Kirchhoff’s Rules
„
„
There are ways in which resistors can
be connected so that the circuits formed
cannot be reduced to a single
equivalent resistor
Two rules, called Kirchhoff’s rules,
can be used instead
Statement of Kirchhoff’s Rules
„
Junction Rule
„
The sum of the currents entering any junction
must equal the sum of the currents leaving that
junction
„
„
A statement of Conservation of Charge
Loop Rule
„
The sum of the potential differences across all the
elements around any closed circuit loop must be
zero
„
A statement of Conservation of Energy
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Mathematical Statement of
Kirchhoff’s Rules
„
„
Junction Rule:
Σ Iin = Σ Iout
Loop Rule:
∑ ΔV = 0
closed
loop
More about the Junction Rule
„
„
„
I1 = I2 + I3
From Conservation
of Charge
Diagram (b) shows
a mechanical analog
2
More about the Loop Rule
„
„
„
„
Start on point a and end on b.
Direction of current, indicates
which point is at a higher
potential.
In (a), the resistor is traversed
in the direction of the current,
the potential across the resistor
is – IR
In (b), the resistor is traversed
in the direction opposite of the
current, the potential across
the resistor is is + IR
Loop Rule, final
„
„
In (c), the source of
(electromotive force)
emf is traversed in the
direction of the emf
(from – to +), and the
change in the electric
potential is +ε
In (d), the source of emf
is traversed in the
direction opposite of the
emf (from + to -), and
the change in the
electric potential is -ε
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Kirchhoff’s Rules Equations
„
„
In order to solve a particular circuit
problem, the number of independent
equations you need to obtain from the two
rules equals the number of unknown
currents
Any fully charged capacitor acts as an
open branch in a circuit (WHY??)
„
The current in the branch containing the
capacitor is zero under steady-state conditions
Problem-Solving Hints –
Kirchhoff’s Rules
„
Draw the circuit diagram and assign labels
and symbols to all known and unknown
quantities. Assign directions to the currents.
„
„
The direction is arbitrary, but you must adhere to
the assigned directions when applying Kirchhoff’s
rules
Apply the junction rule to any junction in the
circuit that provides new relationships among
the various currents
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Kirchhoff’s Rules Illustrated
Kirchhoff’s Rules
Determine the magnitude and direction
of current through the various resistors.
R1
ε1
R2
ε3
R3
ε2
R4
R5
R6
5
Kirchhoff’s Rules
Assume a direction to traverse the
loop.
R1
ε1
R2
ε3
R3
ε2
R4
R5
R6
Kirchhoff’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!!!
6
Kirchhoff’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!!!
Kirchhoff’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!!!
7
Kirchhoff’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!!!
Kirchhoff’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!!!
8
Kirchhoff’s Rules
Assume a direction of current
flow.
R1
ε1
R2
ε3
R3
ε2
R4
R5
I1
R6
Trace out the current. Remember
conservation of charge: I1 = I2 + I3!!!
Kirchhoff’s Rules
Pick a starting point for each loop.
R1
ε1
R2
ε3
R3
ε2
R4
R5
R6
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Kirchhoff’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
Kirchhoff’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
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Kirchhoff’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
= 0
= 0
I3
R2
ε3
R3
ε2
R4
R5
R6
Kirchhoff’s Rules
„
Let’s calculate current and charge for a
small circuit which appears in many,
many electronics.
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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
Applying Kirchoff’s rules:
dq
ε = iR + ; i =
C
dt
dq q
dq
ε = R
+
⇒ εC = RC
+q
dt C
dt
εC − q
− q − εC
dq
dq
dq
⇒
=
⇒
=
εC − q = RC
dt
RC
RC
dt
dt
a
⇒
a
f
− dt
dq
=
RC
q − εC
a
f
f
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Kirchhoff’s Rules (RC – circuit)
a
R
b
C
ε
Applying Kirchoff’s rules:
ε = iR +
leads to the expression:
z
Q
0
q
dq
; i=
C
dt
dq
1
=−
q − εC
RC
z
t
dt'
0
Kirchhoff’s Rules (RC – circuit)
z
Q
0
dq
1
=−
q − εC
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
GH
Q(t ) = εC 1 − e
−
t
RC
I
JK
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Kirchhoff’s Rules (RC – circuit)
F
GH
Q(t ) = εC 1 − e
−
t
RC
I
JK
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.
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Kirchhoff’s Rules (RC – circuit)
Applying Kirchoff’s rules:
0 = iR +
leads to the expression:
z
q
dq
; i=
C
dt
z
1 t
dq
=−
dt '
q
RC 0
F
GH
Q(t ) = εC e
−
t
RC
I ; RC ≡ τ
JK
c
Kirchhoff’s Rules (RC – circuit)
F
GG
H
Q(t ) = εC e
−
t
τc
I
JJ
K
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Verify RC – Circuit Discharge!
F
Q(t ) = εCG e
GH
−
t
τc
I
JJ
K
Since Q is proportional to the potential difference,
we can measure the potential difference across a
capacitor as a function of time! Let’s DO IT (DEMO).
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