2. 2
DEVELOPMENT TEAM OF THE MODULE
WRITERS: MARJORIE A. NARIZ, Master Teacher I
MARIEL BARAGENIO, Teacher III
REX S. LAPID, Teacher III
MARHOUF JAY KUSAIN, Teacher I
CONSOLIDATOR: MARIEL BARAGENIO, Teacher III
LANGUAGE EDITOR: MAE CHRISTINE S. DIANA, Master Teacher I
CONTENT EVANGELINE C. BEDRERO
VALIDATORS: CARMEL MACASINAG
CLARICE PASCUAL
COVER PAGE AIRA MARI CON M. AUSTERO
ILLUSTRATOR:
TEAM LEADER: DR. RAQUEL M. AUSTERO
Education Program Supervisor
Republic of the Philippines
Department of Education
N a t io n a l Ca pit a l Reg io n
Sc h o o l s D iv is io n O f f ic e o f Las Piñ a s Cit y
3. 3
Module 5 RESISTIVITY AND OHM’S LAW
Most Essential Learning Competencies
• Describe the effect of temperature increase on the resistance of a metallic conductor
STEM_GP12EM-IIIe-35
• Describe the ability of a material to conduct current in terms of resistivity and conductivity
(STEM_GP12EM-IIIe-36)
• Apply the relationship of the proportionality between resistance and the length and cross -
sectional area of a wire to solve problems (STEM_GP12EM-IIIe-37)
What’s In
RESISTIVITY
When a voltage is applied to a conductor, an electrical field (𝑬
⃗⃗ ) is created, and charges in the
conductor feel a force due to the electrical field. The current density (𝐽) that results depends on the electrical
field and the properties of the material. This dependence can be very complex. In some materials, including
metals at a given temperature, the current density is approximately proportional to the electrical field. In
these cases, the current density can be modeled as
𝐽 = 𝜎𝐸
⃗ ,
where σ is the electrical conductivity. The electrical conductivity is analogous to thermal conductivity and
is a measure of a material’s ability to conduct or transmit electricity. Conductors have a higher electrical
conductivity than insulators. Since the electrical conductivity is 𝜎 = 𝐽/𝐸
⃗ , the units are
𝜎 =
|𝐽|
|𝐸|
=
𝐴/𝑚2
𝑉/𝑚
=
𝐴
𝑉⋅𝑚
.
Here, we define a unit named the ohm with the Greek symbol uppercase omega, 𝛺. The unit is
named after Georg Simon Ohm. One ohm equals one volt per amp: 1𝛺 = 1𝑉/𝐴. The units of electrical
conductivity are therefore (𝛺 ⋅ 𝑚)−1
.
Conductivity is an intrinsic property of a material. Another intrinsic property of a material is
the resistivity, or electrical resistivity. The resistivity of a material is a measure of how strongly a material
opposes the flow of electrical current. The symbol for resistivity is the lowercase Greek letter rho, ρ, and
resistivity is the reciprocal of electrical conductivity:
𝜌 =
1
𝜎
.
The unit of resistivity in SI units is the ohm-meter (𝛺 ⋅ 𝑚). We can define the resistivity in terms of
the electrical field and the current density.
𝜌 =
𝐸
𝐽
.
The greater the resistivity, the larger the field needed to produce a given current density. The lower
the resistivity, the larger the current density produced by a given electrical field. Good conductors have a
high conductivity and low resistivity. Good insulators have a low conductivity and a high resistivity.
Table 1 lists resistivity and conductivity values for various materials.
Temperature Dependence of Resistivity
Looking at Table 1, you will see a column labeled “Temperature Coefficient.” The resistivity of some
materials has a strong temperature dependence. In some materials, such as copper, the resistivity
increases with increasing temperature. In fact, in most conducting metals, the resistivity increases with
increasing temperature. The increasing temperature causes increased vibrations of the atoms in the lattice
structure of the metals, which impede the motion of the electrons. In other materials, such as carbon, the
resistivity decreases with increasing temperature. In many
materials, the dependence is approximately linear and can be modeled using a linear equation:
𝜌 ≈ 𝜌0[1 + 𝛼(𝑇 − 𝑇0)] ,
4. 4
where 𝜌 is the resistivity of the material at temperature T, 𝛼 is the temperature coefficient of the material,
and 𝝆𝟎 is the resistivity at 𝑻𝟎, usually taken as 𝑇0 = 20.00℃
Note also that the temperature coefficient 𝛼 is negative for the semiconductors listed in Table 1,
meaning that their resistivity decreases with increasing temperature. They become better conductors at
higher temperature, because increased thermal agitation increases the number of free charges available
to carry current. This property of decreasing 𝜌 with temperature is also related to the type and amount of
impurities present in the semiconductors.
Table 1: Resistivities and Conductivities of Various Materials at 20 °C
Material Conductivity, σ
(Ω⋅m)−1
Resistivity, ρ
(Ω⋅m)
Temperature
Coefficient α
(o
C)−1
Conductors
Silver 6.29×107
1.59×10−8
0.0038
Copper 5.95×107
1.68×10−8
0.0039
Gold 4.10×107
2.44×10−8
0.0034
Aluminum 3.77×107
2.65×10−8
0.0039
Tungsten 1.79×107
5.60×10−8
0.0045
Iron 1.03×107
9.71×10−8
0.0065
Platinum 0.94×107
10.60×10−8
0.0039
Steel 0.50×107
20.00×10−8
Lead 0.45×107
22.00×10−8
Manganin
(Cu, Mn. Ni
alloy)
0.21×107
48.20×10−8
0.000002
Constantan
(Cu, Ni
alloy)
0.20×107
49.00×10−8
0.00003
Mercury 0.10×107
98.00×10−8
0.0009
Nichrome
(Ni, Fe, Cr
alloy)
0.10×107
100.00×10−8
0.0004
RESISTANCE
We now consider the resistance of a wire or component.
The resistance is a measure of how difficult it is to pass current
through a wire or component. Resistance depends on the
resistivity. The resistivity is a characteristic of the material used
to fabricate a wire or other electrical component, whereas the
resistance is a characteristic of the wire or component.
To calculate the resistance, consider a section of
conducting wire with cross-sectional area (A), length (L), and
resistivity 𝜌 as shown in Figure 1. A battery is connected across
the conductor, providing a potential difference (ΔV) across it.
The potential difference produces an electrical field that is
proportional to the current density, according to 𝐸
⃗ = 𝜌𝐽.
The magnitude of the electrical field across the segment of the conductor is equal to the voltage
divided by the length, E=V/L, and the magnitude of the current density is equal to the current divided by the
cross-sectional area, J=I/A. Using this information and recalling that the electrical field is proportional to the
resistivity and the current density, we can see that the voltage is proportional to the current:
Material Conductivity, σ
(Ω⋅m)−1
Resistivity, ρ
(Ω⋅m)
Temperature
Coefficient α
(o
C)−1
Semiconductors
Carbon
(pure)
2.86×104
3.50×10−5
-0.0005
Carbon (2.86−1.67)×10−6
(3.5−60)×10−5
-0.0005
Germanium
(pure)
600×10−3
-0.048
Germanium (1−600)×10−3
-0.050
Silicon
(pure)
2300 -0.075
Silicon 0.1 – 2300 -0.07
Insulators
Amber 2.00×10−15
5×1014
Glass 10−9
−10−14
109
−1014
Lucite <10−13
>1013
Mica 10−11
−10−15
1011
−1015
Quartz
(fused)
1.33×10−18
75×1016
Rubber
(hard)
10−13
−10−16
1013
−1016
Sulfur 10−15
1015
TeflonTM
<10−13
>1013
Wood 10−8
−10−11
108
−1011
Figure 1: A potential provided by a battery is applied
to a segment of a conductor with a cross-sectional
area A and a length L.
5. 5
𝐸 = 𝜌𝐽
𝑉
𝐿
= 𝜌
𝐼
𝐴
𝑉 = (𝜌
𝐿
𝐴
)𝐼
The ratio of the voltage to the current is defined as the resistance R:
𝑅 ≡ 𝑉𝐼.
The resistance of a cylindrical segment of a conductor is equal to the resistivity of the material times
the length divided by the area:
𝑅 ≡
𝑉
𝐼
= 𝜌
𝐿
𝐴
.
The unit of resistance is the ohm, 𝛺. For a given voltage, the higher the resistance, the lower the current.
Resistors
A common component in electronic circuits is the resistor. The resistor can be used to reduce current flow
or provide a voltage drop. Figure 2 shows the symbols used for a resistor in schematic diagrams of a circuit.
Two commonly used standards for circuit diagrams are provided by the American National Standard
Institute (ANSI) and the International Electrotechnical Commission (IEC). Both systems are commonly
used.
Figure 2: Symbols for a resistor used in circuit diagrams. (a) The ANSI symbol; (b) the IEC symbol.
Material and Shape Dependence of Resistance
A resistor can be modeled as a cylinder with a cross-sectional area (A) and a length (L), made of a material
with a resistivity ρ as shown in Figure 3. The resistance of the resistor is 𝑅 =𝜌
𝐿
𝐴
.
The most common material used to
make a resistor is carbon. A carbon track is
wrapped around a ceramic core, and two
copper leads are attached. A second type of
resistor is the metal film resistor, which also
has a ceramic core. The track is made from a
metal oxide material, which has semi
conductive properties similar to carbon.
Again, copper leads are inserted into the
ends of the resistor. The resistor is then
painted and marked for identification. A
resistor has four colored bands, as shown in
Figure 4.
Resistances range over many orders of magnitude. Some ceramic insulators, such as those used
to support power lines, have resistances of 1012
Ω or more. A dry person may have a hand-to-foot resistance
of 105
Ω whereas the resistance of the human heart is about 103
Ω A meter-long piece of large-diameter
copper wire may have a resistance of 10−5
Ω, and superconductors have no resistance at all at low
temperatures. As we have seen, resistance is related to the shape of an object and the material of which it
is composed. The resistance of an object also depends on temperature, since R0 is directly proportional
to ρ. For a cylinder, we know 𝑅 =𝜌
𝐿
𝐴
, so if L and A do not change greatly with temperature, R has the same
temperature dependence as ρ. Examination of the coefficients of linear expansion shows them to be
Figure 3: A model of a resistor as a uniform cylinder of length (L) and
cross-sectional area (A). Its resistance to the flow of current is
analogous to the resistance posed by a pipe to fluid flow. The longer the
cylinder, the greater its resistance. The larger its cross-sectional area ,
the smaller its resistance.
Figure 4: Resistor Color Guide
6. 6
about two orders of magnitude less than typical temperature coefficients of resistivity, so the effect of
temperature on L and A is about two orders of magnitude less than on ρ. Thus,
𝑅 = 𝑅0(1 + 𝛼𝛥𝑇)
is the temperature dependence of the resistance of an object, where 𝑅0 is the original resistance (usually
taken to be 𝑇 = 20.00℃) and R is the resistance after a temperature change ΔT. The color code gives the
resistance of the resistor at a temperature of 𝑇 = 20.00℃.
CIRCUIT DIAGRAM
The circuit diagram is a kind of graphical representation of an electrical circuit, showing how
electrical components are connected together. It is usually used by engineers and electricians to explain
elements and paths of an electrical circuit, which is important in design, construction, and maintenance of
electrical and electronic equipment. Circuit diagrams use international standard symbols to provide
schematic diagrams of the circuit and its components. Each symbol is intended to represent some features
of the physical construction of the device.
OHM’S LAW
The current that flows through most substances is directly proportional to the voltage V applied to
it. The German physicist Georg Simon Ohm (1787–1854) was the first to demonstrate experimentally that
the current in a metal wire is directly proportional to the voltage applied:
I∝V.
This important relationship is the basis for Ohm’s law. It can be viewed as a cause-and-effect
relationship, with voltage the cause and current the effect. This is an empirical law, which is to say that it is
an experimentally observed phenomenon, like friction. Such a linear relationship doesn’t always occur. Any
material, component, or device that obeys Ohm’s law, where the current through the device is proportional
to the voltage applied, is known as an ohmic material or ohmic component. Any material or component
that does not obey Ohm’s law is known as a nonohmic material or nonohmic component.
Ohm’s Experiment
In a paper published in 1827, Georg Ohm described an experiment in which he measured voltage across
and current through various simple electrical circuits containing various lengths of wire. A similar experiment
is shown in Figure 5. This experiment is used to observe the current through a resistor that results from an
applied voltage. In this simple circuit, a resistor is connected in series with a battery. The voltage is
measured with a voltmeter, which must be placed across the resistor (in parallel with the resistor). The
current is measured with an ammeter, which must be in line with the resistor (in series with the resistor).
Figure 6: A resistor is placed in a circuit with a battery. The voltage applied varies
from −10.00 V to +10.00 V, increased by 1.00-V increments. A plot shows values of
the voltage versus the current typical of what a casual experimenter might find.
Figure 5: The experimental set-up used to determine if a
resistor is an ohmic or nonohmic device. (a) When the battery
is attached, the current flows in the clockwise direction and the
voltmeter and ammeter have positive readings. (b) When the
leads of the battery are switched, the current flows in the
counterclockwise direction and the voltmeter and ammeter
have negative readings.
7. 7
In this updated version of Ohm’s original experiment, several measurements of the current were
made for several different voltages. When the battery was hooked up as in Figure 5a, the current flowed in
the clockwise direction and the readings of the voltmeter and ammeter were positive. Does the behavior of
the current change if the current flowed in the opposite direction? To get the current to flow in the opposite
direction, the leads of the battery can be switched. When the leads of the battery were switched, the
readings of the voltmeter and ammeter readings were negative because the current flowed in the opposite
direction, in this case, counterclockwise. Results of a similar experiment are shown in Figure 6.
In this experiment, the voltage applied across the resistor varies from −10.00 to +10.00 V, by
increments of 1.00 V. The current through the resistor and the voltage across the resistor are measured. A
plot is made of the voltage versus the current, and the result is approximately linear. The slope of the line
is the resistance, or the voltage divided by the current. This result is known as Ohm’s law:
V=IR
where V is the voltage measured in volts across the object in question, I is the current measured through
the object in amps, and R is the resistance in units of ohms. As stated previously, any device that shows a
linear relationship between the voltage and the current is known as an ohmic device. A resistor is therefore
an ohmic device.
Nonohmic devices do not exhibit a linear relationship between the voltage and the current. One
such device is the semiconducting circuit element known as a diode. A diode is a circuit device that allows
current flow in only one direction. A diagram of a simple circuit consisting of a battery, a diode, and a resistor
is shown in Figure 7. Although we do not cover the theory of the diode in this section, the diode can be
tested to see if it is an ohmic or a nonohmic device.
A plot of current versus voltage is shown
in Figure 8. Note that the behavior of the diode is
shown as current versus voltage, whereas the
resistor operation was shown as voltage versus
current. A diode consists of an anode and a
cathode. When the anode is at a negative
potential and the cathode is at a positive
potential, as shown in part (a), the diode is said
to have reverse bias. With reverse bias, the
diode has an extremely large resistance and
there is very little current flow—essentially zero
current—through the diode and the resistor. As
the voltage applied to the circuit increases, the
current remains essentially zero, until the voltage
reaches the breakdown voltage and the diode conducts current. When the battery and the potential across
the diode are reversed, making the anode positive and the cathode negative, the diode conducts and
current flows through the diode if the voltage is greater than 0.7 V. The resistance of the diode is close to
zero. (This is the reason for the resistor in the circuit; if it were not there, the current would become very
large.) You can see from the graph in Figure 8 that the voltage and the current do not have a linear
relationship. Thus, the diode is an example of a nonohmic device.
Figure 8: When the voltage across the diode is negative and
small, there is very little current flow through the diode. As the
voltage reaches the breakdown voltaAge, the diode conducts.
When the voltage across the diode is positive and greater than
0.7 V (the actual voltage value depends on the diode), the
diode conducts. As the voltage applied increases, the current
through the diode increases, but the voltage across the diode
remains approximately 0.7 V.
Figure 7: A diode is a semiconducting device that allows current
flow only if the diode is forward biased, which means that the
anode is positive and the cathode is negative.
8. 8
Ohm’s law is commonly stated as V=IR, but originally it was stated as a microscopic view, in terms
of the current density, the conductivity, and the electrical field. This microscopic view suggests the
proportionality V∝IV∝I comes from the drift velocity of the free electrons in the metal that results from an
applied electrical field. As stated earlier, the current density is proportional to the applied electrical field.
The reformulation of Ohm’s law is credited to Gustav Kirchhoff, whose name we will see again in the next
chapter.
What’s More
Activity 1: Resistivity and Ohm’s Law Problem Solving
Directions: Solve the following problems and show all pertinent solution.
1. What is the resistance of a 20.0-m-long piece of 12-gauge copper wire having a 2.053-mm
diameter?
2. If the 0.100-mm diameter tungsten filament in a light bulb is to have a resistance of 0.200 Ω at
20ºC, how long should it be?
3. What current flows through a 2.54-cm-diameter rod of pure silicon that is 20.0 cm long, when
1.00 × 103
V is applied to it?
4. An alarm clock draws 0.5 A of current when connected to a 120 volt circuit. Calculate its resistance.
5. What current flows through a hair dryer plugged into a 120 Volt circuit if it has a resistance of 25
ohms?
6. What is the applied voltage of a telephone circuit that draws 0.017amperes through a resistance of
15,000ohms?
Activity 2: Resistor Color Code
A. Using the international color code above, give the value of the following resistors given the colors.
Use the tolerance: Gold ±5%, Silver ±10% and None ±20%.
1. brown, black, orange - ________________
2. blue, gray, brown - ________________
3. yellow, violet, yellow - ________________
4. brown, black, green - ________________
5. red, red, red - ________________
B. Give the color code of the following resistors
1. 1,500 ohm - ________________
2. 1,000,000 ohm - ________________
3. 330 ohm - ________________
4. 2 200 ohm - ________________
5. 1000 ohm - ________________
What I Have Learned
Directions: Read each statement below carefully. Place a T on the line if you think a statement it
TRUE. Place an F on the line if you think the statement is FALSE.
_____1. Electrons are the mobile charge carriers in an electric circuit.
_____2.The resistance of an electric circuit is a measure of the overall amount of hindrance to the flow
of charge through the circuit.
_____3. Electric current is measured in units of Amperes.
9. 11
_____4. A 10-ohm resistor would allow a current of 2 Amperes when 5 Volts is impressed across it.
_____5. The resistance of a conducting wire will increase as the cross-sectional area of the wire is
increased.
_____6. A threefold increase in the resistance of an electric circuit will result in a threefold decrease in
the electric current.
_____7.The electric current in a circuit will triple in value as the electric potential impressed across a
circuit is increased by a factor of three.
_____8.Wider conducting wires are capable of carrying larger currents.
_____9.Suppose a miniature light bulb is connected to a battery in a circuit. A light bulb with a greater
resistance will have a greater current.
_____10.Electric current is equal to the number of Coulombs of charge which move past a point on a
circuit per unit of time
What I Can Do
Search and visit the Ohm’s Law PhET Simulation. Predict how current will change when
resistance of the circuit is fixed and voltage is varied. Then, predict how current will change when voltage
of the circuit is fixed and resistance is varied.
Module 6 RESISTOR IN SERIES AND PARALLEL CONNECTION
Most Essential Learning Competencies
• Evaluate the equivalent resistance, current, and voltage in a given network of resistors connected
in series and/or parallel (STEM_GP12EMIIIg-48)
• Calculate the current and voltage through and across circuit elements using Kirchhoff’s loop and
• junction rules (at most 2 loops only) (STEM_GP12EMIIIg-49)
• Solve problems involving the calculation of currents and potential difference in circuits consisting
of batteries, resistors and capacitors. (STEM_GP12EMIIIg-51)
• Differentiate electric interactions from magnetic interactions (STEM_GP12EMIIIh-54)
• Evaluate the total magnetic flux through an open surface (STEM_GP12EMIIIh-55)
• Describe the motion of a charged particle in a magnetic field in terms of its speed, acceleration,
cyclotron radius, cyclotron frequency, and kinetic energy (STEM_GP12EMIIIh-58)
• Evaluate the magnetic force on an arbitrary wire segment placed in a uniform magnetic field
(STEM_GP12EMIIIh-59)
What’s In
RESISTORS IN SERIES AND PARALLEL CONNECTION
Most circuits have more than one component, called a resistor, that limits the flow of charge in the
circuit. A measure of this limit on charge flow is called resistance. The simplest combinations of resistors
are the series and parallel connections. The total resistance of a combination of resistors depends on
both their individual values and how they are connected.
10. 12
Resistors in Series
Resistors are in series whenever the flow of charge, or the current, must flow through
components sequentially.
Figure 9: Resistors in series connection
The total resistance in the circuit is equal to the sum of the individual resistances, since the
current has to pass through each resistor in sequence through the circuit.
Using Ohm ‘s Law to Calculate Voltage Changes
in Resistors in Series
According to Ohm’s law, the voltage drop, V,
across a resistor when a current flows through it is
calculated by using the equation V=IR, where I is
current in amps (A) and R is the resistance in ohms
(Ω). So the voltage drop across R1 is V1 = I1R1,
across R2 is V2 = I2R2, and across R3 is V3 = I3R3.
The sum of the voltages would equal: VT = V1 + V2 +
V3, based on the conservation of energy and
charge. If we substitute the values for individual
voltages, we get:
VT = I1R1 + I2R2 + I3R3 ; IT = I1 = I2 = I3 = I
so
VT = I(R1+R2+R3)
This implies that the total resistance in a series is equal to the sum of the individual resistances.
RT = R1 + R2 + R3 +…+RN.
Since all of the current must pass through each resistor, it experiences the resistance of each, and
resistances in series simply add up. Since voltage and resistance have an inverse relationship, individual
resistors in series do not get the total source voltage, but divide it. This is indicated in an example of
when two light bulbs are connected together in a series circuit with a battery. In a simple circuit consisting
of one 1.5V battery and one light bulb, the light bulb would have a voltage drop of 1.5V across it. If two
lightbulbs were connected in series with the same battery, however, they would each have 1.5V/2, or
0.75V drop across them. This would be evident in the brightness of the lights: each of the two light bulbs
connected in series would be half as dim as the single light bulb. Therefore, resistors connected in series
use up the same amount of energy as a single resistor, but that energy is divided up between the
resistors depending on their resistances.
Resistors in Parallel
Resistors are in parallel when each resistor is connected
directly to the voltage source by connecting wires having negligible
resistance. Each resistor thus has the full voltage of the source
applied to it.
Each resistor draws the same current it would if it were the
only resistor connected to the voltage source. This is true of the
circuitry in a house or apartment. Each outlet that is connected to
an appliance (the “resistor”) can operate independently, and the
current does not have to pass through each appliance
sequentially.
Figure 10: Resistors connected in a series circuit:
Three resistors connected in series to a battery (left)
and the equivalent single or series resistance (right).
Figure 11: Resistors in parallel connection
11. 13
Ohm ‘s Law and Parallel Resistors
Each resistor in the circuit has the full voltage.
According to Ohm’s law, the currents flowing
through the individual resistors are 𝐼1 =
𝑉/𝑅1, 𝐼2 = 𝑉/𝑅2, and 𝐼3 = 𝑉/𝑅3.
Conservation of charge implies that the total
current is the sum of these currents:
𝐼𝑇 = 𝐼1 + 𝐼2 + 𝐼3
Substituting the expressions for individual
currents gives:
𝐼𝑇 =
𝑉
𝑅1
+
𝑉
𝑅2
+
𝑉
𝑅3
𝑜𝑟 𝐼𝑇 = 𝑉 (
1
𝑅1
+
1
𝑅2
+
1
𝑅3
)
This implies that the total resistance in a parallel circuit is equal to the sum of the inverse of each
individual resistance.
𝑅𝑇 =
1
𝑅1
+
1
𝑅2
+
1
𝑅3
+ ⋯ + R𝑁
This relationship results in a total resistance that is less than the smallest of the individual
resistances. When resistors are connected in parallel, more current flows from the source than would
flow for any of them individually, so the total resistance is lower.
Each resistor in parallel has the same full voltage of the source applied to it, but divide the total
current amongst them. This is exemplified by connecting two light bulbs in a parallel circuit with a 1.5V
battery. In a series circuit, the two light bulbs would be half as dim when connected to a single battery
source. However, if the two light bulbs were connected in parallel, they would be equally as bright as if
they were connected individually to the battery. Because the same full voltage is being applied to both
light bulbs, the battery would also die more quickly, since it is essentially supplying full energy to both
light bulbs. In a series circuit, the battery would last just as long as it would with a single light bulb, only
the brightness is then divided amongst the bulbs.
KIRCHOFF’S LAW
We have just seen that some circuits may be analyzed by
reducing a circuit to a single voltage source and an equivalent
resistance. Many complex circuits cannot be analyzed with the series-
parallel techniques developed in the preceding sections. In this
section, we elaborate on the use of Kirchhoff’s rules to analyze more
complex circuits. For example, the circuit in Figure 5 is known as
a multi-loop circuit, which consists of junctions. A junction, also known
as a node, is a connection of three or more wires. In this circuit, the
previous methods cannot be used, because not all the resistors are in
clear series or parallel configurations that can be reduced. Give it a
try. The resistors R1 and R2 are in series and can be reduced to an
equivalent resistance. The same is true of resistors R4 and R5. But
what do you do then?
Even though this circuit cannot be analyzed using the methods
already learned, two circuit analysis rules can be used to analyze any
circuit, simple or complex. The rules are known as Kirchhoff’s rules,
after their inventor Gustav Kirchhoff (1824–1887).
Figure 12: Resistors connected in a parallel circuit: Three
resistors connected in parallel to a battery (left) and the
equivalent single or parallel resistance (right).
Figure 13: This circuit cannot be
reduced to a combination of series
and parallel connections. However,
we can use Kirchhoff’s rules to
analyze it.
12. 14
Kirchhoff’s First Rule
Kirchhoff’s first rule (the junction rule) applies to the charge entering
and leaving a junction (Figure 6). As stated earlier, a junction, or node, is
a connection of three or more wires. Current is the flow of charge, and
charge is conserved; thus, whatever charge flows into the junction must
flow out.
Although it is an over-simplification, an analogy can be made with
water pipes connected in a plumbing junction. If the wires in Figure 6 were
replaced by water pipes, and the water was assumed to be incompressible,
the volume of water flowing into the junction must equal the volume of
water flowing out of the junction.
Kirchhoff’s Second Rule
Kirchhoff’s second rule (the loop rule) applies to potential differences.
The loop rule is stated in terms of potential V rather than potential energy,
but the two are related since U=qV. In a closed loop, whatever energy is
supplied by a voltage source, the energy must be transferred into other
forms by the devices in the loop, since there are no other ways in which
energy can be transferred into or out of the circuit. Kirchhoff’s loop rule
states that the algebraic sum of potential differences, including voltage
supplied by the voltage sources and resistive elements, in any loop must
be equal to zero. For example, consider a simple loop with no junctions, as
in Figure 7.
The circuit consists of a voltage source and three external load
resistors. The labels a, b, c, and d serve as references, and have no other
significance. The usefulness of these labels will become apparent soon.
The loop is designated as Loop abcda, and the labels help keep track of the voltage differences as we
travel around the circuit. Start at point a and travel to point b. The voltage of the voltage source is added
to the equation and the potential drop of the resistor R1 is subtracted. From point b to c, the potential
drop across R2 is subtracted. From c to d, the potential drop across R3 is subtracted. From
points d to a, nothing is done because there are no components.
Then Kirchhoff’s loop rule states
V − IR1 − IR2 − IR3 = 0.
The loop equation can be used to find the current through the loop:
𝐼 =
𝑉
𝑅1 + 𝑅2 + 𝑅3
=
12.00𝑉
1.00𝛺 + 2.00𝛺 + 3.00𝛺
= 2.00𝐴.
This loop could have been analyzed using the previous methods, but we will demonstrate the power of
Kirchhoff’s method in the next section.
Calculating Current by Using Kirchhoff’s Rules
Find the currents flowing in the circuit in Figure 9.
Strategy: This circuit is sufficiently complex that the currents
cannot be found using Ohm’s law and the series-parallel
techniques—it is necessary to use Kirchhoff’s rules. Currents
have been labeled I1, I2, and I3 in the figure, and assumptions
have been made about their directions. Locations on the
diagram have been labeled with letters a through h. In the
solution, we apply the junction and loop rules, seeking three
independent equations to allow us to solve for the three
unknown currents.
Figure 14: Charge must be conserved, so the
sum of currents into a junction must be equal
to the sum of currents out of the junction.
Figure 15: A simple loop with no junctions.
Kirchhoff’s loop rule states that the
algebraic sum of the voltage differences is
equal to zero.
𝑅3 = 3.00𝛺
𝑅2 = 2.00𝛺
Figure 16: This circuit is combination of series and
parallel configurations of resistors and voltage sources.
13. 15
Solution: Applying the junction and loop rules yields the following three equations. We have three
unknowns, so three equations are required.
Junction c: I1+I2=I3
Loop abcdefa: I1(R1 + R4) − I2(R2 + R5 + R6) = V1 − V3
Loop cdefc: I2(R2 + R5 + R6) + I3R3 = V2 + V3
Simplify the equations by placing the unknowns on one side of the equations.
Junction c: I1 + I2 − I3 = 0
Loop abcdefa: I1(3Ω) − I2(8Ω) = 0.5V − 2.30V
Loop cdefc: I2(8Ω) + I3(1Ω) = 0.6V + 2.30V
Simplify the equations. The first loop equation can be simplified by dividing both sides by 3.00. The
second loop equation can be simplified by dividing both sides by 6.00.
Junction c: I1 + I2 − I3 = 0.
Loop abcdefa: I1(3Ω) − I2(8Ω) = −1.8V.
Loop cdefc: I2(8Ω) + I3(1Ω) = 2.9V.
The results are: I1 = 0.20A, I2 = 0.30A, I3 = 0.50A.
MAGNETISM
All magnets attract iron, such as that in a refrigerator door. However, magnets may attract or repel
other magnets. Experimentation shows that all magnets have two poles. If freely suspended, one pole
will point toward the north. The two poles are thus named the north magnetic pole and the south
magnetic pole (or more properly, north-seeking and south-seeking poles, for the attractions in those
directions).
Universal Characteristics of Magnets and Magnet Poles
• It is a universal characteristic of all magnets that like poles repel and unlike poles attract. (Note
the similarity with electrostatics: unlike charges attract and like charges repel.)
• Further experimentation shows that it is impossible to separate north and south poles in the
manner that + and − charges can be separated.
Misconception Alert: Earth's Geographic North Pole Hides an S
The Earth acts like a very large bar magnet with its south-seeking pole
near the geographic North Pole. That is why the north pole of your
compass is attracted toward the geographic north pole of the Earth—
because the magnetic pole that is near the geographic North Pole is
actually a south magnetic pole! Confusion arises because the geographic
term “North Pole” has come to be used (incorrectly) for the magnetic pole
that is near the North Pole. Thus, “North magnetic pole” is actually a
misnomer—it should be called the South magnetic pole.
Magnetic Field Lines
Since magnetic forces act at a distance, we define a magnetic field to represent magnetic forces.
The pictorial representation of magnetic field lines is very useful in visualizing the strength and direction
of the magnetic field. As shown in Figure 11, the direction of magnetic field lines is defined to be the
direction in which the north end of a compass needle points. The magnetic field is traditionally called
the B - field.
Figure 18: Magnetic field lines are defined to have the direction that a small compass points when placed at a location. (a) If small
compasses are used to map the magnetic field around a bar magnet, they will point in the directions shown: away from the north
Figure 17: Earth as a magnet
14. 16
pole of the magnet, toward the south pole of the magnet. (Recall that the Earth’s north magnetic pole is really a south pole in
terms of definitions of poles on a bar magnet.) (b) Connecting the arrows gives continuous magnetic field lines. The strength of
the field is proportional to the closeness (or density) of the lines. (c) If the interior of the magnet could be probed, the field lines
would be found to form continuous closed loops.
Small compasses used to test a magnetic field will not disturb it. (This is analogous to the way we
tested electric fields with a small test charge. In both cases, the fields represent only the object creating
them and not the probe testing them.) Figure 11 shows how the magnetic field appears for a current
loop and a long straight wire, as could be explored with small compasses. A small compass placed in
these fields will align itself parallel to the field line at its location, with its north pole pointing in the direction
of B. Note the symbols used for field into and out of the paper.
Figure 19: Small compasses could be used to map the fields shown here. (a) The magnetic field of a circular current loop is similar
to that of a bar magnet. (b) A long and straight wire creates a field with magnetic field lines forming circular loops. (c) When the
wire is in the plane of the paper, the field is perpendicular to the paper. Note that the symbols used for the field pointing inward
(like the tail of an arrow) and the field pointing outward (like the tip of an arrow).
Extensive exploration of magnetic fields has revealed a number of hard-and-fast rules. We use
magnetic field lines to represent the field (the lines are a pictorial tool, not a physical entity in and of
themselves). The properties of magnetic field lines can be summarized by these rules:
1. The direction of the magnetic field is tangent to the field line at any point in space. A small
compass will point in the direction of the field line.
2. The strength of the field is proportional to the closeness of the lines. It is exactly proportional
to the number of lines per unit area perpendicular to the lines (called the areal density).
3. Magnetic field lines can never cross, meaning that the field is unique at any point in space.
4. Magnetic field lines are continuous, forming closed loops without beginning or end. They go
from the north pole to the south pole.
The last property is related to the fact that the north and south poles cannot be separated. It is a
distinct difference from electric field lines, which begin and end on the positive and negative charges. If
magnetic monopoles existed, then magnetic field lines would begin and end on them.
Magnetic Field Strength- Force on a Moving Charge in a Magnetic Field
Magnetic fields exert forces on moving charges, and so they exert forces on other magnets, all of
which have moving charges.
Right Hand Rule
The magnetic force on a moving charge is one of the most fundamental known. Magnetic force is
as important as the electrostatic or Coulomb force. Yet the magnetic force is more complex, in both the
number of factors that affects it and in its direction, than the relatively simple Coulomb force. The
magnitude of the magnetic force 𝑭 on a charge q moving at a speed 𝒗 in a magnetic field of
strength 𝑩 is given by
𝐹 = 𝑞𝑣𝐵𝑠𝑖𝑛𝜃
where θ is the angle between the directions of 𝑣 and 𝐵. This force is often called the Lorentz force. In
fact, this is how we define the magnetic field strength 𝐵 - in in terms of the force on a charged particle
moving in a magnetic field. The SI unit for magnetic field strength 𝐵 is called the tesla (T) after the
eccentric but brilliant inventor Nikola Tesla (1856–1943). To determine how the tesla relates to other SI
units, we solve
𝐵 = 𝐹𝑞𝑣𝑠𝑖𝑛𝜃
Because 𝑠𝑖𝑛𝜃 is unitless, the tesla is
1𝑇 =
1𝑁
𝐶 ⋅ 𝑚/𝑠
=
1𝑁
𝐴 ⋅ 𝑚
15. 17
Another smaller unit, called the gauss (G), where 1𝐺 = 10−4
𝑇, is
sometimes used. The strongest permanent magnets have fields near 2 T;
superconducting electromagnets may attain 10 T or more. The Earth’s
magnetic field on its surface is only about 5 × 10−5
𝑇, or 0.5 G.
The direction of the magnetic force F is perpendicular to the plane
formed by 𝑣 and 𝐵, as determined by the right hand rule (RHR), which is
illustrated in Figure 13. RHR states that, to determine the direction of the
magnetic force on a positive moving charge, you point the thumb of the right
hand in the direction of 𝑣, the fingers in the direction of 𝐵, and a perpendicular
to the palm points in the direction of F. One way to remember this is that there
is one velocity, and so the thumb represents it. There are many field lines,
and so the fingers represent them. The force is in the direction you would
push with your palm. The force on a negative charge is in exactly the opposite
direction to that on a positive charge.
Circular Motion of a Charged particle in a Magnetic Field
Magnetic force is always perpendicular to velocity, so that it does no work on the charged particle. The
particle’s kinetic energy and speed thus remain constant. The direction of motion is affected, but not the
speed. This is typical of uniform circular motion. The simplest case occurs when a charged particle
moves perpendicular to a uniform B-field, such as shown in. (If this takes place in a vacuum, the
magnetic field is the dominant factor determining the motion.) Here, the magnetic force (Lorentz force)
supplies the centripetal force
𝐹𝑐 =
𝑚𝑣2
𝑟
.
Noting that
𝑠𝑖𝑛𝜃 = 1
we see that
𝐹 = 𝑞𝑣𝐵.
The Lorentz magnetic force supplies the centripetal force, so these terms are equal:
𝑞𝑣𝐵 =
𝑚𝑣2
𝑟
solving for r yields
𝑟 =
𝑚𝑣
𝑞𝐵
Here, r, called the gyroradius or cyclotron radius, is the radius of curvature of the path of a
charged particle with mass m and charge q, moving at a speed v perpendicular to a magnetic field of
strength B. In other words, it is the radius of the circular motion of a charged particle in the presence of
a uniform magnetic field. If the velocity is not perpendicular to the magnetic field, then v is the component
of the velocity perpendicular to the field. The component of the velocity parallel to the field is unaffected,
since the magnetic force is zero for motion parallel to the field.
Figure 20: Right Hand Rule
Figure 21: Circular Motion of Charged Particle
in Magnetic Field
A negatively charged particle moves in the
plane of the page in a region where the
magnetic field is perpendicular into the page
(represented by the small circles with x’s—like
the tails of arrows). The magnetic force is
perpendicular to the velocity, and so velocity
changes in direction but not magnitude.
Uniform circular motion results.
16. 18
A particle experiencing circular motion due to a uniform magnetic field is termed to be in
a cyclotron resonance. The term comes from the name of a cyclic particle accelerator called a cyclotron,
showed in. The cyclotron frequency (or, equivalently, gyrofrequency) is the number of cycles a particle
completes around its circular circuit every second and can be found by solving for v above and
substituting in the circulation frequency so that
𝑓 =
𝑣
2𝜋𝑟
becomes
𝑓 =
𝑞𝐵
2𝜋𝑚
The cyclotron frequency is trivially given in radians per second by
𝜔 =
𝑞𝐵
𝑚
What’s More
Activity 1. Resistors and Kirchoff’s Law
A. Solve the following problems.
B. Find all the currents and voltages across each resistor and cell in the following circuits.
Activity 2. Kirchoff’s Law
Directions: The following diagrams show a charged particle or a current carrying wire in a magnetic
field. For each diagram use the right-hand rule to draw an arrow on the object that shows the direction
of the magnetic force. Remember that a ⊗ means the direction is into the page and a • means the
direction is out of the page towards you.
17. 19
What I Have Learned
The Right Hand Rule determines the direction of the magnetic field around a current carrying
wire and vice-versa Using your right-hand:
1. Point your thumb in the direction of the conventional _______________.
2. Curl your fingers into a half-circle around the wire, they point in the direction of the
_______________
Also, it can use to determine the directions of magnetic force, conventional current and the
magnetic field. Given any two of these, the third can be found using your right-hand:
1. Point your _______________ in the direction of the charge's velocity, v, or the
direction of the conventional current.
2. Point your _______________ in the direction of the magnetic field, B.
3. Your _______________ now points in the direction of the magnetic force, F.
What I Can Do
Search and visit the Circuit Construction Kit: DC PhET Simulation. Explore it to know the basic
electricity relationships in series and parallel circuits.
Module 7 MAGNETIC FIELDS
Most Essential Learning Competencies
• Calculate the force per unit length on a current carrying wire due to the magnetic field produced
by other current -carrying wires; (STEM_GP12EM - IIIi -63)
• Evaluate the magnetic field vector at any point along the axis of a circular current loop;
(STEM_GP12EM - IIIi -64)
• Solve problems involving magnetic fields, forces due to magnetic fields and the motion of charges
and current -carrying wires in contexts such as, but not limited to, determining the strength of
Earth’s magnetic field, mass spectrometers, and solenoids. (STEM_GP12EM - IIIi -66)
•
18. 20
What’s In
Magnetic Field Due to a Long Straight Wire
Figure 2 shows that the magnetic field surrounding an electric current in a long straight wire, where the
field lines form circles with the wire at the center. The magnetic field B due to the current in a long
straight wire is directly proportional to the current I in the wire and is inversely proportional to the
distance r from the wire:
β= 02πIr
The value of the constant 0, which is permeability of free space, is 0=4πx10-7 T m/A
Force Between Two Parallel Wires
Consider two long parallel wires separated by a distance d as shown in Figure 1. Each wire carries
current I which produces a magnetic field is felt by each other. The force can be expressed as,
F= 02πI1I2dl
Ampere’s Law The magnetic field created by an electric current is proportional to the size of that
electric current with a constant of proportionality equal to the permeability of free space. The law
specifies the magnetic field that is associated with a given current or vice-versa, provided that the
electric field doesn’t change with time.
where the integral on the left is a “path integral”, similar to how we calculate
the work done by a force over a particular path. The circle sign on the integral
means that this is an integral over a “closed” path; a path where the starting
and ending points are the same. Ienc is the net current that crosses the surface
that is defined by the closed path, often called the “current enclosed” by the
path. This is different from Gauss’ Law, where the integral is over a closed
surface (not a closed path, as it is here). In the context of Gauss’ Law, we
refer to “calculating the flux of the electric field through a closed surface”;
in the context of Ampere’s Law, we refer to “calculating the circulation of
the magnetic field along a closed path”.
19. 21
Figure 2 (a) compasses placed near a long straight current-carrying wire indicates that field lines form
circular loops centered on the wire. (b) RHR 2 (Right Hand Rule 2) states that if the right-hand thumb
points in the direction of the current, the fingers curl in the direction of the field.
Ampere’s Laws Applications
Sample Problem #1: The two wires of a
2.0-m long appliance cord are 3.0mm
apart and carry a current of 8.0A dc.
Calculate the force one wire exerts on
the other.
Solution:
The formula
F= 02πI1I2dl
Each wire is in the magnetic field of the
other when the current is on, so we can
write 02π=2.0x10-7 T m/A
Substitute the given to the formula
F=(2.0x10-
7TmA)8.0A2(2.0m)(3.0x10-
3m)=8.5x10-3N
The currents are in opposite directions (one
toward the appliance, the other is away from
it), so the force will be repulsive and tend to
spread the wires apart.
Sample Problem #2: Compute the magnetic field of a
long straight wire that has a circular loop with a radius
of 0.05m. 2amp is the reading of the current flowing
through this closed loop.
Solution:
Given
R = 0.05m
I = 2A
μ0 = 4π×10-7N/A2
Ampere’s law formula is
∮B dl→=μ0I
In the case of long straight wire
Magnetic Field Produced by a Solenoid
20. 22
A solenoid is a long coil of wire (with many turns and loops). Its shape allows the field inside the
solenoid to strong and uniform. However, the field just outside the coils is nearly zero.
The magnetic field inside of a current-carrying solenoid is very uniform in direction and magnitude.
Only near the ends does it begin to weaken and change direction. The field outside has similar
complexities to flat loops and bar magnets, but the magnetic field strength inside a solenoid is simply,
where n is the number of loops per unit length of the solenoid (n = N/l, with N being the number of
loops and l the length). Note that B is the field strength anywhere in the uniform region of the interior
and not just at the center. Large uniform fields spread over a large volume are possible with solenoids.
Biot-Savart Law refers to an equation describing the magnetic field generated by a constant electric
current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric
current. Biot–Savart law is consistent with both Ampere’s Law and Gauss’s Law. It was created by two
French physicists, Jean Baptiste Biot and Felix Savart, who derived the mathematical expression for
magnetic flux density at a point due to a nearby current-carrying conductor, in 1820. These scientists
concluded that any current element projects a magnetic field into the space around it.
The Biot-Savart law equation that we will use is,
Figure 5
Image Source: Magnetic Fields Produced by Currents: Ampere’s Law | Physics (lumenlearning.com)
21. 23
What’s More
Direction: Solve the following problems. Include all your pertinent solutions and
express your final answers in the correct number of significant figures.
1. The two wires of a 3.0-m long appliance cord are 3.6mm apart and carry a
current of 7.0A dc. Calculate the force one wire exerts on the other.
2. Compute the magnetic field of a long straight wire that has a circular loop with
a radius of 0.04m. 3.5A is the reading of the current flowing through this closed
loop.
What I Have Learned
What do you think is the importance of Biot-Savart Law in our understanding of
magnetic fields?
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
22. 24
What I Can Do
Research on other practical applications of magnetic fields in the industry, technology,
and engineering. List atleast two applications for each field?
References:
Ling, S. J., Moebs, W., & Sanny, J. (2016, October 6). University Physics Volume 2.
Retrieved from
https://openstax.org/books/university-physics-volume-2/pages/1-introduction
Ling, S. J., Moebs, W., & Sanny, J. (2016, October 6). University Physics Volume 2.
Retrieved from
https://openstax.org/books/university-physics-volume-2/pages/1-introduction
Hewes, J. (2016). Circuit Symbols. Retrieved 2020, from
https://electronicsclub.info/circuitsymbols.htm
Lumen Learning (2013). Resistors in Series and Parallel. Retrieved from
https://courses.lumenlearning.com/boundless-physics/chapter/resistors-in-
series-and-parallel/
Other online reference:
Magnetic Fields Produced by Currents: Ampere’s Law | Physics (lumenlearning.com)
Biot-Savart Law (gsu.edu)
Biot-Savart Law - Statement, Formula, Examples, Importance, Problems (byjus.com)
