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Series and Parallel
Resistances and
Kirchhoff’s Laws
Objectives
•
Calculate the equivalent resistance for resistors connected in both
series and parallel combinations.
•
Construct series and parallel circuits of lamps (resistors).
•
Observe and explain relative lamp brightness in series and parallel
circuits.
•
Apply Kirchhoff’s first and second laws.
•
Calculate the current and voltage for resistor circuits connected in
parallel.
•
Calculate the current and voltage for resistor circuits connected in
series.
Physics terms
•
series circuit
•
parallel circuit
•
equivalent resistance
•
junction
•
loop
Equations
Equivalent resistance for
resistors connected in series
Equivalent resistance for
resistors connected in parallel
Equations
Kirchhoff’s laws:
current law:
voltage law:
Series circuits
A series circuit has only one path for the flow of electric current.
I
I
I
Parallel circuits
In a parallel circuit the electric current can split apart and
come back together again.
V
V
Series circuits
The battery and the resistors in these circuits are identical.
What is different about these circuits?
Series circuits
The circuit with 3 resistors is 3 times harder for
the current to flow through.
I1
I2
I3
When you add resistors in series, equivalent resistance increases
and current flow decreases: I3 < I2 < I1
Equivalent resistance
What is “equivalent resistance”?
• Equivalent resistance is the total combined resistance
of a group of resistors.
• Equivalent resistance has the value of a single resistor
that could replace the multiple resistors in a circuit,
while keeping total current through the circuit the same.
Equivalent resistance
These two circuits have the same total resistance . . .
. . . so they will have the same total current, I.
Equivalent resistance: series
For resistors in series, you can find the equivalent
resistance by simply adding up the individual
resistances:
Engaging with the concepts
A 10 Ω resistor, a 15 Ω resistor,
and a 5 Ω resistor are connected
in series.
What is the equivalent
resistance of this arrangement?
10
15
5
Engaging with the concepts
A 10 Ω resistor, a 15 Ω resistor,
and a 5 Ω resistor are connected
in series.
What is the equivalent
resistance of this arrangement?
30
10 Ω + 15 Ω + 5 Ω = 30 Ω
10
15
5
Engaging with the concepts
Two strings of tree lights, each
with a resistance of 150 Ω, are
connected together in series.
What is the Req?
150
150
0
Engaging with the concepts
Two strings of tree lights, each
with a resistance of 150 Ω, are
connected together in series.
What is the Req?
300 Ω
When you add resistors in series
does Req increase or decrease?
Does total current increase or
decrease?
300
150
150
0
Engaging with the concepts
Two strings of tree lights, each
with a resistance of 150 Ω, are
connected together in series.
What is the Req?
300 Ω
When you add resistors in series
does Req increase or decrease?
Req increases
Does total current increase or
decrease?
I decreases
450
150
150
150
What is the Req of each circuit?
Req = _______
Req = _______
What is the Req of each circuit?
Req = 10 Ω
Req = 15 Ω
How much current flows?
Req = 10 Ω
Req = 15 Ω
How much current flows?
1A
0.67 A
Parallel circuits
A parallel circuit has more paths for electric current to flow.
The circuit on the right lets twice as much current flow.
What is the Req for the parallel circuit?
R
2R
½R
Parallel circuits
In the circuit with parallel resistors:
total current flow doubles because total resistance is halved.
What is the Req for the parallel circuit?
R
2R
½R
Equivalent resistance: parallel
When you add resistors in parallel, total resistance goes DOWN.
To find the Req you must add the inverse of the resistances:
Equivalent resistance: parallel
If you have two 4 Ω resistors in parallel, what is
the equivalent resistance?
A. ½ Ω
B. 2 Ω
C. 4 Ω
D. 8 Ω
Equivalent resistance: parallel
Don’t forget to
flip the fraction
at the end!
If you have two 4 Ω resistors in parallel, what is
the equivalent resistance?
A. ½ Ω
B. 2 Ω
C. 4 Ω
D. 8 Ω
Engaging with the concepts
A 10 Ω resistor and a 15 Ω
resistor are connected in
parallel. What is the Req of
this arrangement?
10
Equivalent resistance
15
Engaging with the concepts
A 10 Ω resistor and a 15 Ω
resistor are connected in
parallel. What is the Req of
this arrangement? 6 ohms
6
Equivalent resistance
When resistors are connected
in parallel, total resistance
decreases.
10
15
Engaging with the concepts
Two strings of lights, each
with a resistance of 150 Ω,
are connected together.
What is the Req when the
strings are connected in
parallel?
150
Equivalent resistance
150
Engaging with the concepts
Two strings of lights, each
with a resistance of 150 Ω,
are connected together.
What is the Req when the
strings are connected in
parallel? 75 Ω
75
150
150
Equivalent resistance
The total resistance is always less than any
of the individual resistors added in parallel!
What is the Req of each circuit?
These are all 10 Ω resistors. What is the Req in each case?
Req = ____
Req = ____
Req = ____
What is the Req of each circuit?
These are all 10 Ω resistors. What is the Req in each case?
Req = 10 Ω
Req = 5 Ω
Req = 3.3 Ω
Adding resistors in parallel makes the total resistance decrease.
How much current flows?
Apply Ohm’s law ( V = IR ) to find the total current through the battery.
Req = 10 Ω
I = ____
Req = 5 Ω
I = ____
Req = 3.3 Ω
I = ____
How much current flows?
Apply Ohm’s law ( V = IR ) to find the total current through the battery.
Req = 10 Ω
I=3A
Req = 5 Ω
I=6A
Req = 3.3 Ω
I=9A
Analogy for resistors in series
What happens to the potential
energy of the water as it passes
through each water wheel?
How is this analogous to
the voltage drops across
each resistor?
Voltage drops: resistors in series
Half as much current flows in the series circuit.
What are the voltage drops across the two resistors in series?
Voltage drops: resistors in series
10 V
drop
10 V
drop
Single resistor
Resistor in series
Voltage drops: resistors in parallel
Twice as much current flows in the parallel circuit.
What are the voltage drops across the resistors in parallel?
Voltage drops: resistors in parallel
4A
4A
20 V
drop
Single resistor
Resistor in parallel
Each resistor in parallel has the full 20 V drop.
Assessment
1. Two resistors with resistances of 10 Ω and 30 Ω are
connected in series with a 20 volt battery.
a) What is the equivalent resistance of the circuit?
b) What is the current flow through the circuit?
Assessment
1. Two resistors with resistances of 10 Ω and 30 Ω are
connected in series with a 20 volt battery.
a) What is the equivalent resistance of the circuit?
b) What is the current flow through the circuit?
Assessment
1. Two resistors with resistances of 10 Ω and 30 Ω are
connected in series with a 20 volt battery.
a) What is the equivalent resistance of the circuit?
b) What is the current flow through the circuit?
Assessment
2. Two resistors with resistances of 10 Ω and 30 Ω are
connected in parallel with a 20 volt battery.
a) What is the equivalent resistance of the circuit?
b) What is the current flow through the circuit?
Assessment
2. Two resistors with resistances of 10 Ω and 30 Ω are
connected in parallel with a 20 volt battery.
a) What is the equivalent resistance of the circuit? 7.5 Ω
b) What is the current flow through the circuit?
Assessment
2. Two resistors with resistances of 10 Ω and 30 Ω are
connected in parallel with a 20 volt battery.
a) What is the equivalent resistance of the circuit? 7.5 Ω
b) What is the current flow through the circuit?
Assessment
3. If you connect these two identical resistors as shown, will they
together draw more or less current than one of the resistors
alone?
Assessment
3. If you connect these two identical resistors as shown, will they
together draw more or less current than one of the resistors
alone?
They will draw more current, because they are connected in
parallel. There are now two different, but identical paths for
electricity to flow, so the current doubles!
Kirchhoff’s laws
Two fundamental laws apply to
ALL electric circuits.
These are called Kirchhoff’s laws,
in honor of German physicist
Gustav Robert Kirchhoff
(1824–1887).
Kirchhoff’s current law
Kirchhoff’s first law is the current law.
•It is a rule about electric current.
•It is always true for ALL circuits.
Kirchhoff’s current law
The current law is also known as
the junction rule.
A junction is a place where three
or more wires come together.
This figure shows an enlargement
of the junction at the top of the
circuit.
Kirchhoff’s current law
Current Io flows INTO the junction.
Currents I1 and I2 flow OUT of the
junction.
What do you think the current law
says about I, I1, and I2?
Kirchhoff’s current law
Kirchhoff’s current law:
The current flowing INTO a junction
always equals the current flowing OUT
of the junction.
Kirchhoff’s current law
Example:
Why is the current law true?
Why is this law always true?
Conservation of charge
Why is this law always true?
It is true because electric charge
can never be created or destroyed.
Charge is ALWAYS conserved.
Applying the current law
This series circuit has NO
junctions.
The current must be the same
everywhere in the circuit.
Current can only change at a
junction.
Applying the current law
A 60 volt battery is connected
to three identical 10 Ω
resistors.
10 Ω
What are the currents
through the resistors?
60 V
10 Ω
10 Ω
Applying the current law
A 60 volt battery is connected
to three identical 10 Ω
resistors.
10 Ω
What are the currents
through the resistors?
Req = 30 Ω
60 V
I = 60 V/30 Ω
= 2 amps through each resistor
10 Ω
10 Ω
Applying the current law
This series circuit has two junctions. Find the missing current.
?
Applying the current law
This series circuit has two junctions. Find the missing current.
I2 = 2 A
2 amps
Applying the current law
How much current flows into the upper junction?
Applying the current law
How much current flows into the upper junction?
I=4A
4 amps
Kirchhoff’s voltage law
Kirchhoff’s second law is the voltage law.
•It’s a rule about voltage gains and drops.
•It is always true for ALL circuits.
Kirchhoff’s voltage law
The voltage law is also known as the loop rule.
A loop is any complete path around a circuit.
This circuit has only ONE loop.
Pick a starting place. There is only ONE
possible way to go around the circuit and
return to your starting place.
Kirchhoff’s voltage law
This circuit has more than one loop.
Charges can flow up through the
battery and back through R1.
That’s one loop.
Can you describe a second loop
that charges might take?
Kirchhoff’s voltage law
This circuit has more than one loop.
Charges can flow up through the
battery and back through R1.
That’s one loop.
Can you describe a second loop
that charges might take?
Charges can flow up through the battery
and back through R2. That’s another loop.
Kirchhoff’s voltage law
Kirchhoff’s voltage law says that
sum of the voltage gains and drops
around any closed loop must equal
zero.
Kirchhoff’s voltage law
If this battery provides a 30 V gain,
what is the voltage drop across each
resistor?
Assume the resistors are identical.
30 V
Kirchhoff’s voltage law
If this battery provides a 30 V gain,
what is the voltage drop across each
resistor?
-10 V
Assume the resistors are identical.
+30 V
-10 V
-10 V
10 volts each!
Applying Kirchhoff’s voltage law
A 60 V battery is connected in series
with three different resistors.
Resistor R1 has a 10 volt drop.
Resistor R2 has a 30 volt drop.
-10 V
60 V
-30 V
What is the voltage across R3?
?
Applying Kirchhoff’s voltage law
A 60 V battery is connected in series
with three different resistors.
Resistor R1 has a 10 volt drop.
Resistor R2 has a 30 volt drop.
-10 V
60 V
-30 V
What is the voltage across R3?
-20 V
20 volts
Applying Kirchhoff’s voltage law
What if a circuit has more than one loop?
Treat each loop separately.
The voltage gains and drops
around EVERY closed loop
must equal zero.
Applying Kirchhoff’s voltage law
A 30 V battery is connected in
parallel with two resistors.
What is the voltage across R1?
30 V
Applying Kirchhoff’s voltage law
A 30 V battery is connected in
parallel with two resistors.
What is the voltage across R1?
R1 must have a 30 V drop.
30 V
Applying Kirchhoff’s voltage law
A 30 V battery is connected in
parallel with two resistors.
What is the voltage across R1?
R1 must have a 30 V drop.
What is the voltage across R2?
30 V
Applying Kirchhoff’s voltage law
A 30 V battery is connected in
parallel with two resistors.
What is the voltage across R1?
R1 must have a 30 V drop.
What is the voltage across R2?
R2 also has a 30 V drop.
30 V
Why is the voltage law true?
Why is this law always true?
Why is the voltage law true?
Why is this law always true?
This law is really conservation of energy for circuits.
All the electric potential energy gained
by the charges must equal the energy
lost in one complete trip around a loop.
Conservation of energy
All the electrical energy gained
by passing through the battery
is lost as charges pass back
through the resistors.
All the gravitational potential
energy gained by going up a
mountain is lost by going
back to your starting place.
Assessment
1. A current I = 4.0 amps flows
into a junction where three
wires meet.
I1 = 1.0 amp. What is I2?
Assessment
1. A current I = 4.0 amps flows
into a junction where three
wires meet.
I1 = 1.0 amp. What is I2?
Use the junction rule: I2 = 3.0 amps
Assessment
2. A 15 volt battery is connected in
parallel to two identical resistors.
a) What is the voltage across R1?
b) If R1 and R2 have different
resistances, will they have
different voltages?
Assessment
2. A 15 volt battery is connected in
parallel to two identical resistors.
a) What is the voltage across R1?
15 volts (use the loop rule)
b) If R1 and R2 have different
resistances, will they have
different voltages?
Assessment
2. A 15 volt battery is connected in
parallel to two identical resistors.
a) What is the voltage across R1?
15 volts (use the loop rule)
b) If R1 and R2 have different
resistances, will they have
different voltages?
They will still both have
a 15 V drop.
Assessment
3. Two 30 Ω resistors are connected in parallel with a 10 volt battery.
a) What is the total resistance of the circuit?
a) What is the voltage drop across each resistor?
c) What is the current flow through each resistor?
Assessment
3. Two 30 Ω resistors are connected in parallel with a 10 volt battery.
a) What is the total resistance of the circuit? 15 ohms
a) What is the voltage drop across each resistor?
c) What is the current flow through each resistor?
Assessment
3. Two 30 Ω resistors are connected in parallel with a 10 volt battery.
a) What is the total resistance of the circuit? 15 ohms
a) What is the voltage drop across each resistor? 10 volts
Each resistor is in its own loop with the 10 V battery,
so each resistor has a voltage drop of 10 V.
c) What is the current flow through each resistor?
Assessment
3. Two 30 Ω resistors are connected in parallel with a 10 volt battery.
a) What is the total resistance of the circuit? 15 ohms
a) What is the voltage drop across each resistor? 10 volts
Each resistor is in its own loop with the 10 V battery,
so each resistor has a voltage drop of 10 V.
c) What is the current flow through each resistor? 0.33 amps
Assessment
4. Two 5.0 Ω resistors are connected in series with a 30 volt battery.
a) What is the total resistance of the circuit?
a) What is the current flow through each resistor?
c) What is the voltage drop across each resistor?
Assessment
4. Two 5.0 Ω resistors are connected in series with a 30 volt battery.
a) What is the total resistance of the circuit? 10 ohms
a) What is the current flow through each resistor?
c) What is the voltage drop across each resistor?
Assessment
4. Two 5.0 Ω resistors are connected in series with a 30 volt battery.
a) What is the total resistance of the circuit? 10 ohms
a) What is the current flow through each resistor? 3.0 amps
The circuit has only one branch,
so current flow is the same
everywhere in the circuit.
c) What is the voltage drop across each resistor?
Assessment
4. Two 5.0 Ω resistors are connected in series with a 30 volt battery.
a) What is the total resistance of the circuit? 10 ohms
a) What is the current flow through each resistor? 3.0 amps
The circuit has only one branch,
so current flow is the same
everywhere in the circuit.
c) What is the voltage drop across each resistor? 15 volts
Use the loop rule: