Kirchhoffs Laws

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Kirchhoffs Laws
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Yesterday

• Ohms Law VIR
• Ohms law isnt a true law but a good
  approximation for typical electrical circuit
  materials
• Resistivity ?1/? (Conductivity) Property of
  the material
• Resistance proportional to resistivity and
  length, inversely proportional to area

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Question 1
Two cylindrical resistors are made from the same
material, and they are equal in length. The first
resistor has diameter d, and the second resistor
has diameter 2d.
Compare the resistance of the two cylinders.
a) R1 gt R2 b) R1 R2
c) R1 lt R2
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Question 1

• a
• b
• c

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Question 1
Two cylindrical resistors are made from the same
material, and they are equal in length. The first
resistor has diameter d, and the second resistor
has diameter 2d.
Compare the resistance of the two cylinders.
a) R1 gt R2 b) R1 R2
c) R1 lt R2

• Resistance is proportional to Length/Area

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Question 2
Two cylindrical resistors are made from the same
material, and they are equal in length. The first
resistor has diameter d, and the second resistor
has diameter 2d.
If the same current flows through both resistors,
compare the average velocities of the electrons
in the two resistors
a) v1 gt v2 b) v1 v2
c) v1 lt v2
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Question 2

• a
• b
• c

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Question 2
Two cylindrical resistors are made from the same
material, and they are equal in length. The first
resistor has diameter d, and the second resistor
has diameter 2d.
If the same current flows through both resistors,
compare the average velocities of the electrons
in the two resistors
a) v1 gt v2 b) v1 v2
c) v1 lt v2
Current ? Area ? Current Density Current Density
? average velocity of electrons I is the same
A1ltA2 ? v1gtv2
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Resistors in Series

• What is the same effective single resistance to
  two resistances in series?
• Whenever devices are in SERIES, the current is
  the same through both.

• By Ohms law, the Voltage difference across
  resistance R1 is
• Across R2 is
• Total voltage difference
• ?the effective single resistance is

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Another (intuitive) way

• Consider two cylindrical resistors with lengths
  L1 and L2

• Put them together, end to end to make a longer
  one...

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The Worlds Simplest (and most useful)
circuitVoltage Divider
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Question 3
Two resistors are connected in series to a
battery with emf E. The resistances are such
that R1 2R2. The currents through the
resistors are I1 and I2 and the potential
differences across the resistors V1 and V2. Are

• I1gtI2 and V2E
• I1I2 and V2 E
• I1I2 and V21/3E
• I1ltI2 and V21/2E
• I1ltI2 and V21/3E

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Resistors in Parallel

• Very generally, devices in parallel have the same
  voltage drop

I
a
I1
I2

• Current through R1 is I1.
• Current through R2 is I2.

V
R1
R2
I
d

• But current is conserved

Þ
Þ
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Another (intuitive) way
Consider two cylindrical resistors with
cross-sectional areas A1 and A2
Put them together, side by side to make one
fatterone,
Þ
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Kirchhoffs First RuleLoop Rule or
Kirchhoffs Voltage Law (KVL)
"When any closed circuit loop is traversed,
the algebraic sum of the changes in potential
must equal zero."

• This is just a restatement of what you already
  know that the potential difference is
  independent of path!

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Rules of the Road

• Our convention
• Voltage gains enter with a sign, and voltage
  drops enter with a - sign.
• We choose a direction for the current and move
  around the circuit in that direction.
• When a battery is traversed from the negative
  terminal to the positive terminal, the voltage
  increases, and hence the battery voltage enters
  KVL with a sign.
• When moving across a resistor, the voltage drops,
  and hence enters KVL with a - sign.

e1
- e2
0
- IR1
- IR2
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Current in a Loop
Start at point a (could be anywhere) and assume
current is in direction shown (could be either)
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Question 3

• Consider the circuit shown.
• The switch is initially open and the current
  flowing through the bottom resistor is I0.
• Just after the switch is closed, the current
  flowing through the bottom resistor is I1.
• What is the relation between I0 and I1?

(a) I1 lt I0
(b) I1 I0
(c) I1 gt I0
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Question 3

• a
• b
• c

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Question 3

• Consider the circuit shown.
• The switch is initially open and the current
  flowing through the bottom resistor is I0.
• Just after the switch is closed, the current
  flowing through the bottom resistor is I1.
• What is the relation between I0 and I1?

(a) I1 lt I0
(b) I1 I0
(c) I1 gt I0

• From symmetry the potential (Va-Vb) before the
  switch is closed is Va-Vb 12V.
• Therefore, when the switch is closed, potential
  stays the same and NO additional current will
  flow!
• Therefore, the current before the switch is
  closed is equal to the current after the switch
  is closed.

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Question 3

• Consider the circuit shown.
• The switch is initially open and the current
  flowing through the bottom resistor is I0.
• After the switch is closed, the current flowing
  through the bottom resistor is I1.
• What is the relation between I0 and I1?

12V
R
a
I
12V
12V
R
b

• Write a loop law for original loop

12V 12V - I0R - I0R 0 I0 12V/R

• Write a loop law for the new loop

12V - I1R 0 I1 12V/R
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Kirchhoffs Second RuleJunction Rule or
Kirchhoffs Current Law (KCL)

• In deriving the formula for the equivalent
  resistance of 2 resistors in parallel, we applied
  Kirchhoff's Second Rule (the junction rule).
• "At any junction point in a circuit where the
  current can divide (also called a node), the sum
  of the currents into the node must equal the sum
  of the currents out of the node."

• This is just a statement of the conservation of
  charge at any given node.

• The currents entering and leaving circuit nodes
  are known as branch currents.
• Each distinct branch must have a current, Ii
  assigned to it

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How to use Kirchhoffs Laws
A two loop example

• Assume currents in each section of the circuit,
  identify all circuit nodes and use KCL.

(1) I1 I2 I3

• Identify all independent loops and use KVL.

• e1 - I1R1 - I2R2 0
• I2R2 - e2 - I3R3 0
• e1 - I1R1 - e2 - I3R3 0

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How to use Kirchoffs Laws

• Solve the equations for I1, I2, and I3
• First find I2 and I3 in terms of I1

Now solve for I1 using eqn. (1)
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Lets plug in some numbers
e1 24 V
e 2 12 V
R1 5W R23W R34W
Then, and
I12.809 A I2 3.319 A, I3
-0.511 A
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Junction Demo
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Summary

• Kirchhoffs Laws
• KCL Junction Rule (Charge is conserved)
• Review KVL (V is independent of path)
• Non-ideal Batteries Power
• Discharging of capacitor through a Resistor

Reading Assignment Chapter 26.6
Examples 26.17,18 and 19
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Two identical light bulbs are represented by the
resistors R2 and R3 (R2 R3 ). The switch S is
initially open.
2) If switch S is closed, what happens to the
brightness of the bulb R2?
a) It increases b) It decreases
c) It doesnt change
3) What happens to the current I, after the
switch is closed ?
a) Iafter 1/2 Ibefore b) Iafter Ibefore c)
Iafter 2 Ibefore
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I
R2
R1
Four identical resistors are connected to a
battery as shown in the figure.
R4
R3
E
5) How does the current through the battery
change after the switch is closed ?
Before Rtot 3R Ibefore 1/3
E/R After R23 2R R423 2/3 R
Rtot 5/3 R Iafter 3/5 E/R
a) Iafter gt Ibefore b) Iafter Ibefore c)
Iafter lt Ibefore