Transcript Kirchhoff`s rules and RC Circuits

Kirchhoff’s Rules Illustrated
Kirchoff’s Rules
Determine the magnitude and direction
of current through the various resistors.
R1
ε1
R2
ε3
R3
ε2
R4
R5
R6
Kirchoff’s Rules
Assume a direction to traverse the
loop.
R1
ε1
R2
ε3
R3
ε2
R4
R5
R6
Kirchoff’s Rules
Assume a direction of current
flow.
I1
R1
ε1
R2
ε3
R3
ε2
R4
R5
R6
Trace out the current. Remember
conservation of charge!!!
Kirchoff’s Rules
Assume a direction of current
flow.
I2
R1
ε1
R2
ε3
R3
ε2
R4
R5
R6
Trace out the current. Remember
conservation of charge!!!
Kirchoff’s Rules
Assume a direction of current
flow.
ε1
R2
ε3
I2
R1
R3
ε2
R4
R5
R6
Trace out the current. Remember
conservation of charge!!!
Kirchoff’s Rules
Assume a direction of current
flow.
R1
ε1
R2
ε3
R3
ε2
R4
R5
R6
I2
Trace out the current. Remember
conservation of charge!!!
Kirchoff’s Rules
Assume a direction of current
flow.
R1
ε1
I3 R2
ε3
R3
ε2
R4
R5
R6
Trace out the current. Remember
conservation of charge!!!
Kirchoff’s Rules
Assume a direction of current
flow.
R1
ε1
R2
ε3
R3
ε2
R4
I1
R5
R6
Trace out the current. Remember
conservation of charge: I1 = I2 + I3!!!
Kirchoff’s Rules
Pick a starting point for each loop.
R1
ε1
R2
ε3
R3
ε2
R4
R5
R6
Kirchoff’s Rules
Traverse the loop in the direction YOU
have chosen. End where you start!!
R1
ε1
R2
ε3
R3
ε2
R4
R5
R6
Keep track of all the Potential differences
encountered and sum to zero.
-I1 R1 + -I3 R3 + ε2
+
-I1 R6 +
-I1 R4
+ ε1
= 0
Kirchoff’s Rules
Traverse the right loop in the direction YOU
have chosen. End where you start!!
R1
ε1
R2
ε3
R3
ε2
R4
R5
R6
Keep track of all the Potential differences
encountered and sum to zero.
ε3 + -I2 R5 - ε2 + I3 R3 + -I2 R2
= 0
Kirchoff’s Rules
Summary:
-I1 R1 + -I3 R3 + ε2
-I1 R4
-I1 R6 +
+
+ ε1
ε3 + -I2 R5 - ε2 + I3 R3 + -I2 R2
I1 =
I2 +
R1
ε1
R2
I3
ε3
R3
ε2
R4
R5
R6
= 0
= 0
Kirchhoff’s Rules (RC – circuit)
a
R
b
C
ε
As soon as switch is thrown into position a, there is a current flow
throughout the entire circuit.
After a very long time, current stops flowing through resistor R. This
is equivalent to stating that the potential difference across R is zero
after a long time after the switch is thrown to position a. (Charge
stops “flowing” and is stored in the capacitor).
Kirchhoff’s Rules (RC – circuit)
a
R
b
C
ε
q
dq
  iR  ; i 
C
dt
dq q
dq
 R

 C  RC
q
dt C
dt
Applying Kirchoff’s rules:
dq
C  q  RC
dt
C  qf dq
a


RC

dt
 dt
dq

RC
q  C
a
f
a f
 q  C
dq


RC
dt
Kirchhoff’s Rules (RC – circuit)
a
R
b
C
ε
Applying Kirchoff’s rules:
  iR 
q
dq
; i
C
dt
leads to the expression:
z
Q
0
dq
1

q  C
RC
z
t
dt'
0
Kirchhoff’s Rules (RC – circuit)
z
Q
0
dq
1

q  C
RC
z
t
dt'
0
a
R
b
C
ε
Thus the total charge on the capacitor builds up over time, and the
current through the circuit comes to a halt! (All potential difference is
across the capacitor and none over the resistor.)
The total charge is expressed as:
F
I
Q(t )  CG
1 e
J
H K

t
RC
Kirchhoff’s Rules (RC – circuit)
F
I
Q(t )  CG
1 e
H J
K

t
RC
Kirchhoff’s Rules (RC – circuit)
a
R
b
C
ε
What about discharging the capacitor through the resistor R? Wait a
really long time (t >> RC), and switch S to position b.
Kirchhoff’s Rules (RC – circuit)
Applying Kirchoff’s rules:
0  iR 
leads to the expression:
q
dq
; i
C
dt
z
z
dq
1 t

dt '
0
q
RC
F
I
Q(t )  CG
e
; RC  
J
H K

t
RC
c
Kirchhoff’s Rules (RC – circuit)
F
Q(t )  CG
e
G
H

t
c
I
J
J
K
Verify RC – Circuit Discharge!
F
Q(t )  CG
e
G
H

t
c
I
J
J
K