Resistors and Capacitors in the Same Circuit!!. So far we have assumed resistance (R), electromotive force or source voltage (ε), potential (V), current.

Resistors and Capacitors in the Same Circuit!!

So far we have assumed resistance (R), electromotive force or source voltage (ε), potential (V), current (I), and power (P) are constant.

When charging or discharging a capacitor I, V, P change with time, we will use lower-case i, v, and p for the instantaneous values of current, voltage, and power.

Previous assumptions and

how they are changing.

ε S

cR C

ba

R - C in SeriesInitial conditions

An ideal source (r=0)

At time t=0 we will close the switch. The instantaneous current at this time is i=0 and the instantaneous charge on the capacitor is q = 0 because at this instant there has been no chance for charge to build on the capacitor…yet!

R - C in SeriesCharging the Capacitor

εS

cR Cba

i=0

+q

-q

Remember: Current through a

series circuit is the same through all parts.

Total voltage input is equal to the sum of the voltage drops.

At time t=0, Vbc = 0,

So Vab = εTherefore, at t=0,

RR

VI abo

As capacitor C charges,Vbc , so Vab ,Which causes I , so… bcab VV

R - C in SeriesCharging the Capacitor

After a long time C will be fully charged and i = 0, so Vab = 0 and Vbc = ε

During charging…let q be the charge on the capacitor, and i be the current at any time tChoose “+” current for a “+” charge on the left capacitor plate…

At any time t, instantaneous voltages across the resistor and capacitor are…

Vab = iR and Vbc =q/C

εS

cR Cba

i=0

+q

-q

R - C in SeriesUsing Kirchoff’s Law

Then solve for i…

C

qiR

RC

qI

RC

q

RRi C

q

0

εS

cR Cba

i=0

+q

-q

Remember: At t=0, q=0 so initial current I0 = ε/R.

As q , q/RC , capacitor charge approaches final Qf and i to zero.

When i=0, … so… RC

Q

Rf

CQ f

R - C in SeriesNow for the Calculus!

dtRCCq

dq 1

Cq

RCRC

q

Rdt

dqi

1

ε S

cR Cba

i=0 +q

-q

Now integrate both sides…

Rearrange equation to put all terms with q on one side and everything else on the other…

tqdt

RCCq

dq00

1

Use “u-substitution” to evaluate the integral…

RC

t

C

Cq

ln Eliminate the “ln”…

RCt

eC

Cq

R - C in SeriesResults of the Calculus!

RCt

RCt

eIeRdt

dqti o

)(

)1()1()( RCt

RCt

eQeCtq f

ε S

cR Cba

i=0 +q

-q

To find the instantaneous current, remember that i = dq/dt

At any time t during the charging, the instantaneous charge is…

R - C in SeriesThe Graphs

i

t

Io

Io/2Io/e

RC

q

QfQf/e

Qf/2

RC

)1()( RCt

eQtq f

RCt

eIti o

)(

When time t=RC (yes…the product of resistance and capacitance)…

ffe

e

QQq

IIi

632.01

368.01

01

0

Or about 37% of the original current

Or about 63% of the original voltage

The product of R and C is called the time constant of the circuit. The time constant is a measure of how fast the capacitor charges. The symbol for time constant is τ (greek letter “tau”). It is calculated as RC because a that time the exponent of the “e” function becomes “-1” which gives us the equations above.

RC

NOTE:If the time constant is small, the capacitor will charge quickly.If the resistance is small, current is small, so the capacitor charges more quickly. If the time constant is large, it will take more time for the capacitor to charge.

At t=10τ, I = 0.000045I0 or current is approximately zero.

R-C CircuitsDischarging the capacitor

S

cR C

ba

+Q0

-Q0

Remove battery to discharge capacitor

Assume the capacitor, C, is fully charged, Q0

At t = 0, close the switch…

Initial Conditions at time t=0….

0Qq RC

QI 0

0

and

R-C CircuitsFinding the instantaneous charge during capacitor

discharge

Assume the capacitor, C, is fully charged, Q0

The time-varying current, i, …

RCt

eQtq

RC

t

Q

q

0

0

)(

ln

tq

Qdt

RCq

dqRC

q

dt

dqi

0

10

S

cR C

ba

+q -qii

Now integrate and

solve for q(t)

R-C CircuitsFinding the instantaneous current during capacitor

dischargeThe time-varying charge, q, …

RCt

RCt

RCt

eIeRC

Q

dt

dqti

eQtq

00

0

)(

)(

S

cR C

ba

+q -qii

q

t

Qo

Qo/2Qo/e

RCi

t

Io

Io/2Io/e

RC

R-C CircuitsEnergy & Power

During charging…The power of the battery, resistor and capacitor are as follows…

Cq

bccapacitor

resistor

battery

iiVP

RiP

iP

2

So…the total power of the circuit is…

CqiRii 2

Where ½ the energy is stored in the capacitor and the other ½ is dissipated by the resistor.