1. No category

Honors Physics
Unit 9: Electricity & Circuits
Name: _______________
Static Investigation with Scotch Tape
Work with a partner at your table, with each person completing the questions below on your individual packets. Write
your answers after discussing with your table. If you have two pairs, compare results.
1.
Have each person in your pair tear off a piece of Scotch tape about
15 cm long. Separately, put a ruler flat on the tabletop, and press
the pieces of tape firmly onto the tabletop, with just the end of the
tape on the ruler, as shown to the right. Then both of you simultaneously
pick up the tape end of the rulers quickly, peeling the tape off the table. Hold
the rulers by the other end so that your hands aren’t close to the tape. Do
NOT let the tape touch anything, if it does, retape it to the table and start
again.
Ruler
Ruler&
Tape
&&&&&Tape&
A) Gradually bring the two tapes toward one another without letting them touch. Move them so that first the sticky
sides face each other, then the non-sticky sides, then the sticky side faces the non-sticky side. Describe how the
tapes behave.
B) Does the behavior of the tapes depend on which side (sticky or non-sticky) of the tape faces the other?
C) How does the distance between the tapes affect their interaction?
D) Describe the behavior of the following pairs of tape when they are brought near (but not touching) each other:
i) Two T tapes
ii) Two B tapes
iii) A T and a B tape
Now remove the four tapes and begin again, this time stacking four pieces of tape on top of one another, labeling them B
(the bottom tape, 2, 3, and T (the top tape). Peel them off the table as before, then peel them apart in any order.
E) Describe the behavior of the following pairs of tape when they are brought near (but not touching) each other:
i) Tapes B and 2
ii) Tapes B and 3
iii) Tapes B and T
iv) Tapes 2 and 3
v) Tapes 2 and T
vi) Tapes 3 and T
F) Does tape 2 behave consistently like tape B or tape T? If so, which?
Honors Physics
Unit 9: Electricity & Circuits
G) Does tape 3 behave consistently like tape B or tape T? If so, which?
We use the term charge to describe the property of matter that makes the tapes behave the way they do.
H) What can you conclude about how many different types of charge the tapes exhibit in these experiments, and how
these types of charge interact?
Electric Charge
One property of the particles that make up atoms is their electric charge, which comes in two varieties: positive and
negative. The unit of electric charge is the coulomb (C) named after French physicist Charles Coulomb.
Electric charge exists in multiples of the elementary charge, which is the charge carried by a proton (+) or an
electron (-), which means that electric charge is quantized. The elementary charge (e) is 1.6 x 10-19 C. Each electron
has -1.6 x 10-19 C of charge and each proton has +1.6 x 10-19 C of charge.
Problems
1. The mass of an electron is 9.1 x 10-31 kg. If you could pack together one gram of electrons (about the mass of an
aspirin tablet), how many coulombs of charge would it have?
2. How many electrons are there in 1 C of charge?
Electric charge is a quantity that is conserved; that is, in any isolated system the total amount of charge (the algebraic
sum of positive and negative charge) remains the same.
3. If you could squeeze an electron so that it was half its original volume, what would happen to its charge?
Coulomb's Law
The force of repulsion or attraction depends on the magnitudes of the two charges and their distance apart in much
the same way that the gravitational force between two masses depends on the magnitudes of the two masses and
their distance apart.
The electric force between two charges q1 and q2 a distance r apart is given by Coulomb's Law:
F=
kq1q2
r2
where the proportionality constant k has the approximate value of 9 x 109. What are the units of k?
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Honors Physics
Unit 9: Electricity & Circuits
4m
1. Two small charges are positioned as shown in the diagram to the right.
a. Find the force on the 3 µC charge due to the 1 µC charge.
1 µC
3 µC
b. Now find the force on the 1 µC charge due to the 3 µC charge.
2. What would the force between a gram of electrons on the earth and a gram of electrons on the moon, 3.85 x 108 m
away?
3. Would this force be attractive or repulsive? Is this force very strong?
The force is along a line joining the two charges. Electric force is a vector, like any other force. If there are more
than two charges, the force on each is the vector sum of the forces due to the others. This means if the line of force
between more than 2 charges is not along an axis, you would have to break up at least one of the forces into
component vectors.
4.
The three charges at right are separated by r = 2 cm and q = 1 nC. What is the
magnitude and direction of the force felt by the negative charge?
-2q
r
r
q
q
r
The Electric Field
Consider an electric charge +Q by itself in a region of space. Now consider a point
P
Q
P which is a distance r away from the charge. We know that if we put another
•
positive charge at P, it would feel an electric force F. The possibility of a force
r
exists everywhere around Q, whether or not we put charges there to feel them. The
electric field is defined based on this idea.
Place a small positive charge qo (called the test charge) at point P. The electric field E at that point is a vector
pointing in the direction of the electric force F on the test charge. The electric field E is defined as the electric force
E=
divided by the charge qo.
F
qo . The units of electric field are N / C.
3
Honors Physics
5.
Unit 9: Electricity & Circuits
Using the definition for force from Coulomb’s law, find E below:
6.
We can think of the gravitational force the same way as the electric force. Each mass causes a gravitational
field, and other masses feel a force from the field.
A) To the!right is a table listing the definition and units for the electric
field E . Fill in the!corresponding boxes in the table for the
gravitational field g .
B) What is the value of the gravitational field of the earth near its
surface?
This applet allows you to set up a distribution of charge and see the resulting electric field:
http://www.cco.caltech.edu/~phys1/java/phys1/EField/EField.html
Electric Field of a Point Charge
Q
Consider an electric charge +Q by itself in a region of space, and a point P
which is a distance r away from the charge.
1.
If a small positive charge q (called a test charge) were placed at
point P, what would the magnitude of the electric force be on it?
P
•
r
2.
What is the magnitude of the electric field at P?
C) If a larger test charge were used, or a negative charge instead of a positive one, would that affect your answer to
part B)? If so, how? If not, why not?
E) The boxes below represent the space near two point charges; one positive and one negative. In each case, points 1
and 2 are equal distances from the charge, point 3 is twice that distance away, and point 4 is three times that
distance away. At each of the four labeled points in the boxes, draw vectors representing the electric field. The
length of the vectors should be proportional to the strength of the electric field at that point.
1•
•2
1•
3•
•2
3•
•4
Single positive charge
•4
Single negative charge
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Honors Physics
Unit 9: Electricity & Circuits
Electric Field Lines
The electric field is a set of vectors defined at each point in space. The vectors you’ve drawn above are only a few
of infinitely many electric field vectors surrounding the charges. Since it's impossible to draw all those vectors
(they'd all be on top of one another and be confusing), we have a different way to represent the field. A field line is a
line drawn in such a way that it is always tangent to the field vectors.
In the boxes below, draw lines that start at the charges and continue to the edge of the box. Somewhere along each line
draw an arrowhead indicating the direction of the electric field along the line. These are called electric field lines.
1•
•2
1•
3•
•2
3•
•4
Single positive charge
•4
Single negative charge
The field lines not only give information about the direction of the field, they also indicate its strength. When field
lines are close together, the field is strong, and when they spread farther apart, the field is weaker.
6.
Combine the two charges into the box below: draw both a positive and a negative charge and for each location
you choose, draw both vectors to indicate the force from each charge. Then draw the composite vector at that
location to indicate total force felt at that point. Choose at least 6 different locations.
5
Honors Physics
7.
Unit 9: Electricity & Circuits
Sketch the field lines in the plane of the paper for each of the following scenarios.
-2q
F’
8.
What would the 2 graphs above look like if the left charge was twice the magnitude of the right charge?
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Honors Physics
9.
Unit 9: Electricity & Circuits
What would the field lines look like for a vertical line of charge that went to infinity?
10. The diagram to the right shows two field lines in a region of space that cross at
point P. There are at least two reasons why this could never happen.
A) What would this mean about the direction of the electric field at point P?
This can’t happen!
P
B) What would this mean about the magnitude of the electric field at point P?
11. The diagram to the right shows two regions of space where a
cloud obscures what’s inside. There are electric field lines
pointing into Cloud 1, and pointing out from Cloud 2.
Describe what must be going in inside each cloud.
12.
Cloud 1
An infinite sheet of positive charge is shown where it intersects the page in
the diagram at right.
a. What direction is the electric field at points P1, P2, and P3? Draw the
Cloud 2
P1
vectors indicating the field at those points.
b. What do the field lines look like on both sides of the sheet? Draw them.
P2
c. How does the strength of the field vary with distance away from the
sheet? Explain your answer using the field lines you drew in b.
P3
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Honors Physics
Unit 9: Electricity & Circuits
Conductors and Insulators
Materials differ in their ability to allow charge to move through them. Materials through which charge moves easily
(such as most metals) are called conductors, and materials that inhibit the flow of charge (such as dry air, glass,
plastics, and rubber) are called insulators.
++ ++
+ + +
+
Charged
Insulator
1. A positively charged insulator is brought near an uncharged conductor as shown to the
right. Remember that an uncharged conductor has equal amounts of positive and
negative charges. Would these two objects attract, repel or neither? Explain your
reasoning using Coulomb’s Law and the fact that charges move freely through a
conductor.
2. The drawing to the right represents a positively charged solid metal ball. Initially, the excess
charge is spread evenly throughout the ball.
A) Given that the excess positive charges repel each other and move easily through the
conductor, describe what would happen to them. (Note: This would happen almost
instantly.)
Uncharged
Conductor
++
+ + +++ ++
+ + ++ + +
+ + ++ ++ +
+ + ++ +
+ + ++ +
+
B) What can you conclude about the electric field inside the solid metal ball once the charges are at equilibrium?
3. Identical conducting spheres 1 and 2 have equal amounts of charge q and
repel with a force F as shown in figure I to the right. A third identical
sphere 3 is initially neutral.
a. If the third sphere is touched to the first, as shown in figure II, how
much charge is on each sphere?
2
1
I
2
1
3
b. If the third sphere is now touched to the second, as shown in figure
III, how much charge is on each sphere?
II
1
c. If the third sphere is removed, the first two now repel with a force
F'. What is F' in terms of F?
2
3
III
1
I
V
2
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Honors Physics
Unit 9: Electricity & Circuits
Electric Potential Energy and Electric Potential
Electric Potential Energy (UE) is seen when a positive charge is brought close to another positive charge. These
charges feel a repulsive force due to each other, and this force has the potential to release energy; if the charges are
released, they will move away from each other. In the study of electricity, this value does not have as much
importance as a related variable, that of electric potential, which sounds the same but is significantly different.
The electric potential is a field, like the electric field E, but unlike the electric field the electric potential is a scalar
field rather than a vector field. Recall that to define a field we have to have a recipe for determining the field's value
at every point.
The recipe for electric potential is similar to the recipe for electric field: To find the electric potential at a given
point, place a positive test charge q at the point and measure its UE. Then divide the UE by the magnitude of the test
charge q.
The electric potential is symbolized by V, V = UE / q
and since UE = kq/r2 then
V=
kq
r
Electric potential really means electric potential energy per unit charge. The units of electric potential are measured
in volts.
4.
What are the SI units of the Volt, based on the definition given above? Prove your answer using units.
Recall that the electric field due to a distribution of point charges is the vector sum of the fields due to each one.
Since electric potential is a scalar field, the electric potential due to a distribution of point charges is the algebraic
sum of potentials due to each one.
5.
Points A, B, C and D are located as shown in the diagram,
at various distances from two positive charges of magnitude 2
µC. The “grid lines” are 1 meter apart.
A) Rank the four points in terms of the electric potential at the
point, from greatest to least. Explain how you determine
the ranking.
2 µC
A
2 µC
B
1m
C
D
B) Determine the electric potential V at the point that has the greatest electric potential.
C) Find the UE of a proton (charge = 1.6 x 10-19 C) placed at the point with the greatest electric potential.
D) Sketch vectors on the diagram to indicate the approximate direction of the electric field (if any) at each of the
four points.
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Honors Physics
Unit 9: Electricity & Circuits
E) Determine the electric potential V at point D, showing your calculation.
The two positive charges are now replaced with negative charges of the same magnitude.
G) Rank the four points in terms of the electric potential at the point, from greatest (most positive) to least (most
negative). Explain how you determine the ranking.
Potential Difference
When calculating gravitational potential energy, mgh, we can choose the
zero level anywhere we want. What is important in a practical sense is the
difference between the potential energies at two locations. Similarly, it is
often more important to know the difference in electrical potential (V1- V2)
between two points. This difference is called the potential difference ΔV
and it is measured in volts. The instrument that can measure the potential
difference between two points is called a voltmeter.
6.
from
http://electron9.phys.utk.edu/phys136d/modules/m5/Potential.htm
Two points A and B are located 100 cm and 110 cm from a positive
1 µC charge. If a voltmeter's probes are placed at A and B, what would it read?
Electric Force, Field, Potential Energy, and Potential
The chart below may help you to remember the relationships connecting the four electrostatic quantities we have
discussed:
Electric Force
!!!!!!!!"
a vector
(
)
per unit
charge
Units: N
times
distance
Units: J
(
)
Units: N/C = V/m
per unit
distance
= Work
Electric Potential Energy
(a scalar )
Electric Field
!!!!!!!!"
a vector
per unit
charge
Potential
= Gradient
Electric Potential
(a scalar )
Units: J/C = V
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Honors Physics
Unit 9: Electricity & Circuits
Introduction to Circuits
You are given a battery, a small bulb, and a small piece of wire (without alligator clips).
By experimenting with these, attempt to get the bulb to light. You should be able to come
up with four different arrangements that make the bulb light.
A)
Make a sketch of these four arrangements.
The term used for an arrangement in which the bulb lights is a closed circuit, a complete circuit, or simply a circuit.
When a connection is broken, so that the bulb no longer lights, the circuit is said to be open. Start with a complete circuit
with the battery, bulb and wire. Open it at some connecting point, and insert a metal washer between the two connecting
points.
B)
Does the bulb light?
C)
Instead of a washer, use a plastic poker chip. Does the bulb light?
Materials such as metals that allow the bulb to light are called conductors, and materials that prevent the bulb from
lighting are called insulators.
D)
What is the difference between a “complete” circuit that includes a washer, and a “complete” circuit that includes
a plastic poker chip? Should they both be called complete? Explain the difference.
Examine the light bulb closely. The part that glows is called the filament. Two wires lead from the filament
into the base (where they can’t be seen). Examine the base and the small tip at the bottom of the bulb.
E)
The two wires from the filament are attached with solder – pronounced “sod-der” – which is dull,
not shiny. On the diagram to the right, indicate where these attachment points are. !
Since the filament is a conductor, we say that there is a conducting path from one attachment point to the
other, which includes the filament. At the bottom of the base is a circular piece of insulating material.
Examine a light bulb socket.
F)
Describe how the socket works in terms of the parts of the bulb and the concepts of
insulator, conductor, and “conducting path.” Identify parts in your description on the diagram to
the right.
From now on, use the socket to connect to the bulb rather than the
bare bulb, and use the battery holder and the wires with alligator
clips to make your connections. Some materials, such as pencil
lead, fall in between the categories “insulator” and “conductor”.
You are given a piece of 9 mm pencil lead in the tray. Use it with
the battery (in the holder), bulb (in the socket) and wires with
alligator clips to form a complete circuit.
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Honors Physics
Unit 9: Electricity & Circuits
G)
Connect the pencil lead between two alligator clips as shown in figure I.
Does the bulb light?
H)
Move one clip closer to the other as shown in figure II. Is there any difference in the behavior of the bulb? If so,
describe what happens.
A Scientific Model for Electric Circuits
The behavior of a circuit is explained in terms of the flow of electric charge through a conducting loop. Charge passing
through the battery gains electrical potential energy (Ue), and delivers that energy to the various elements of the circuit,
such as the bulb (which converts the Ue to heat and light). The charge eventually returns to the battery to obtain more Ue.
The charges are very numerous, and behave as a “conveyor belt” for the Ue. When the loop is broken, the “conveyor
belt” stops.
A)
Look at your battery to determine its rating in Volts. Using the definition of a Volt, write a sentence describing
what is happening to a one-coulomb sample of charge in your complete circuit, according to the description above.
The brightness of the bulb is determined by the rate at which energy UE is converted into light (and heat). This rate of
energy per unit time is power and is measured in J/s, or Watts.
B) Using the pencil lead, you found that you could make the bulb dimmer with the same battery. According to the
model above of an electric circuit, what does it mean when the bulb gets dimmer?
The rate of flow of electric charge is called current. Current measures how much charge passes a point in the circuit per
unit time, and is measured in Coulombs per second (C/s). One Coulomb per second is defined as one ampere after the
French physicist and mathematician André-Marie Ampère.
C) According to the model of an electric circuit, what two factors could be responsible for the bulb getting dimmer?
Circuit diagrams allow us to represent a circuit on paper using
symbols instead of drawing pictures. We use the symbols to the right
for the battery and bulb, and for a circuit consisting of a battery and
bulb, with wires represented by lines. On the battery, the long line
represents the positive terminal and the short block represents the
negative terminal.
Set up a two-bulb circuit with the bulbs connected one after the other as shown to the right.
Bulbs connected one after the other are said to be connected in series.
D)
How does the brightness of one bulb compare to the other?
E)
Switch the order of the bulbs. Does it matter in what order the bulbs are connected?
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Honors Physics
F)
Unit 9: Electricity & Circuits
How does the brightness of these bulbs compare to the brightness of a bulb in a single-bulb circuit?
Set up a two-bulb circuit with the bulbs connected as shown to the right. Bulbs
connected this way are said to be connected in parallel.
G)
How does the brightness of one bulb compare to the other?
H)
Switch the order of the bulbs. Does it matter in what order the bulbs are connected?
I)
How does the brightness of these bulbs compare to the brightness of a bulb in a single-bulb circuit?
J)
How does the current in either of these bulbs compare to the current through the bulb in a single-bulb circuit?
K)
What does this imply about the current through the battery in this parallel circuit compared to either bulb?
An instrument used to measure the current at a point in a circuit is called an
ammeter. An ammeter has an input terminal, usually colored red and labeled “+”,
and an output terminal colored black and labeled “–” as shown to the right.
Two students make the following statements about the two-bulb series circuit at the
top of this page, explaining why the bulbs are dimmer than the bulb in a single-bulb
circuit:
Student 1: “The bulbs are dimmer because when the charges go through the first bulb, they give
up half their energy, then when they go through the second bulb, they give up the rest. So each
bulb only gets half the energy.”
Student 2: “The bulbs are dimmer because the current in this circuit slows down to half the
speed of the current in the single-bulb circuit.”
L)
Explain how one could use an ammeter and/or a voltmeter to test each student’s statement.
M)
Is it possible that both students are correct? Explain.
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Honors Physics
Unit 9: Electricity & Circuits
Emf: Electromotive Force
In an electric circuit, there must be a device that converts non-electric energy into electric energy; this device is
called a source of emf. The letters stand for "electromotive force," which is an old-fashioned and inaccurate term -emf is not a force at all.
Some sources of emf:
• Batteries convert chemical energy into electrical energy
• Generators convert mechanical energy into electrical energy
• Solar cells convert electromagnetic energy into electrical energy
• Thermocouples convert thermal energy into electrical energy
We will mainly be using batteries as our source of emf and instead of using the old fashioned
+ –
term, we will use the term potential as the battery is a source of electric potential ! electrical
energy per unit charge.
Batteries are drawn as shown, with the high potential at the positive, or longer line and the low
V
potential at the shorter, thicker line. For your average 1.5 V battery that means there is a 1.5 V
Source of emf
difference in potential between the bottom and the top of the battery.
Current
We think of charge flowing in a circuit the way water flows in a hose. The rate of flow (gallons per minute in a
hose) is called current. In an electrical circuit current is the rate of flow of electric charge, so it would be measured
in coulombs per second. The letter I is used for current:
I=
Δq
Δt
where Δq is the amount of charge that flows and Δt is the amount of time it takes. Again using units, check the
dimensions of the Ampere:
Direct current (DC) refers to current that flows continuously in the same direction. Alternating current (AC) is
current that reverses direction at regular intervals, this is the electricity we use in our classrooms and homes. We
only consider DC in this course.
Resistance
We have seen that materials differ in their ability to conduct electricity. This is true even of materials we have been
calling conductors. Given a conductor, we can ask how much potential (voltage) is needed to make charge flow at
the rate of 1 Ampere (ie. 1 Coulomb per second). The answer will be different for different conductors. Those for
which it takes a large potential to generate 1 A of current are said to have a high resistance to the flow of charge.
Those that require only a low potential have a low resistance. In fact, we define resistance as the ratio of potential
difference (in Volts) to the current that results (in Amperes). The symbol for resistance is R, and the definition is:
R=
V
I
or, more commonly, V = IR
Resistance is thus measured in Volts per Ampere,
and one Volt per Ampere is defined as one Ohm, named after Georg Ohm.
The symbol for an Ohm is the Greek letter capital omega: Ω. Dimensionally:
1Ohm = 1
Volt
V
or 1Ω = 1
Ampere
A
In circuits, the wires are made of conductors with almost negligible resistance. The conductors in the circuit that
don't have negligible resistance are called resistors, and we symbolize them in circuits with the symbol
.
Many resistors obey Ohm's Law, which states that the resistance of the resistor is a constant, independent of the
particular current or voltage. We will assume that all resistors including light bulbs obey Ohm's Law.
14
Honors Physics
Unit 9: Electricity & Circuits
When charges flow through a resistor, they lose electrical potential energy, which is given off as heat. The same
amount of charge that goes in one end comes out the other (otherwise it would build up in the resistor and it would
explode!). The energy each charge has, however, is less than when it went in.
Example:
An automobile headlight with a resistance of 30 Ω is placed across a 12 V battery. What is the current through the
circuit?
Using the equation
R=
V
V
I=
R so
I we solve for I:
I=
12V
30Ω
I = 0.4 A
Practice Problems:
1. A transistor radio uses 2.0 x 10-4 A of current when it is operated by a 3.0 V battery. What is the resistance of the
radio circuit?
2. An ammeter measures 3.00 A of current in a circuit containing 10.0 Ω of resistance. What is the EMF of the
battery that is the power source of the circuit?
Electrical Power
Recall that power is the rate of consumption of energy or the rate of production of energy. Power is measured in
Joules per second, and one Joule per second is one Watt:
1Watt = 1
Joule
second
When we speak of electrical power in a circuit, we refer either to the rate at which energy is supplied to the circuit
(by the source of emf), or to the rate at which this energy is used or given off (for example, as heat in a resistor).
Looking at the units tells us a lot about electrical power in a circuit. Consider:
1. Write out what a Watt is using units that include one of the following: (ex for b: W = AV since P = IV)
a) Coulombs
b) Amps
c) Volts
d) Ohms
To find the power of a battery (the rate at which it supplies energy to a circuit), we multiply its emf (in Volts) by the
current through it (in Amperes). The result is the power in Watts. Similarly, for a resistor, the rate at which it
dissipates heat is the potential difference across it multiplied by the current through it. In short:
P = IV
Combining the definition of electrical power with Ohm's Law allows us to write two other relationships involving
power:
2. Write out the relationship of power to current and resistance:
3. Write out the relationship of power to voltage and resistance:
15
Honors Physics
Unit 9: Electricity & Circuits
Practice Problems:
1. A 6 V battery delivers a 0.50 A current to an electric motor.
(a) What power is consumed by the motor?
(b) If the motor runs for 5.0 minutes, how much electric energy is used?
2. The current through the starter motor of a car is 210 A. If the battery supplies 12-V to the motor, what electric
energy is supplied to the motor in 10.0 s?
3. A lamp draws a current of 0.50 A when it is connected to a 120-V source.
(a) What is the resistance of the lamp?
(b) What is the power consumption of the lamp?
4. A 75-W bulb is connected to 120-V.
(a) What is the current through the bulb?
(b) What is the resistance of the bulb?
Confused with the new units? See How Stuff Works: What are amps, watts, volts and ohms?
http://www.howstuffworks.com/question501.htm
R
R2
1
Resistors in Series and Parallel
All of I
All of I
Resistors connected in series
Two resistors in a circuit can be connected to each other in one of
two ways. If they are connected in such a way that any current
going through one must go through the other, as in the top diagram
to the right, they are connected in series.
If they are connected in such a way that the current splits, some
going through one resistor and the rest going through the other, as in
the bottom diagram to the right, they are said to be connected in
parallel.
Our next task is to determine what is the equivalent resistance of two
resistors connected each of these two ways.
Some of I
R1
R2
All of I
All of I
The rest of I
Resistors connected in parallel
16
Honors Physics
Unit 9: Electricity & Circuits
The Series Rule
Consider a simple circuit with a battery and two resistors in series, as shown to the
right. By conservation of energy, the energy E given to each Coulomb of charge
q by the battery is equal to the energy dissipated by that same charge as heat in
R1
R2
resistor R1 (E1) plus the energy dissipated in R2 (E2): E = E1 + E2 . Not only
is the energy equal, the energy per unit charge is equal:
E E1 E2
=
+
q
q
q
+
-
But energy per unit charge is emf or potential difference:
By Ohm's Law, V = IR, so
V = V1 + V2
V
IR = IR1 + IR2
where R is the equivalent resistance of the series combination, and the I's are the same since they are in series.
Dividing by I gives the series rule for resistors:
R = R1 + R2
which says simply that if resistors are connected in series, the combined resistance is just the sum of the individual
resistances. The same is true no matter how many resistors are connected. R = R1 + R2 + … Rn
Practice Problems:
1. A 3 Ω resistor is connected to a 6 Ω resistor and a 4 Ω resistor in series. What is the total resistance?
2. You have just one 12 Ω resistor and many 4 Ω resistors and you want a total of 24 Ω of resistance. What would
you do? Prove your method mathematically.
The Parallel Rule
Now consider the circuit to the right. Since the current splits, we know that the current
I through the battery is the sum of the currents through the resistors:
R1
I = I1 + I2
V
I=
R in each case, and the potential difference across each resistor is
By Ohm's Law,
the same as the emf of the battery, since the wire keeps everything it connects to at the
same potential. So:
V V
V
=
+
R R1 R2
R2
+
-
1
1
1
=
+
V
Dividing by V gives the parallel rule for resistors: R R1 R2 , which says that to
find the equivalent resistance of a parallel combination, take the reciprocal of each
resistor, add up the reciprocals, then take the reciprocal of the result. Again, the same holds true for three or more
resistors in parallel.
17
Honors Physics
Unit 9: Electricity & Circuits
Concept Questions:
1. Consider two resistors R1 and R2 with R1 < R2.
A) If R1 and R2 are connected in series, is the total resistance of the combination smaller than R1, larger than R2, or
between R1 and R2? Explain.
B) If R1 and R2 are connected in parallel, is the total resistance of the combination smaller than R1, larger than R2, or
between R1 and R2? Explain.
Ammeters and Voltmeters
An ammeter is an instrument that measures current, and a voltmeter is an
instrument that measures potential difference or voltage. In many ways,
these instruments are opposites of one another.
In the first diagram, an ammeter is inserted into the circuit so that all of
the current to be measured passes through it. The ammeter measures the
current at the point where it is inserted.
2.
Should the ammeter have high or low resistance in order to accurately
measure the current without affecting it? Should it be inserted into the
circuit in series or in parallel?
A
2
V
Potential difference doesn't exist at a point. It exists between two points
as it measures the difference in potential between those two points. In
the second diagram, the probes of a voltmeter are touched to the two
points between which we want to measure the potential difference.
3.
Should the voltmeter have high or low resistance in order to accurately
measure the potential difference without affecting it? Should it be
inserted into the circuit in series or in parallel?
1
1.
Practice Problems:
How much charge is transferred by a current of 2.5 A in one minute?
18
Honors Physics
2.
Unit 9: Electricity & Circuits
A 12 V battery is connected in series with a 3 Ω lamp and a 6 Ω lamp.
A) Sketch the circuit in the space to the right.
B) Find the current in the 6 Ω lamp.
3. A 6 V battery is used to light a 40 Ω lamp. The lamp will overheat if more than 0.1 A of current passes through it.
How could you limit the current in the lamp to 0.1 A?
4.
A 10 Ω electric iron, a 20 Ω toaster, and a 120 Ω lamp are connected in parallel to a 120 V emf.
A) Sketch the circuit in the space at right.
B) What is their combined resistance?
C)
What is the current in each?
D)
What is the current through the battery?
5.
The same electric iron, toaster, and lamp are connected in series with a 120
V emf.
A) Sketch the circuit.
B) What is their combined resistance?
C)
What is the potential difference across each?
19
Honors Physics
6.
Unit 9: Electricity & Circuits
Four 20 Ω resistors are wired in parallel and the combination is connected to a 1.5 V battery.
A) What is the current in each resistor?
B) What is the current through the battery?
2Ω
7. A 4 V battery with negligible internal resistance is connected to three resistors
as shown.
A)
B)
C)
Find the current in the 2 Ω resistor.
Find the potential difference across the 4 Ω resistor.
Find the current in the 6 Ω resistor.
+
4V
4Ω
6Ω
-
8.
15 Ω
The ammeter in the circuit to the right reads 0.3 A.
A) What is the emf of the battery?
B) What is the current in the 60 Ω resistor?
+
20 Ω
60 Ω
-
A
9.
Using the circuit at right, answer these questions.
A) What is the equivalent resistance of R1 + R2?
B) What is the equivalent resistance of R3 + R4?
C) What is the total resistance in this circuit?
D) What is the current through the battery?
20