TX00CY32-3012
Project in Electrical and Automation Engineering
Magnetism / Period 4 / 2019
PART 1 : Basics of Magnetism
Berit Mannfors / berit.mannfors@metropolia.fi
1
INTRODUCTION
In practice, all essential electrical processes are electromagnetic.
It was experimentally observed that:
• Electric currents produce magnetic fields
• Magnets exert forces on moving charges (e.g. on wires with electric
current).
Magnetism is widely utilized, for example in:
• Meters
• Motors
• Loudspeakers
• Computer memory units
Basically, magnetic materials are divided into three categories depending
on their magnetic properties:
• Ferromagnetic (iron, nickel, cobalt, e.g. refrigerator magnets)
• Paramagnetic (e.g. platinum, aluminum, oxygen)
• Diamagnetic (e.g. carbon, copper, plastic, water)
2
1.1. Magnetic Dipoles and Magnetic Interaction
Electric charges (electric monopoles) are a source of electric fields but
magnetic fields are created by magnetic dipoles.
WHY NOT BY MAGNETIC MONOPOLES?
The theory of QED (quantum electrodynamics) by Paul Dirac found that
charge is not conserved if not magnetic monopoles existed during the first
seconds of the birth of the universe, and explained the absence of magnetic
monopoles because of the strong intermonopole interaction which created
magnetic dipoles.
As an analogy to electric dipoles as two equal but opposite charges at a
short distance from each other, a magnetic dipole is a system of two
opposite magnetic monopoles, the magnetic South pole (S) and the
magnetic North pole (N).
Thus, halving one magnet produces two magnetic dipoles.
3
The magnetic North pole (N) of a magnet, when freely suspended, is the one
that points to the Earth’s North magnetic pole.
As an analogy to electric fields, the direction of a magnetic field at any point
in space:
• Is along the (geometric) tangent of a magnetic field line
• Points toward the North magnetic pole inside the magnet
• Points toward the South magnetic pole outside the magnet.
Magnetic field lines are closed lines!
Similarly, the strength of a magnetic field is proportional to the density of
field lines through a surface under study, and related to the magnetic flux.
𝐕𝐬
𝐍
The SI unit of the strength of the magnetic field is tesla, 𝐓 = 𝐦𝟐 = 𝐀𝐦 .
A non-SI unit, gauss (G), is also used: 𝟏 𝐆 = 𝟏𝟎−𝟒 𝐓 .
4
By analogy to electric point charges, magnetic dipoles also interact:
Interaction between magnetic dipoles can be attractive or repulsive:
Opposite poles attract each other while poles of the same type repel
each other.
5
Earth’s magnetic field
Earth’s magnetic field originates from
strong circulating electric currents because
of moving ions in the melted outer iron
core. The inner core is at solid state because
of high pressure. When Earth rotates about
its axis, the liquid iron core follows the
motion with a slower pace.
The south magnetic pole is rather close to
Earth’s geographic north pole (at the moment on
the North-Eastern coast of Canada and moving
South-eastward) whereas Earth’s north magnetic
pole is close to Earth’s geographic south pole.
Earth’s magnetic field is known to have reversed
(http://en.wikipedia.org/wiki/Earth%27s_magne
tic_field).
Northern lights are caused by interactions of
charged particles from the Sun with Earth’s
magnetic field. The strength of Earth’s magnetic
flux on the surface of Earth varies from about
30 T (T = tesla) to 60 T.
6
1.2. Magnetic Force
MAGNETIC FORCE ON A MOVING CHARGE IN A MAGNETIC FIELD:
ത 𝐹ത𝐸 = 𝑞 𝐸ത , the
By analogy to the electric force on charge 𝑞 in an electric field of 𝐸,
magnetic force on charge 𝑞 moving at velocity 𝑣ҧ in a magnetic field of strength 𝐵
is given as a vector product:
ഥ 𝑩 = 𝒒ഥ
ഥ
𝑭
𝒗×𝑩
The magnitude of the magnetic force is 𝐹𝐵 = 𝑞𝑣𝐵 sin 𝜃, and the direction of the
force is obtained with the right-hand rule as shown in the figure:
7
Arrows (e.g.  , ) are used to give directions of vectors (magnetic fields,
currents, forces, etc.) on the plane, crosses () through the plane from front
to back (‘arrow tails’, arrow leaving you), and points () through the plane
from back to front (‘arrow points’, arrow approaching you).
When both electric and magnetic fields are present, a particle with a charge
of 𝑞 moving at velocity 𝑣ҧ experiences an electromagnetic (EM) Lorentz force:
ഥ 𝑬𝑴 = 𝒒 𝑬
ഥ+𝒗
ഥ
ഥ×𝑩
𝑭
Lorentz forces are utilized in applications where deflection of charge is
desired (e.g. particle accelerators, oscilloscopes, mass spectrometers).
Based on Newton’s equation of motion:
ഥ 𝑬𝑴 = 𝒒 𝑬
ഥ+𝒗
ഥ = 𝒎ഥ
ഥ×𝑩
𝑭
𝒂
8
MAGNETIC FORCE ON A CURRENT-CARRYING WIRE IN A MAGNETIC FIELD:
Electrons (charge = e) in a wire of length 𝐿 and cross-sectional area 𝐴 flow
at a drift velocity of 𝑣ҧ𝑑 in the direction opposite to the electric current 𝐼 ҧ .
Assuming 𝑁 electrons drifting in the wire, the magnetic force is
𝐹ത𝐵 = 𝑁 ∙ −𝑒 𝑣ҧ𝑑 × 𝐵ത
𝑑𝑞
The drift velocity is obtained by considering the electric current 𝐼 =
and
𝑑𝑡
electron density 𝑛 = 𝑁/volume (𝑁 = number of electrons in the wire):
𝐼=
𝑁∙ +𝑒
𝑡𝑖𝑚𝑒
=
𝑛∙𝐴𝐿 ∙𝑒
𝑡𝑖𝑚𝑒
= 𝑛𝐴𝑒 ∙
𝐿
𝑡𝑖𝑚𝑒
= 𝑛𝐴𝑒 ∙ 𝑣𝑑 =
𝑁𝑒𝑣𝑑
𝐿
.
By substituting the drift velocity 𝑣ҧ𝑑 ,
the magnetic force on a wire carrying
a current of 𝐼 becomes as follows:
ഥ𝑩 = 𝑰 ഥ
ഥ
𝑭
𝑳×𝑩
9
1.3. The Hall Effect
The Hall effect is a phenomenon where, in a magnetic field a small voltage is induced across a
thin slab carrying a current. The Hall effect is utilized in, for example, sensors to measure
magnetic fields, strong currents or weak forces. The phenomenon can also be utilized to
determine the type and the number of charge carriers in the material of the slab.
A metal slab (of length 𝐿, width 𝑤, and height ℎ), carrying a current of 𝐼 in the direction of the
slab’s longest side, is placed in a uniform and perpendicular magnetic field of 𝐵ത in the direction
of the slab’s width. Charge carriers in the slab are affected by the magnetic force and driven to
one side of the slab, causing polarization and production of an electric field and a small voltage
across the slab.
A magnetic force is created, which carries
the positive charge carriers upward.
Motion of charge carries causes polarization,
which induces a small Hall electric field, 𝐸ത𝐻 ,
and a Hall voltage, ∆𝑉𝐻 .
10
For positive charge carriers of 𝒒:
ത
The magnitude of the magnetic force on the moving positive charge carriers is (𝑣ҧ𝑑 ⊥ 𝐵)
𝐹𝐵 = 𝑞𝑣𝑑 𝐵
The magnetic force polarizes the slab and induces an electric force with magnitude of
𝐹𝐸 = 𝑞𝐸𝐻
In a stationary state, charge carriers no longer move  the forces are equal.
Then, the Hall electric field is
𝐸𝐻 = 𝑣𝑑 𝐵
The Hall field induces the Hall voltage, ∆𝑉𝐻 = 𝐸𝐻 ∙ ℎ = 𝑣𝑑 𝐵ℎ , across the slab.
𝐼
Implementing the drift velocity of charge carriers, 𝑣𝑑 , by 𝑛𝑞𝐴 (see page 9), the Hall
voltage yields information about the density and sign of charge carriers 𝑛 in the slab as
well as the applied magnetic field and electric current:
∆𝑉𝐻 =
𝐼𝐵
𝑛𝑞𝑤
11
1.4. Magnetic Field and Ampère’s law
Moving charges and electric currents create magnetic fields.
𝑄
An analogy to finding electric fields by Gauss’ law: 𝛷𝐸 = ‫𝐸 ׯ‬ത ∙ 𝑑𝐴ҧ = 𝜀𝑒𝑛𝑐 ,
0
magnetic fields can be derived from Ampère’s law:
ഥ ∙ 𝒅ഥ
𝑳 = 𝝁𝟎 𝑰𝒆𝒏𝒄
‫𝑩ׯ‬
Since magnetic field lines are closed lines, Ampère’s law does not yield the
magnetic flux through a surface.
In Ampère’s law,
• The enclosed current, 𝐼𝑒𝑛𝑐 , is the current inside magnetic field lines
• 𝝁𝟎 = 𝟒𝝅 ∙ 𝟏𝟎−𝟕 𝐓𝐦/𝐀, the permeability of free space (vacuum, also used
for air), called the magnetic constant, yields the response to an external
electric field.
Note that other fields of technology use the word of permeability for other
purposes.
12
1.5. Magnetic Field and the Biot-Savart Law
The figure shows a part of a current-carrying wire:
ത from
The Biot-Savart law is used for calculation of arising magnetic fields, 𝑑 𝐵,
steady currents, 𝐼 :
𝝁𝟎 𝑰𝒅𝒍ҧ × 𝒓ො
ഥ=
𝒅𝑩
𝟒𝝅 𝒓𝟐
In the equation:
• 𝑑 𝑙 ҧ = length element (a small piece of wire)
• 𝑟 = distance from the element to the point 𝑃 where the magnetic is
calculated
• 𝑟Ƹ = unit vector in the direction of the distance 𝑟.
The vector product contains the mutual direction of the length element and the
distance vector, viz, the angle of 𝜃.
13
EXAMPLE 1: Magnetic Field of a Long Current-Carrying
Straight Wire
Using Ampère’s law to obtain the magnetic field from a long and straight
wire carrying a current of 𝐼, the line integral is calculated along the circular
magnetic field lines around the wire.
From Ampère’s law, at a distance of 𝑅 from the wire,
Φ𝐵 = ‫𝐵 ׯ‬ത ∙ 𝑑 𝐿ത = 𝐵 ∙ 2𝜋𝑅 = 𝜇0 𝐼𝑒𝑛𝑐 = 𝜇0 𝐼

𝝁𝟎 𝑰
𝑩=
𝟐𝝅𝑹
The permeability 𝜇0 is replaced by 𝜇 = 𝜇𝑟 𝜇0 in a media other than vacuum
or air around the wire. The relative permeability, 𝜇𝑟 , is a material constant.
14
EXAMPLE 2: Magnetic Interaction Between Parallel Wires
The magnetic field due to a wire with a current of 𝐼 affects another
current-carrying wire, causing an interaction force between the wires.
𝜇 𝐼
The magnetic field, 𝐵2 = 0 2 , created by the wire on the right at a
2𝜋𝑑
distance of 𝑑 from the other wire, exerts a magnetic force on the wire on
𝜇 𝐼 𝐼 𝐿
the left: 𝐹𝐵21 = 𝐼1 𝐿ത × 𝐵ത2 = 0 1 2 .
2𝜋𝑑
15
EXAMPLE 3: Magnetic Field by Current-Carrying Coils
Coils or inductors are electric components made of wax-insulated wire
wound in a cylindrical form. Long straight coils are called solenoids and those
in the shape of doughnuts are called toroids.
When an electric current flows through the wire of a coil, a magnetic field is
produced inside the coil (with a minor field outside a short solenoid).
In many applications, ferromagnetic material is inserted inside the coil to
enhance the storage of energy.
The magnetic flux through the coil depends on the magnetic field inside the
coil, the coil area as well as the mutual directions of the magnetic field and
coil surface area vector as explained in the following pages.
16
For the most typical coils:
• The magnetic field inside a long solenoid of length 𝐿, with N turns, and
carrying a current of 𝐼:
𝑩𝒔𝒐𝒍𝒆𝒏𝒐𝒊𝒅
𝝁𝟎 𝑵𝑰
≈
𝑳
• The magnetic field inside a toroid of an average radius of 𝑅, with N turns,
and carrying a current of 𝐼:
𝑩𝒕𝒐𝒓𝒐𝒊𝒅
𝝁𝟎 𝑵𝑰
=
𝟐𝝅𝑹
17
1.6. Gauss Law of Magnetism
By analogy to an electric flux (𝑞𝑒𝑛𝑐 is the charge inside the Gauss surface),
Φ𝐸 = න 𝐸ത ∙ 𝑑𝐴ҧ =
𝑞𝑒𝑛𝑐
𝜀
the Gauss law can be written also for magnetism.
It was believed earlier that magnetic monopoles don’t exist. Then, the Gauss law
was given as follows (magnetic field lines are closed lines):
Φ𝐵 = ර 𝐵ത ∙ 𝑑𝐴ҧ = 0
Because of very strong interaction between magnetic monopoles, individual
magnetic monopoles cannot be isolated. However, by analogy to electric flux,
ഥ ∙ 𝒅𝑨
ഥ = 𝝁 ∙ 𝒒𝒎
𝚽𝑩 = ර 𝑩
This can be easily derived for a magnetic flux through a sphere, when the magnetic
𝜇 𝑞
field created by a stationary magnetic monopole 𝑞𝑚 at a distance of 𝑟 is 𝐵 = 4𝜋 𝑟𝑚2 .
18
1.7. Magnetic Dipole Moment and Torque
By analogy to an electric dipole moment (𝑝ҧ = 𝑞𝑑 𝑢ො 𝑑 ), a magnetic dipole
moment, 𝜇ҧ , is produced by a current-carrying loop:
ഥ
ഥ = 𝑰𝑨
𝝁
In an electric field, electric forces are exerted on the charges of the electric dipole,
producing a rotational motion of the electric dipole toward a full alignment of the
dipole parallel to the field (i.e., toward the situation where the electric dipole is
parallel to the field: 𝑝ҧ ↑↑ 𝐸ത ).
Similarly, in a magnetic field, a current loop responds to an external magnetic
field by experiencing a torque which produces a rotational motion of the loop
toward the full alignment of the magnetic dipole moment parallel to the
field: 𝜇ҧ ↑↑ 𝐵ത .
19
Far from the current-carrying loop, the magnetic field created by the loop is a
dipole field, as shown in the following figure:
The torque 𝜏ҧ of a dipole is given as follows:
ഥ
ഥ×𝑩
𝝉ത = 𝝁
20
1.8. Electromagnetic Induction
Electromagnetic induction is a process in which an opposing voltage and
current is produced in a current loop by a varying magnetic flux.
Electromagnetic induction is utilized in, e.g., transformers, electric motors,
microphones and loudspeakers.
A magnetic flux is defined by analogy to an electric flux:
ഥ ∙ 𝒅𝑨
ഥ = 𝑩𝑨 cos 𝜽
𝚽𝑩 = න 𝑩
The SI unit of the magnetic flux is the weber, Wb:
Wb = T  m2
Two phenomena exist due to varying magnetic flux:
1) An electric current is induced in a conductor that is in a relative
motion to a magnetic field.
2) An electric field and, therefore, an electric current is induced in a
conductor by a temporally varying magnetic field (one of Maxwell’s
electromagnetic laws, utilized in generators and motors).
21
For simplicity, a loop conductor is placed in a perpendicular magnetic field of 𝐵ത .
Then the magnetic flux through the loop is:
Φ𝐵 = ‫𝐵 ׬‬ത ∙ 𝑑 𝐴ҧ = 𝐵𝐴 cos 0° = 𝐵𝐴
When the magnetic flux changes with time, an
opposing voltage is induced. The magnitude of the
induced voltage is given by the rate of the change
of the magnetic flux (Faraday’s law):
𝜺𝒊𝒏𝒅
𝒄𝒉𝒂𝒏𝒈𝒆 𝒐𝒇 𝒎𝒂𝒈𝒏𝒆𝒕𝒊𝒄 𝒇𝒍𝒖𝒙
𝒅𝚽𝑩
=−
=−
𝒕𝒊𝒎𝒆 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍
𝒅𝒕
The magnetic flux changes, when
• The strength of the magnetic field, 𝐵, varies
• The loop area, 𝐴, varies
• The angle 𝜃, i.e., the mutual direction of the loop surface vector, 𝐴,ҧ and
ത varies.
the magnetic field, 𝐵,
22
The following picture shows the induced currents in the loop (red arrows) due to
changes in the magnetic flux by a varying perpendicular magnetic field:
The initial magnetic field, 𝐵ത𝑖𝑛𝑖𝑡𝑖𝑎𝑙 , induces a current 𝐼 in the loop (left).
When the strength of the magnetic field decreases, the magnetic flux decreases, and
an additional current (red arrow) is induced parallel to the original current (middle).
When the strength of the magnetic field increases, the magnetic flux increases, and
an additional current (red arrow) is induced opposite to the original current (right).
23
1.9. Magnetization and Hysteresis
Magnetic materials are categorized according to their ability to respond to
external magnetic fields: Ferromagnetic materials respond strongly while
paramagnetic and diamagnetic materials respond weakly. In addition,
ferromagnetic and paramagnetic materials are attracted to magnets while
diamagnetic materials are repelled by magnets.
Ferromagnetism is caused by strong organization of
electrons’ magnetic moments, called electronic
spins. (Electrons “orbiting” the atomic nuclei in the
material represent small current loops, being
small magnets.)
Ferromagnetic materials contain areas, called
magnetic domains, where the electronic spins
are aligned almost parallel, giving rise to local net
magnetic dipoles (intrinsic magnetization).
When a ferromagnetic material is placed in an
external magnetic field, the magnetic dipoles of the
domains align parallel to the field.
24
Paramagnetism differs from ferromagnetism in the strength of electronic dipole-dipole
interactions. Electronic dipole moments are in more random directions and only weakly
aligned in an external magnetic field.
Diamagnetism is a state of zero dipole moment: Assume two electrons in a same
energy state. Based on quantum mechanical rules, the electrons in the same energy
state “spin” in opposite directions, their magnetic dipoles being opposite to each other.
This results in a zero net magnetic dipole moment, and the repulsive interaction with an
external magnetic field.
When an external magnetic field changes, a voltage and, therefore, an electric field is
induced. The induced electric field exerts an additional force on the spinning electrons,
slowing down one of the electrons and speeding up the other (Newton’s mechanics).
Huge currents are induced in superconductors which have almost zero electric
resistance, eliminating the effect of external magnetic fields  magnetic levitation.
25
Hysteresis is an irreversible phenomenon which shows a differently varying
ഥ
magnetic field strength, 𝐵, of a material when an external magnetic field, 𝐻,
increases or decreases. This is because of the
ഥ of the material,
amount of magnetization, 𝑀,
ഥ : CHECK
which is included in 𝐵ത but not in 𝐻
ഥ = 𝜇0 1 + 𝜒𝑚 𝐻
ഥ = 𝜇0 1? +𝜇𝑟 𝐻
ഥ.
𝐵ത = 𝜇 𝐻
When the material is placed in
ഥ the
an external magnetic field, 𝐻,
field strength 𝐵 increases until
its maximum value (saturation).
When the external field is decreased,
the magnetic field strength
decreases more slowly and reaches
a value of residual magnetism at the zero field.
When the field is reversed and increasing, the field
strength again increases but is decreased differently
by a decreasing field.
Materials which can hold their magnetization are hard
magnetic materials while soft magnetic materials
cannot hold their magnetization (see the right figure).
26
Problems
Problem 1.1. The figure shows a proton in a magnetic
field. For which of the three proton
velocities shown will the magnetic force be:
(a) Largest?
(b) Smallest?
(c) What will be the direction
of the forces in the given cases?
Problem 1.2. Three protons enter a 0.10-T
magnetic field at 2.0 Mm/s,
as shown in the figure.
Find the magnetic forces (magnitudes
and directions) for all protons.
27
Problem 1.3. Earth’s magnetic field at the equator is about 0.3 G, parallel to the
surface of the Earth and points from South to North. Find the
magnetic force on an electron moving at a thermal speed of 106 m/s :
(a) Vertically up from the surface.
(b) Horizontally to the east.
Problem 1.4. Electric and magnetic fields can be used to deflect particles from their
original direction of motion. Consider a space with perpendicular and
uniform electric and magnetic fields. An electron enters this space but
is not deflected.
(a) What must be the electron’s velocity not to deflect? (The method is
used in velocity selectors.)
(b) When is the velocity of the electron smallest?
Problem 1.5. Based on Newton’s second law, forces are needed to change the
motion of particles. Considering a magnetic force on a uniformly
ഥ), what does a non-zero acceleration of
moving charged particle (𝒒, 𝒗
the particle mean?
28
Problem 1.6. An electron moving in perpendicular direction to a 0.10-T magnetic field
experiences an acceleration of 6.0  1015 m/s2.
(a) What is its speed?
(b) By how much does its speed change in 1 ns?
Problem 1.7. The magnitude of Earth’s magnetic field is about 0.5 G near Earth’s
surface.
(a) What is the maximum possible force on an electron with kinetic
energy of 1 keV?
(b) Compare the results of (a) with the gravitational force on the
electron.
Problem 1.8. In one experiment, a conducting bar carrying a 4.1 kA current will pass
through a 1.3-m long region of 12-T magnetic field at 60° angle with the
bar. Must the bar be clamped in place or not? To answer the question,
find the magnetic force experienced by the bar.
29
Problem 1.9. The picture on the right shows a charged
particle moving in a uniform and
perpendicular magnetic field.
How does the particle continue
its motion (speed, direction)?
Problem 1.10. A proton moves at a constant velocity along a circular path of radius
12.7 cm in a uniform and perpendicular magnetic field of 1.2 T. Then,
𝟐𝝅𝑹
the time of one revolution is 𝑻 = 𝒗 . How long does it take from the
proton to complete one revolution?
(Hint: The acceleration of circular motion is given by
the radius of the circular path.)
𝒗𝟐
𝑹
, where 𝑹 is
Problem 1.11. Radioastronomers detect electromagnetic radiation at a frequency of
42 MHz from an interstellar gas cloud. If the radiation results from
electrons spiraling in a magnetic field, what is the field strength?
30
Problem 1.12. Microwaves in a microwave oven are produced by electrons circling in
a magnetic field at a frequency of 2.4 GHz.
(a) What is the magnetic field strength?
(b) The electron’s motion takes place inside a special tube called a
magnetron. If the magnetron can accomodate electron orbits with
maximum diameter 2.5 mm, what is the maximum electron
energy?
(https://www.explainthatstuff.com/how-magnetrons-work.html)
Problem 1.13. The figure shows a flexible wire passing
ഥ.
through a strong magnetic field 𝑩
The wire is deflected as shown in the
figure. Is the current in the wire
flowing:
(a) To the right?
(b) To the left?
31
Problem 1.14. Consider the magnetic force on the systems shown in the following
picture. What happens to the systems (individual charges, currentcarrying wires)?
Problem 1.15. Consider the following case of parallel
wires, both carrying a current of 𝑰.
(a) In which directions do the
currents flow in the wires?
(b) If a strong magnetic field from
front to back is then applied to
the wires, how do the wires interact?
32
Problem 1.16. In standard household wiring, parallel wires about 1 cm apart carry
currents of about 15 A. What is the force per unit length between
these wires?
Problem 1.17. A wire 1.0 mm in diameter carries 5.0 A current distributed uniformly
over its cross section. What are the field strengths:
(a) At 0.10 mm from its axis?
(b) At the wire’s surface?
(c) At the 1.0 cm distance from its axis?
Problem 1.18. A superconducting solenoid has 3300 turns/m and carries a 4.1-kA
current. What is the field strength in the solenoid?
Problem 1.19. Jupiter has the strongest magnetic field in our solar system, about
14 G at its poles. Assuming the field is a dipole field, what is Jupiter’s
magnetic dipole moment? The mean radius of Jupiter is 69.1 ∙ 106 m.
33
Problem 1.20. An electric motor contains a 250-turn circular coil of a diameter of
6.2 cm. If it develops a maximum torque of 1.2 Nm at a current of
3.3 A, what is the magnetic field strength?
Problem 1.21. A prosthetic ankle includes a miniature electric motor containing a
150-turn circular coil that is 15 mm in diameter. The motor needs to
develop a maximum torque of 3.1 mNm. The strongest magnets that
will fit in the prosthetic ankle produce a 220-mT field. What must be
the current in the motor’s coil?
Problem 1.22. A coaxial cable consists of an inner conductor of a diameter of 1.0 mm,
and a hollow concentric outer conductor of a diameter of 1.0 cm.
A 100-mA current flows down the inner conductor and back along the
outer conductor. From the cable axis, what is the magnetic field
strength:
(a) At a 0.10-mm distance?
(b) At a 5.0-mm distance?
(c) At a 2.0-cm distance?
34
Problem 1.23. Earth’s magnetic field in the Northern Hemisphere is about 60 T in
magnitude, makes an angle of about 60 relative to the ground, and
points toward geographic North. Assume power lines in the area
carry a current of 500 A from West to East.
(a) What is the magnitude of the magnetic force/km on the power
lines?
(b) Into which direction might the power lines curve?
Problem 1.24. Electric motors used in disk drives or power tools consist of a current
loop in a magnetic field. Assume a 30 cm  20 cm rectangular loop
carrying a current of 90 A. The loop is placed between the magnetic
South and North poles of two magnets, yielding a 50 mT magnetic
field between the poles and through the loop.
(a) Find the magnetic force on each side of the loop, when the loop
area is in the plane of the field.
(b) For a DC motor to function, the direction of current is reversed
after each revolution. How does this affect the motor.
35
Problem 1.25. In the US, some farmers have stolen electricity from power lines by
utilizing electromagnetic induction. In one such case, the farmer had
strung a rectangular loop vertically beneath a power line. The power
line carried an alternating current (AC) of 60 Hz.
An AC current can be given as follows: 𝑰 𝒕 = 𝑰𝟎 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 . Assume
the amplitude of 𝑰𝟎 = 𝟏𝟎 𝐤𝐀 and the frequency of 𝒇 = 𝟔𝟎 𝐇𝐳.
Factor 𝒕 in the equation is the time in seconds.
(a) Find an estimate to a varying magnetic flux through a 20 m  2 m
loop if the power line was 10 meters above the midpoint of the
loop.
(b) Find the induced varying voltage in the loop.
(c) What is the varying induced current in the loop if the resistance
of the loop is 0.10 ?
Problem 1.26. If the loop of the previous problem were mounted in a horizontal
rather than vertical plane at the same distance from the power line,
would the induced current increase slightly, decrease slightly, remain
the same, or become essentially zero?
36