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