Agenda

• Today– Finish Chapter 26: RC Circuits

• Freedom?

• Post-Freedom– Magnetism & Induction: 27-29

• Post-Post Freedom– Finish induction & Review, Exam II

Capacitors in Circuits

• Series– 1/CT = 1/C1 + 1/C2 …– Effective Distance increased, reduces CT

• Parallel– CT = C1 + C2 + C3…– Effective Area increased, increases CT

• Effectively opposite of Resistors• What is voltage dependence for R’s?

– Current

• What is voltage dependence for C’S?– Charge

• I = dQ/dt ….

What Happens Here?

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-

Lamp

Start with Switch Open?Close Switch

What Happens Here?

+

-

Lamp

Start with Switch Open?Close SwitchWhat occurs now? Changes with Time?At t=0, cap’s are shortsFilling with charge, no opposition to flowAt t= infinity, cap’s are opensFilled with chargeAfter a long time….

What Happens Here?

+

-

Lamp

Start with Switch Open?Close SwitchWhat occurs now? Changes with Time?At t=0, cap’s are shortsFilling with charge, no opposition to flowAt t= infinity, cap’s are opensFilled with chargeAfter a long time….Voltage on Capacitor = Voltage of CellNo current flow, Lamp is offNow take out battery, replace with switch

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-

What Happens Here?

Lamp

What Happens when switch is closed?Path emerges for charge to flowLight back on for some timeDims, then off as charge dissipates

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-

Discharge Circuit

Lamp

Capacitor Begins with Voltage VQ=VCLamp = Resistor value RAssume constant R for simplicityDefine states: Initial & FinalInitial (t=0) switch JUST closedFinal (t=infinity) looong time afterInitial, Current max or min?MaxFinal, Current max or min?Min (zero)Now we have “boundary values”Next up: DiffEQ

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Discharge Circuit

Lamp

Q=VCInitial: VC = V, I=I0

Final: VC = 0, I = 0Examine Voltage LoopVC + IR = 0R Lamp resistanceHolds true for any timeQ/C = -IRQ/C = -R(dQ/dt)-(Q/C)dt = RdQ-dt/(RC) = dQ/QIntegrate Both Sides (tau) = RC [Time Constant]

+

-

Discharge Circuit

Lamp

Q=VCInitial: VC = V, I=I0, q = Q0 Final: VC = 0, I = 0, q = 0-dt/(RC) = dQ/QIntegrate Both Sides (tau) = RC [Time Constant]

+

-

1

1 ln 2

dqdt

q

tC q C

Book does same by choosing integration limitsHere: Use constants to match boundary conditions

Discharge Circuit

Lamp

Q=VCInitial: VC = V, I=I0, q = Q0 Final: VC = 0, I = 0, q = 0-dt/(RC) = dQ/QIntegrate Both Sides (tau) = RC [Time Constant]

+

-

1

1 ln 2

1 ln

exp 1

exp

dqdt

q

tC q C

tD q

tD q

tq A

What should A be?

Discharge Circuit

Lamp

Q=VCInitial: VC = V, I=I0, q = Q0 Final: VC = 0, I = 0, q = 0-dt/(RC) = dQ/QIntegrate Both Sides (tau) = RC [Time Constant]

+

-

0

1

1 ln 2

1 ln

exp 1

exp

exp

dqdt

q

tC q C

tD q

tD q

tq A

tq Q

Charge not usually too usefulVoltage?

Discharge Circuit

0

1

1 ln 2

1 ln

exp 1

exp

exp

dqdt

q

tC q C

tD q

tD q

tq A

tq Q

Lamp

Q=VCInitial: VC = V, I=I0, q = Q0 Final: VC = 0, I = 0, q = 0-dt/(RC) = dQ/QIntegrate Both Sides (tau) = RC [Time Constant]

+

-

Charge not usually too usefulVoltage?

0

0

exp

exp

Qq t

C C

tV V

Discharge Circuit

0

1

1 ln 2

1 ln

exp 1

exp

exp

dqdt

q

tC q C

tD q

tD q

tq A

tq Q

Lamp

Q=VCInitial: VC = V, I=I0, q = Q0 Final: VC = 0, I = 0, q = 0-dt/(RC) = dQ/QIntegrate Both Sides (tau) = RC [Time Constant]

+

- Charge not usually too usefulCurrent?

0

0

0

exp

exp

exp

tV V

VV t

R R

tI I

Charge Circuit?

Lamp

+

-

Pretty Similar…

Problems? Freedom?

• Today– Finish Chapter 26: RC Circuits

• Freedom?

• Post-Freedom– Magnetism & Induction: 27-29

• Post-Post Freedom– Finish induction & Review, Exam II