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Class 9
Today we will:
• learn about current, voltage, and power in
circuits.
• learn about resistance of materials and how
resistance depends on geometry and temperature.
• introduce Ohm’s law.
Lesson 4
Circuits and Resistance
Current, Voltage, and Power in
Simple Circuits
Current
• Benjamin Franklin didn’t know if current was caused
by positive charges moving or negative charges
moving, so he took a guess… and got it wrong.
Current
• Current was defined as the direction positive charge
in a wire would travel.
Current
• In reality, negative charges are moving in the opposite
direction to the current.
Current
• However, we usually ignore that and talk about
“positive charge carriers” in a wire that move in the
direction of the current.
Current
• Current is the charge that moves past a point in the
wire per unit time.
Ne dq
i

t dt
Current
• Units of current are amperes or amps.
1 coulomb
• 1 ampere 
sec
• Car batteries deliver several hundred amperes.
• Most electronic circuits run on a few mA.
Voltage
•Voltage is electric potential in circuits.
•Voltage is provided by batteries or generators.
•A battery pushes electrons out the negative terminal
and sucks electrons into the positive terminal.
Voltage
•If we attach a wire to the positive terminal, a few go
into the battery, leaving a positive charge on the
surface of the wire.
•The electrons stop moving when the surface charge
on the wire pulls the electrons in the wire with the
same force as the positive charge on the battery.
+
++++++++++++
Circuits
•If we connect a wire from the positive to the negative
terminal of the battery, current will continue to flow
through the circuit.
–
– –
–
–
– – ––
–
–
+
+
+
+
+
+
+
++ + + +
Circuits
•Charge remains on the surface of the wire. The
surface charge is positive near the positive terninal
and negative near the negative terminal.
•The charge density is greatest near the terminals of
the battery.
•Current flows uniformly through the entire crosssection of the wire.
–
– –
–
–
– – ––
–
–
+
+
+
+
+
+
+
++ + + +
Circuits
• A 10V battery gives 10eV of energy to each electron
that passes through it. U  qV
• Collisions with atoms in the wire cause each electron
to lose 10eV of energy every time it goes around the
circuit.
–
– –
–
–
– – ––
–
–
+
+
+
+
+
+
+
++ + + +
Ground
•The ground acts like a huge conductor.
•Current can flow into the ground or out of the ground
without any limits.
•The two circuits below are equivalent.
+
+
Circuits
Definitions:
•An open circuit is one where there is an open switch
or a broken wire so that no current flows.
•A closed circuit is one in which there is a continuous
path for current to flow from positive to negative.
•A short circuit is one where there is an unintentional
current path to ground. Currents, sometimes large,
flow where they should not, leading to shock and fire
hazards.
Circuits
What good does flowing charge do?
•Produces heat, light.
•Produces magnetic fields – used in motors, vibrators,
etc. to give mechanical power.
•Produces electromagnetic radiation – radio waves for
communication.
•Electronics: amplification, logic, light detection,
radiation detection, cathode-ray tubes, etc.
Power
Each time an electron goes through a battery, it gains
energy. The total energy gained per second is:
# of electrons
P  U of each electron 
second
charge
1
 U of each electron 

second charge per electron
1
 eV  i   iV
e
P=iV
Where does the energy go?
Where does the energy go?
•Electrons collide with other
electrons in atoms and quickly
reach terminal velocity – so they
don’t keep gaining kinetic energy.
Where does the energy go?
• In a wire, it goes to heat.
P=iV
•In other devices it can go to
light, mechanical energy, energy
of radiation fields, etc.
Resistance
Resistance in a Wire
V
Definition: R(V , I ) 
I
•In general R is a function of I,
and V.
•For many materials R is nearly a
constant.
•When R is a constant, we call
the material “ohmic.”
Resistors
•Devices to increase the resistance in part
of a circuit.
•Made of graphite chunks, wire wound
around a core, etc.
•Used to
•Produce heat or light
•Adjust current flow and voltages in
circuits.
Resistors
• Even if we don’t want the resistance, we
often need to account for resistance in
cables, electronic devices, etc.
Ohm’s Law
We assume resistors have constant V.
V  IR
Resistance has units of ohms, written as
an upper case omega.
1V
1 
1A
Typical resistances range from a few
ohms to several megohms.
Graphite Resistors
•Resistance is color coded.
0
1
2
3
4
5
second digit
2
10%
7
8
tolerance
6
# of zeros
5
first digit
6
9
5%
10%
R=5600 Ω (+/-10%)
20%
What Affects Resistance?
What Affects Resistance?
•Material
•Length
•Cross-sectional area
•Temperature
Resistance and Geometry
• One block has V, I, R.
Resistance and Geometry
• Take two blocks with I going through each.
• Voltage is
V   2V
Current is
V  2V

 2R
• Resistance is R 
I
I
I  I
Resistance and Geometry
• Take two blocks with I going through each.
• Voltage is
V
Current is 2 I
V V
R


• Resistance is R 
I  2I 2
Resistance and Resistivity

R
A
• ρ is the resistivity. It depends on the material
from which the resistor is made. The units of
resistivity are Ωm.
• σ = 1/ ρ is the conductivity
Resistance and Temperature
We assume that resistance varies
linearly with temperature.
R  a  bT
Resistance and Temperature
R  R0 1   T  T0 
If T = T0, then R =R0.
Resistance and Temperature
R  R0 1   T  T0 
• α is the temperature coefficient of
resistivity (resistance).
• α is usually positive.
• α is negative for graphite.
Class 10
Today we will:
• learn how to determine if two resistors are in
series or parallel.
• find out how resistors combine when connected
in series and parallel.
• work examples of series-parallel reduction to
find current, voltage and power in resistance
networks.
Resistors in Series
Have the Same Current
• Take two resistors with I going through each.
• Voltage is
V   V1  V2 Current is I   I
Resistance is
V  V1  V2 V1 V2
R 

   R1  R2
I
I
I
I
Resistors in Series
I ,V1 , R1
I ,V2 , R2
V  V1  V2
R  R1  R2

I ,V , R
Resistors in Parallel
Have the Same Voltage
• Take two blocks with I going through each.
• Voltage is
V V
Current is
I   I1  I 2
Resistance is
1
I  I1  I 2 I1 I1 1
1


   
R V 
V
V V R1 R2
Resistors in Parallel
Have the Same Voltage
I1 ,V , R1

I 2 ,V , R2
I  I1  I 2
1 1
1
 
R R1 R2
I ,V , R
A Test for Resistors in Series
Look at the wire connecting the two resistors. Is
there anything at all (a circuit element or a
junction) along this wire?
no
The resistors are
connected in
series.
yes
The resistors are
NOT connected in
series.
A Test for Resistors in Parallel
Look at the wire connecting one end of the first resistor to one end of
the second resistor. Is there a circuit element (a junction is OK and
usually there are junctions) along this wire?
yes
no
The resistors are NOT connected in parallel.
Look at the wire connecting the other end of the first resistor to the
other end of the second resistor. Is there a circuit element along this
wire?
yes
The resistors are NOT
connected n parallel.
no
The resistors are
connected n
parallel.
Series-Parallel Quiz
Answer the following six
questions
to see if you understand what
series and parallel mean.
Resistors A and B are in
1. series
2. parallel
3. neither
Resistors A and B are in
1. series
2. parallel
3. neither
Resistors A and B are in
1. series
2. parallel
3. neither
Resistors A and B are in
1. series
2. parallel
3. neither
Resistors A and B are in
1. series
2. parallel
3. neither
Resistors A and B are in
1. series
2. parallel
3. neither
Quiz Answers
1. series
2. neither
3. neither
4. parallel
5. series
6. parallel
Series- Parallel Reduction
• Find a combination in series or parallel.
• Combine resistors into a single equivalent
resistor.
•Repeat until there is only one resistor.
•The voltage across the resistor is the same as
the voltage across the battery.
Series- Parallel Reduction
• Find V, I, R, P for the last step.
•Bootstrap your way back to the beginning,
diagram by diagram.
• What if there are resistors that aren’t in series or
parallel?
--- You’ll need to use Kirchoff’s Laws which we’ll
learn later.
Now we’ll work some examples…
Find all the currents, voltages, and powers
2A
24V,2A
2A
2A
2A
2A, 4V
2A
2A, 20V
4V
4V
2A
2A, 20V
4V, 2/3 A
4V, 4/3 A
2A
2A, 20V
2/3 A
2/3 A
4V, 4/3 A
2A
2A
2A
2/3 A, 8/3 V
2/3 A, 4/3 V
4V, 4/3 A
2A
2A, 4V
2A, 16V
16/9 W
2/3 A, 8/3 V
8/9 W
2/3 A, 4/3 V
4V, 4/3 A 16/3 W
2A
48 W
2A, 16V
32 W
2A, 4V
8W
Using Meters
• Ammeters measure current. they must be paced in
series with other circuit elements so current flows through
them. Ammeters should have very small voltage.
• Voltmeters measure voltage. To measure the voltage
between two points, you connect the two leads of the
meter to those points. Therefore voltmeters are placed in
parallel. Voltmeters should have large voltage.
Real Batteries
• Real batteries have internal resistance. When
they are placed in a circuit we can represent
them as a resistor in series with an ideal battery.
There’s a voltage drop
across the internal
resistance.
This means that the full
voltage of the ideal
battery isn’t available to
the circuit.
Class 11
Today we will:
• discuss Kirchoff’s loop and node equations.
• learn how to determine the number of loop and
the number of node equations we will need.
• write Kirchoff’s equations for a sample circuit.
Kirchoff’s Junction Rule
• Current into a junction equals current out of a
junction.
• Comes from conservation of charge.
Kirchoff’s Junction Rule
• It’s like water in pipes – the water flowing into a
junction must flow out again.
Kirchoff’s Loop Rule
• The net change in voltage around a closed loop
is zero.
• Comes from Conservation of energy
V  0
Kirchoff’s Loop Rule
• It’s like moving a ball around any closed path,
the change in gravitational potential energy is
zero.
U  0
Applying Kirchoff’s Laws
This is a “turn the crank” approach – but it works well.
1. Mark each junction.
A
B
D
C
2. Label all currents – between each pair of
junctions. Choose a direction – it doesn’t
have to be right!
Here there are 6 currents.
A
I1
I3
B
I6
D
I4
I5
I2
C
3. Number of junction equations: If there are N
junctions, there are N-1 junction equations.
Here there are 4 junctions, so there are 3
junction equations.
A
I1
I3
B
I6
D
I4
I5
I2
C
4. Write the junction equations:
Current in = Current out.
A : I 3  I 6  I1
B : I1  I 4  I 2
C : I 2  I5  I6
A
I1
I3
B
I6
D
I4
I5
I2
C
5. Number of loop equations: Number of
currents – Number of junction equations.
A : I 3  I 6  I1
Here: 6-3=3, so we need 3 loops.
B : I1  I 4  I 2
C : I 2  I5  I6
A
I1
I3
B
I6
D
I4
I5
I2
C
6. At least one loop must cover every circuit
element.
A : I 3  I 6  I1
B : I1  I 4  I 2
C : I 2  I5  I6
A
I1
I3
1
2
B
D
I4
I5
3
I2
C
I6
7. Put a plus or minus on every resistor (side
current goes in is +) and battery (+ is
positive terminal).
A : I 3  I 6  I1
B : I1  I 4  I 2
C : I 2  I5  I6
A
I1
+
+
I3
++
1
2
B
+
I2
D
+
I4
3
+
C
I5
+
I6
8. Write the loop equations.
1 :  6  2 I 3  1I1  4 I 4  0
2 :  3I 6  2 I 3  6  6 I 5  7 I 6  0
A : I 3  I 6  I1
3 :  6 I 5  4 I 4  5I 2  0
B : I1  I 4  I 2
C : I 2  I5  I6
A
I1
+
+
I3
++
1
2
B
+
I2
D
+
I4
3
+
C
I5
+
I6
8. Write the loop equations.
When you go around the loop, ignore the current
arrows! Follow the loop in the direction of the loop
arrow!
A
I1
+
+
I3
++
1
2
B
+
I2
D
+
I4
3
+
C
I5
+
I6
8. Solve the system of equations using your
favorite method. I’ll usually just ask for the
equations.
1 :  6  2 I 3  1I1  4 I 4  0
2 :  3I 6  2 I 3  6  6 I 5  7 I 6  0
3 :  6 I 5  4 I 4  5I 2  0
A : I 3  I 6  I1
B : I1  I 4  I 2
C : I 2  I5  I6