PHYS 221 General Physics II
DC Circuits, Kirchoff’s Rules
Spring 2015
Assigned Reading: 19.4
Lecture 7
Review: Ohm’s Law, Resistance, & Resistivity
• The constant of proportionality between I and V is the
electrical resistance,
V
I
R
• The value of the resistance of a conductor depends only
on its composition, size, and shape
L
R
A
• The resistivity, ρ, depends only on the material used to
make the conductor
Phys 221 Spring 2014
Lecture 07
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Predicting Current in Circuits
How could one predict currents running in a
complicated circuit?
Answer: Kirchhoff’s Rules
Phys 221 Spring 2014
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How much current will flow in a circuit?
Analogy:
Want to predict I
Potential difference V
defines amount of the
current that flows in
a circuit
Phys 221 Spring 2014
• If water height difference (potential
energy) is maintained, the flow rate
of water will not change
• The flow rate of water will depend
on the water height difference
Lecture 07
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Kirchhoff’s Rules
1.Kirchhoff’s Loop Rule:
 V  0
Change in PE of a charge
as it travels around a
complete circuit loop must
be zero
Phys 221 Spring 2014
2. Kirchhoff’s Junction Rule:
I  0
This is the consequence of
conservation of charge:
(charge cannot be created
or destroyed).
I1 + I2 = I3
so: I1 + I2 - I3 = 0
Lecture 07
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Resistor Arrangement
“In series”
When two or more electrical devices
are wired so that the same current
flows through each device.
“In parallel”
Phys 221 Spring 2014
When two or more electrical devices
are wired so that the potential
difference across them is the same.
Current splits between the devices at
the input and recombines at the
output.
Lecture 07
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Resistors in series
Analogy: equivalent resistance should be larger than any of
the resistances connected in series.
Kirchhoff’s First Rule (“Loop Rule” or “Kirchhoff’s
Voltage Law”)
The algebraic sum of the changes in potential
encountered in a complete traversal of any loop of
circuit must be zero.


I
Move
around
circuit:
 
Phys 221 Spring 2014
R1
R2
-IR1
-IR2
Lecture 07
 
=0
8

Rules

I
R1
R2
Voltage Gains enter with a + sign.
Voltage Drops enter with a - sign.


a
- +
b
Moving from a to b
b in the direction of
arrow then +

a
I
Moving from a to b
In the direction
opposite to the
current
V=+IR
Note:
 

- +
b
a
Moving from a to b
a Opposite arrow
then -

b
I

Moving from a to b
In direction of the
current then
V=-IR
Always points from negative to positive
9
Loop Example

R1
b
a
+
R4
f
I
I
c
R2
V
n
0 
+
-
d
e
R3

IR1  IR2   2  IR3  IR4  1  0
loop
I
1   2
R1  R2  R3  R4
Phys 221 Spring 2014
If 1 < 2 , I will be negative,
i.e. it will flow clockwise,
opposite to path
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Kirchhoff’s Second Rule (Junction Rule or
“Kirchhoff’s Current Law”)
The sum of the current entering any junction must be equal
to the sum of the currents leaving that junction.
I1
I2
I1  I 2  I 3
I3
Phys 221 Spring 2014
Lecture 07
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Resistors in Series
I
-

R2
+
R1
I
•
When a potential difference, V, is
applied across resistances in series,
the resistances have identical current.
•
Resistances in series can be replaced
with an equivalent resistance, REQ, that
has the same current I and the same
total potential difference V as the
actual resistances.
R3
I
  IR1  IR2  IR3  0
I
  I (R1  R2  R3 )  0
+
-

  I (Rseries )  0
R
Rseries  R1  R2  R3
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Resistors in parallel
Analogy: equivalent resistance should be smaller than any of
the resistances connected in parallel
Phys 221 Spring 2014
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R3
R2
R1

-
+
Resistors in Parallel
*In parallel, the resistances all have the same potential differences.
* Resistances in parallel can be replaced with an equivalent resistance, REQ,
that has the same potential difference, V, and the same total current, I,
as the actual resistances.
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Resistors in parallel
• ΔV across all resistors is the same:
  I1 R1  I 2 R2  I 3 R3
I1 

I2 
R1

I3 
R2

R3
• Total current: I = I1 + I2 + I3
Equivalent Circuit

Req


R1


R2


R3
1
1
1
1

 

Req R1 R2 R3
Phys 221 Spring 2014
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i>Clicker question
Consider the circuit shown:
What is the relation between
a
Va -Vd and Va -Vc ?
50
b
I2
I1
12V
20 
80 
d
(A) (Va -Vd) < (Va -Vc)
c
(B) (Va -Vd) = (Va -Vc)
(C) (Va -Vd) > (Va -Vc)
Phys 221 Spring 2014
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i>Clicker question
Consider the circuit shown:
What is the relation between
I1 and I2?
a
50
b
I2
I1
12V
20 
80 
d
c
(A) I1 < I2
(B) I1 = I2
(C) I1 > I2
Phys 221 Spring 2014
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Summary of Resistor & Capacitor Combinations
Resistors
Capacitors
Series R  R
 i
eq
n
1
1

Ceq i 1 Ci
n
i 1
C1C2
Ceq 
for n  2
C1  C2
Parallel
n
1
1

Req i 1 Ri
n
Ceq   Ci
i 1
R1 R2
Req 
for n  2
R1  R2
Phys 221 Spring 2014
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Summary of Simple Circuits
Resistors in series:
Req  R1  R2  R3  ...
Current through is same;
Resistors in parallel:
Voltage drop across is IRi
1
1
1
1
 

 ...
Req R1 R2 R3
Voltage drop across is same;
Phys 221 Spring 2014
Current through is V /Ri
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i>Clicker question
You are to connect resistors R1 and R2 (with R1>R2)
to a battery, first individually, then in series and then in
parallel. Rank those arrangements according to the
amount of current through the battery, smallest first.
Phys 221 Spring 2014
A.
R1, Series, R2, Parallel
B.
Parallel, R2, R1, Series
C.
R1, Parallel, R2, Series
D.
Series, R1, R2, Parallel
E.
R2, Series, Parallel, R1
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Example
1
2
Loop 1: -12  3I  4I1  0
3 equations with 3
unknowns: I1 = 1.5 A,
I2 = 0.5 A, I = 2.0 A
Loop 2: +5  2I 2  4I1  0
Junction"b": I  I1  I 2
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Tips on Using Kirchhoff’s Rules
• When you are given a circuit, you must first carefully
analyze circuit topology.
– Find all independent loops and junctions
– Assign branch currents (i.e., assume directions)
– Decide on a navigation direction around the loops
• Use Kirchhoff’s First Rule for all independent loops
in the circuit.
• Use Kirchhoff’s Second Rule for all independent
junctions in the circuit.
• A negative current in your results is OK! It only
implies that the current actually flows in the opposite
(to the assumed) direction.
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