Physics 212 Lecture 19, Slide 1 Physics 212 Lecture 19 LC and RLC Circuits.

Physics 212 Lecture 19, Slide Physics 212 Lecture 19, Slide 11

Physics 212Physics 212Lecture 19Lecture 19

LC and RLC CircuitsLC and RLC Circuits


C

I

L Q

LC Circuit

C

QVC

++

--

dt

dILVL

++

--

Circuit Equation:Circuit Equation: 0dt

dIL

C

Q

dt

dQI

LC

Q

dt

Qd

2

2

Qdt

Qd 22

2

LC

1

wherewhere


CL

Same thing if we notice thatSame thing if we notice that and and1

kC

m L

Qdt

Qd 22

2

LC

1

k

x

m

FF = -kx

aax

dt

xd 22

2

m

k


CL

Time Dependence

II++++

----


What is the potential difference across the inductor at What is the potential difference across the inductor at t = 0 t = 0 ?? A) VA) VLL = 0 = 0 B) VB) VLL = Q = Qmaxmax/C /C C) VC) VLL = Q = Qmaxmax/2C /2C

Pendulum…Pendulum…

At time At time t = 0t = 0 the capacitor is the capacitor is fully charged with fully charged with QQmaxmax and the and the current through the circuit is current through the circuit is 00.. L C

The voltage across the capacitor is Qmax/C Kirchhoff's Voltage Rule implies that must also be equal to the voltage across the inductor

Checkpoint 1a

since VL = VC


At time At time t = 0t = 0 the capacitor is the capacitor is fully charged with fully charged with QQmaxmax and and the current through the circuit the current through the circuit is is 00..

L C

What is the potential difference across the What is the potential difference across the inductor when the current is maximum ?inductor when the current is maximum ? A) VA) VLL = 0 = 0 B) VB) VLL = Q = Qmaxmax/C /C C) VC) VLL = Q = Qmaxmax/2C /2C

dI/dt is zero when current is maximum

Checkpoint 1b


At time At time t = 0t = 0 the capacitor is the capacitor is fully charged with fully charged with QQmaxmax and and the current through the circuit the current through the circuit is is 00..

L C

How much energy is stored in the capacitor How much energy is stored in the capacitor when the current is a maximum ?when the current is a maximum ? A) U = QA) U = Qmaxmax

22/(2C) /(2C) B) U = QB) U = Qmaxmax

22/(4C) /(4C) C) U = 0 C) U = 0

Total Energy is constant !ULmax = ½ LImax

2

UCmax = Qmax2/2C

I = max when Q = 0

Checkpoint 1c


The capacitor is charged such The capacitor is charged such that the top plate has a that the top plate has a charge charge +Q+Q00 and the bottom and the bottom plate plate -Q-Q00. At time . At time t=0t=0, the , the switch is closed and the circuit switch is closed and the circuit oscillates with frequencyoscillates with frequency = = 500500 radians/s radians/s.. What is the value of the capacitor C?What is the value of the capacitor C?

A) C = 1 x 10A) C = 1 x 10-3-3 F F B) C = 2 x 10B) C = 2 x 10-3-3 F F C) C = 4 x 10C) C = 4 x 10-3-3 F F

L C

L = 4 x 10L = 4 x 10-3-3 H H = = 500 rad/s500 rad/s

Checkpoint 2a

LC

1 3

34210

)104)(1025(

11

LC

++++----


Which plot best represents Which plot best represents the energy in the inductor the energy in the inductor as a function of time as a function of time starting just after the switch starting just after the switch is closed? is closed?

Energy proportional to Energy proportional to I I 22 UU cannot be negative cannot be negative

21

2LU LI

Current is changing Current is changing U ULL is not constant is not constant

InitialInitial current is zerocurrent is zero

Checkpoint 2b

L C

closed at t=0closed at t=0

+Q+Q00

-Q-Q00


When the energy stored When the energy stored in the capacitor reaches in the capacitor reaches its maximum again for its maximum again for the the first time after t=0first time after t=0, , how much charge is how much charge is stored on the top plate of stored on the top plate of the capacitor? the capacitor?

Checkpoint 2c

L C

closed at t=0closed at t=0

+Q+Q00

-Q-Q00

A) +QA) +Q00

B) +QB) +Q00 /2 /2C) 0C) 0D)D) -Q-Q00/2/2E)E) -Q-Q00

Q is maximum when current goes to zeroQ is maximum when current goes to zero

dt

dQI

Current goes to zero twice during one cycleCurrent goes to zero twice during one cycle

CC

RR

LL

I(t)0L C RV V V

0d I QL I Rdt C

Use I = dQ/dt and divide by L:

2

20

d Q Q

dt LC

dQ R

dt L

20 cos '

RtLQ t Q e t

Add resistanceAdd resistance


Damped oscillationsDamped oscillations


CalculationCalculationThe switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened.

What is QMAX, the maximum charge on the capacitor?

• Conceptual Analysis– Once switch is opened, we have an LC circuit – Current will oscillate with natural frequency 0

• Strategic Analysis– Determine initial current– Determine oscillation frequency

– Find maximum charge on capacitor

CC

RR

LLVV


CalculationCalculation

What is IL, the current in the inductor, immediately AFTER the switch is opened? Take positive direction as shown.

(A) IL < 0 (B) (B) IL = 0 (C) (C) IL > 0

Current through inductor immediately Current through inductor immediately AFTERAFTER switch is opened switch is opened IS THE SAME ASIS THE SAME AS

the current through inductor immediately the current through inductor immediately BEFOREBEFORE switch is opened switch is opened

BEFORE switch is opened:BEFORE switch is opened:all current goes through inductor in direction shownall current goes through inductor in direction shown

CC

RR

LLVV

IILL

The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened.


CalculationCalculationThe switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened.

The energy stored in the capacitor immediately after the switch is opened is zero.

(A) TRUE (B) (B) FALSE

CC

RR

LLVV

VVCC=0=0

BUT: VBUT: VLL = V = VCC since they are in parallel since they are in parallel

VVCC = 0 = 0

IILL(t=0+) > (t=0+) > 00

IILL

dIdILL/dt ~ 0 /dt ~ 0 V VLL = 0 = 0

BEFOREBEFORE switch is opened: switch is opened:

VVCC = 0 = 0VVCC cannot change abruptly cannot change abruptly

AFTERAFTER switch is opened: switch is opened:

IMPORTANT: NOTE DIFFERENT CONSTRAINTS AFTER SWITCH OPENEDIMPORTANT: NOTE DIFFERENT CONSTRAINTS AFTER SWITCH OPENEDCURRENTCURRENT through through INDUCTORINDUCTOR cannot change abruptly cannot change abruptlyVOLTAGEVOLTAGE across across CAPACITORCAPACITOR cannot change abruptly cannot change abruptly

UUCC = ½ CV = ½ CVCC22 = 0 !! = 0 !!



What is the direction of the current immediately after the switch is opened?

(A) clockwise (B) (B) counterclockwise

CC

RR

LLVV



BEFOREBEFORE switch is opened: Current moves switch is opened: Current moves downdown through L through L

IILL

AFTERAFTER switch is opened: Current continues to move down through L switch is opened: Current continues to move down through L

IILL(t=0+) > (t=0+) > 00

VVCC(t=0+) = (t=0+) = 00




What is the magnitude of the current right after the switch is opened?

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



CC

RR

LLVV

VVLL = 0 = 0

BEFORE switch is opened:BEFORE switch is opened: CC

RR

LLVV

VVLL=0=0

IILL

IILL IILL

V = IV = ILLRR

IILL(t=0+) > (t=0+) > 00

VVCC(t=0+) = (t=0+) = 00


L

CVIo

C

L

R

VIo 2

R

VIo

R

VIo 2



What is Qmax, the maximum charge on the capacitor during the oscillations?

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

CC

RR

LLVV

IILL(t=0+) (t=0+) =V/R=V/R

VVCC(t=0+) = (t=0+) = 00

IILL

Hint:Hint: Energy is conserved Energy is conserved


CVQ2

1max LC

R

VQ max CVQ max

LCR

VQ max

CCLL

IImaxmax

When I is max When I is max (and Q is 0) (and Q is 0)

2

2

1LIU

CC

When Q is max When Q is max (and I is 0) (and I is 0)

LLQQmaxmax

C

QU

2max

2

1

C

QLI

2max2

2

1

2

1

LCR

VLCIQ maxmax


Follow-Up 1Follow-Up 1The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened.

Is it possible for the maximum voltage on the capacitor to be greater than V?

(A) YES (B) NO(B) NO

CC

RR

LLVV

IImax max =V/R=V/R

IILL

max

V LV

R C

We can rewrite this condition in terms of the resonant frequency:We can rewrite this condition in terms of the resonant frequency:

0L R OROR0

1R

C

We will see these forms again when we study AC circuits!!We will see these forms again when we study AC circuits!!

max

VQ LC

R

max

VQ LC

R VVmaxmax can be greater than V can be greater than V IFIF::

LR

C

Physics 212 Lecture 19, Slide 1 Physics 212 Lecture 19 LC and RLC Circuits.

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