PS 250: Lecture 15 Kirchhoff’s Rules and

RC Circuits

J. B. Snively October 2nd, 2015

Today’s Class

Kirchhoff’s Rules for Circuit Calculations R-C Circuits Summary

The algebraic sum of the currents into an junction (“node”) is zero:

The algebraic sum of the potential differences in any loop is zero:

Kirchhoff’s Rules

XI = 0

XV = 0

I1 I2

I3

V1+

- I

V2

V3

Loop Rule:

...RNR2R1

Vsrc+

-

Sum of the potential differences across each source and resistor equals zero:

I

XV = Vsrc � IR1 � IR2 � ...IRN = 0

Junction Rule:

RN...R2R1V+

-

Sum of the currents entering and leaving a junction point (“node”) equals zero:

XI = Isrc �

V

R1� V

R2� ...

V

RN= 0

Today’s Class

Kirchhoff’s Rules for Circuit Calculations R-C Circuits Summary

Capacitor’s initial charge q=0, vbc=q/C=0

Initial Current = I = Vab/R Capacitor behaves initially like a short circuit!

Charging a Capacitor At t=0...

E+

-

Vab=E

Vbc=0CR

t=0

Capacitor’s final charge Qf=CE, Vbc=Qf/C=E

Final Current = 0 Charged capacitor behaves like an open circuit!

Charging a Capacitor At t=∞...

E+

-

Vab=0

Vbc=ECR

t=∞

Charging a Capacitor What happens in between?

E+

-

Vab=iR

Vbc=q/CCR

t

E � iR� q

C= 0Apply Loop Rule:

Re-arrange: i =dq

dt=

ER

� q

RC= � 1

RC(q � CE)

Charging a Capacitor What happens in between?

Solve for q via Integration (not shown):

dq

dt= � 1

RC(q � CE)Consider Differential Equation:

q = CE(1� e�t/RC) = Qf (1� e�t/RC)

Then, solve for i:

i =dq

dt=

ERe�t/RC = I

o

e�t/RC

Discharging a CapacitorVab=iR

Vbc=q/CCR

t

Apply Loop Rule:

Re-arrange:

iR = � q

C

i =dq

dt= � q

RC

Discharging a Capacitor

Solve for q via Integration (not shown):

Consider Differential Equation:

Then, solve for i:

dq

dt= � q

RC

q = Qo

e�t/RC

i =dq

dt= � Q

o

RCe�t/RC = I

o

e�t/RC

R-C Circuits Quick Math + Examples = Fun!

Summary / Next Class:

Read ahead through 27.1-27.2

Work on next Homework / Mastering Physics