge (=CV) Capacitor discharges through both resistorsCapacitor discharges through both resistors


CalculationCalculationIn this circuit, assume V, C, and Ri are known.C initially uncharged and then switch S is closed.

What is the voltage across the capacitor What is the voltage across the capacitor after a long time ?after a long time ?

– Circuit behavior described by Kirchhoff’s Rules: • KVR: Vdrops = 0 • KCR: Iin = iout

– S closed and C charges to some voltage with some time constant

– Determine currents and voltages in circuit a long time after S closed

V

R1 R2

C

R3

S

35


CalculationCalculation

V

R1 R2

C

R3

S

Immediately after S is closed: what is I2, the current through C

what is VC, the voltage across C?

(A) (A) Only I2 = 0 (B) (B) Only VC = 0 (C) (C) Both I2 and VC = 0 (D) (D) Neither I2 nor VC = 0

• Why?– We are told that C is initially uncharged (V = Q/C)– I2 cannot be zero because charge must flow in order to charge C

In this circuit, assume V, C, and Ri are known.C initially uncharged and then switch S is closed.


37



V

R1 R2

C

R3

S

• Why?– Draw circuit just after S closed

(knowing VC = 0)

• Immediately after S is closed, what is I1, the current through R1 ?

V

R1 R2

S

R3

VC = 0

I1

– R1 is in series with the parallel combination of R2 and R3



39

(A) (B) (C) (D) (E)(A) (B) (C) (D) (E)

1 3

V

R R1

V

R1 2 3

V

R R R 1 2 3

1 2 2 3 1 3

R R RVR R R R R R

32

321 RR

RRR

V



V

R1 R2

C

R3

S

• Why?– After a long time in a static

circuit, the current through any capacitor approaches 0 !

– This means we redraw circuit with open circuit in middle leg

After S has been closed “for a long time”, what is IC, the current through C ?

(A) (B) (C)(A) (B) (C)

1

V

R 0

V

R1

R3IC = 0 VC

I



2

V

R

41



V

R1 R2

C

R3

S

• Why??

After S has been closed “for a long time”, what is VC, the voltage across C ?

(A) (B) (C) (D) (E)(A) (B) (C) (D) (E)

V3

1 3

RVR R 0

2

2 31

2 3

RV

R RR

R R

2

1 2

RVR R

– VC = V3 = IR3 = (V/(R1+R3))R3



V

R1

R3VC

I I

43


ChallengeChallengeIn this circuit, assume V, C, and Ri are known.C initially uncharged and then switch S is closed.

V

R1 R2

C

R3

S

• Strategy– Write down KVR and KCR for the circuit when S is closed

• 2 loop equations and 1 node equation – Use I2 = dQ2/dt to obtain one equation that looks like simple

charging RC circuit ( (Q/”C”) + “R”(dQ/dt) – “V” = 0 )– Make correspondence: “R” = ?, and “C” = ?, then = “R” ”C”

What is c, the charging time constant?

We get: CRR

RRRc

31

312


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

0

Q t

Q

0

t

RCQ t Q e

t

RC

““Fraction of initial Fraction of initial charge that remains”charge that remains”

““How many time constants worth How many time constants worth of time that have elapsed”of time that have elapsed”

45

How do exponentials work?How do exponentials work?


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

0

Q t

Q

t

RC = 1

RC = 2

Time constant:Time constant: = = RCRC

The bigger The bigger is, is,the longer it takes to getthe longer it takes to getthe same change…the same change…

0

t

RCQ t Q e

47


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

RC = 1

RC = 2

Which circuit has the largest time constant?

A) Circuit 1

B) Circuit 2

C) Same

49

= R= RequivequivCC

Checkpoint 2aCheckpoint 2a

The two circuits shown below contain identical capacitors that hold the same charge at t = 0. Circuit 2 has twice as much resistance as circuit 1.


Checkpoint 2bCheckpoint 2b


Which of the following statements best describes the charge remaining on each of the the two capacitors for any time after t = 0?A. Q1 < Q2 B. Q1 > Q2 C. Q1 = Q2 D. Q1 < Q2 at first, then Q1 > Q2 after long time E. Q1 > Q2 at first, then Q1 < Q2 after long time




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

RC = 1

RC = 2

Q = QQ = Q00ee-t/RC-t/RC

Look at plot !!!Look at plot !!!




Charge capacitor – store a logical “1”Charge capacitor – store a logical “1”

Discharge capacitor – store a logical “0”Discharge capacitor – store a logical “0”

Capacitor Capacitor dischargesdischarges through resistance through resistance between plates. between plates.

Only holds Q for < 1 msec.Only holds Q for < 1 msec.

““Dynamic” random access memoryDynamic” random access memory

Charge Q must be “refreshed” constantly, Charge Q must be “refreshed” constantly, So memory is called dynamic.So memory is called dynamic.


1. Capacitor is an open circuit for dc (direct current). 1. Capacitor is an open circuit for dc (direct current).

R

+Q+Q

V CIC

-Q-Q

VVCC = Q/C and I = Q/C and ICC = dQ/dt. If I = dQ/dt. If ICC isis flowing then Q is changing, so V flowing then Q is changing, so VCC is changing. is changing.

But in a dc circuit, nothing changes with time, so we must have IBut in a dc circuit, nothing changes with time, so we must have ICC = 0. = 0.

For t For t

V C

R

IC = 0

VVCC = V = V

Situation once things stop changing.Situation once things stop changing.

Physics 212 Lecture 11, Slide 1 Physics 212 Lecture 11 RC Circuits Change in schedule Exam 2 will be...

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