Module 4 electricity 2023 (1)

MINISTRY OF EDUCATION AND TRAINING
NONG LAM UNIVERSITY
FACULTY OF CHEMICAL ENGINEERING AND FOOD TECHNOLOGY

Course: Physics 1

Module 4: Electricity and magnetism

Instructor: Dr. Nguyen Thanh Son

Academic year: 2022-2023


Contents

Module 4: Electricity and magnetism
4.1 Electromagnetic concepts and law of conservation of electric charge
4.1.1. Electromagnetic concepts
4.1.2. Law of conservation of electric charge
4.2 Electric current
4.2.1. Electric current
4.2.2. Electric current density
4.3 Magnetic interaction - Ampère’s law
4.3.1. Magnetic interaction
4.3.2. Ampère’s law for magnetic field
4.4 Magnetic intensity
4.4.1. Magnetic intensity
4.4.2. Relationship between magnetic intensity and magnetic induction
4.5 Electromagnetic induction
4.5.1. Magnetic flux
4.5.2. Faraday’s law of electromagnetic induction

4.6 Magnetic energy
4.6.1. Energy stored in a magnetic field
4.6.2. Magnetic energy density

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4.1 Electromagnetic concepts and law of conservation of electric charge

4.1.1. Electromagnetic concepts
• A magnetic field is a vector field which can exert a magnetic force on moving electric charges
and on magnetic dipoles (such as permanent magnets). When placed in a magnetic field,
magnetic dipoles tend to align their axes parallel to the magnetic field. Magnetic fields surround
and are created by electric currents, magnetic dipoles, and changing electric fields. Magnetic
fields also have their own energy, with an energy density proportional to the square of the field
magnitude.

• Magnetic field forms one aspect of electromagnetic field. A pure electric field in one reference
frame will be viewed as a combination of both an electric field and a magnetic field in a moving
reference frame. Together, electric and magnetic fields make up electromagnetic field, which is
best known for underlying light and other electromagnetic waves.
• Electromagnetism describes the relationship between electricity and magnetism.
Electromagnetism is essentially the foundation for all of electrical engineering. We use
electromagnets to generate electricity, store memory on computers, generate pictures on a
television screen, diagnose illnesses, and in just about every other aspect of our lives that
depends on electricity.
• Electromagnetism works on the principle that an electric current through a wire generates a
magnetic field. We already know that a charge in motion creates a current. If the movement of

the charge is restricted in such a way that the resulting current is constant in time, the field thus
created is called a static magnetic field. Since the current is constant in time, the magnetic field
is also constant in time. The branch of science relating to constant magnetic field is called
magnetostatics, or static magnetic field. In this case, we are interested in the determination of (a)
magnetic field intensity, (b) magnetic flux density, (c) magnetic flux, and (d) the energy stored in
the magnetic field.
♦ Linking electricity and magnetism
• There is a strong connection between electricity and magnetism. With electricity, there are
positive and negative charges. With magnetism, there are north and south poles. Similar to
electric charges, like magnetic poles repel each other, while unlike poles attract.
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• An important difference between electricity and magnetism is that in electricity it is possible to
have individual positive and negative charges. In magnetism, north and south poles are always
found in pairs. Single magnetic poles, known as magnetic monopoles, have been proposed
theoretically, but a magnetic monopole has never been observed.
• In the same way that electric charges create electric fields around them, north and south poles
will set up magnetic fields around them. Again, there is a difference. While electric field lines
begin on positive charges and end on negative charges, magnetic field lines are closed loops,
extending from the south pole to the north pole and back again (or, equivalently, from the north
pole to the south pole and back again). With a typical bar magnet, for example, the field goes
from the north pole to the south pole outside the magnet, and back from south to north inside the
magnet.
• Electric fields come from electric charges. So do magnetic fields, but from moving charges, or
currents, which are simply a whole bunch of moving charges. In a permanent magnet, the
magnetic field comes from the motion of the electrons inside the material, or, more precisely,
from something called the electron spin. The electron spin is a bit like the Earth spinning on its

axis.
• The magnetic field is a vector; the same way the electric field is. The electric field at a
particular point is in the direction of the force that a positive charge would experience if it were
placed at that point. The magnetic field at a point is in the direction of the force that a north pole
of a magnet would experience if it were placed there. In other words, the north pole of a
compass points in the direction of the magnetic field that exerts a force on the compass.
• The symbol for magnetic field induction or magnetic flux density is the letter B. The SI unit of
B is the tesla (T).
• One of various manifestations of the linking between electricity and magnetism is
electromagnetic induction (see Section 4.5). This involves generating a voltage (an induced
electromotive force) by changing the magnetic field that passes through a coil of wire.
• In other words, electromagnetism is a two-way link between electricity and magnetism. An
electric current creates a magnetic field, and a magnetic field, when it changes, creates a voltage.
The discovery of this link led to the invention of transformer, electric motor, and generator. It
also explained what light is and led to the invention of radio.
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4.1.2. Law of conservation of electric charge
• Electric charge
•

There are two kinds of charge, positive and negative.

•

Like charges repel; unlike charges attract.

•

Positive charge results from having more protons than electrons; negative charge results
from having more electrons than protons.

•

Charge is quantized, meaning that charge comes in integer multiples of the elementary
charge e.

•

Charge is conserved.

• Probably everyone is familiar with the first three concepts, but what does it mean for charge to
be quantized? Charge comes in multiples of an indivisible unit of charge, represented by the
letter e. In other words, charge comes in multiples of the charge of the electron or the proton. A
proton has a charge of +e, while an electron has a charge of -e. The amount of electric charge of
any object is only available in discrete units. These discrete units are exactly equal to the amount
of electric charge that is found on the electron or the proton.
• Electrons and protons are not the only things that carry charge. Other particles (positrons, for
example) also carry charge in multiples of the electronic charge. Putting "charge is quantized" in
terms of an equation, we say:
q = ne

(79)

where q is the symbol used to represent electric charge, while n is a positive or negative integer
(n = 0, ±1, ±2, ±3, …), and e is the elementary charge, 1.60 x 10-19 coulombs.
• Table of elementary particle masses and charges:

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♦ The law of conservation of electric charge
• The law of conservation of charge states that the net charge of an isolated system remains
constant. This law is inherent to all processes known to Physics.
• In other words, electric charge conservation is the principle that electric charges can neither
be created nor destroyed. The quantity of electric charge of an isolated system is always
conserved.
• If a system starts out with an equal number of positive and negative charges, there is nothing
we can do to create an excess of one kind of charge in that system unless we bring in some
charge from outside the system (or remove some charge from the system). Likewise, if
something starts out with a certain net charge, say +100 e, it will always have +100 e unless it is
allowed to interact with something external to it.
♦ Electrostatic charging
• Forces between two electrically-charged objects can be extremely large. Most things are
electrically neutral; they have equal amounts of positive and negative charge. If this was not the
case, the world we live in would be a much stranger place. We also have a lot of control over
how things get charged. This is because we can choose the appropriate material to use in a given
situation.
• Metals are good conductors of electric charge, while plastics, wood, and rubber are not. They
are called insulators. Charge does not flow nearly as easily through insulators as it does through
conductors; that is the reason why wires you plug into a wall socket are covered with a
protective rubber coating. Charge flows along the wire, but not through the coating to you.
• In fact, materials are divided into three categories, depending on how easily they will allow
charge (i.e., electrons) to flow along them. These are:
•

conductors, metals, for example,

•

semi-conductors, silicon is a good example, and

•

insulators, rubber, wood, plastics, for example.

• Most materials are either conductors or insulators. The difference between them is that in
conductors, the outermost electrons in the atoms are so loosely bound to their atoms that they are
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free to travel around. In insulators, on the other hand, the electrons are much more tightly bound
to their atoms, and are not free to flow. Semi-conductors are a very useful intermediate class, not
as conductive as metals but considerably more conductive than insulators. By adding certain
impurities to semi-conductors in the appropriate concentrations, the conductivity can be wellcontrolled.
• There are three ways that objects can be given a net charge. These are:
1. Charging by friction - this is useful for charging insulators. If you rub one material with
another (say, a plastic ruler with a piece of paper towel), electrons have a tendency to be
transferred from one material to the other. For example, rubbing glass with silk or saran
wrap generally leaves the glass with a positive charge; rubbing PVC rod with fur generally
gives the rod a negative charge.
2. Charging by conduction - useful for charging metals and other conductors. If a charged
object touches a conductor, some charge will be transferred between the object and the

conductor, charging the conductor with the same sign as the charge on the object.
3. Charging by induction - also useful for charging metals and other conductors. Again, a
charged object is used, but this time it is only brought close to the conductor, and does not
touch it. If the conductor is connected to ground (ground is basically anything neutral that
can give up electrons to, or take electrons from, an object), electrons will either flow on to it
or away from it. When the ground connection is removed, the conductor will have a charge
opposite in sign to that of the charged object.
• Electric charge is a property of the particles that make up an atom. The electrons that surround
the nucleus of the atom have a negative electric charge. The protons which partly make up the
nucleus have a positive electric charge. The neutrons which also make up the nucleus have no
electric charge. The negative charge of the electron is exactly equal and opposite to the positive
charge of the proton. For example, two electrons separated by a certain distance will repel one
another with the same force as two protons separated by the same distance, and, likewise, a
proton and an electron separated by the same distance will attract one another with a force of the
same magnitude.
• In practice, charge conservation is a physical law that states that the net change in the amount
of electric charge in a specific volume of space is exactly equal to the net amount of charge
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flowing into the volume minus the amount of charge flowing out of the volume. In essence,
charge conservation is an accounting relationship between the amount of charge in a region and
the flow of charge into and out of that region.
Mathematically, we can state the law as
q(t2) = q(t1) + qin – qout

(80)

where q(t) is the quantity of electric charge in a specific volume at time t, q in is the amount of
charge flowing into the volume between time t1 and t2, and qout is the amount of charge flowing
out of the volume during the same time period.
• Another statement for this law is the net electric charge of an isolated system remains
constant.
• The simple version of (80) is
q = constant

for an isolated system

(80’)

• The SI unit of electric charge is the coulomb (C).
4.2 Electric current

4.2.1. Electric current
• Electric current is the flow of electric charge, as shown in Figure 51.
The moving electric charges may be either electrons or ions or both.
• Whenever there is a net flow of charge through some region, an
electric current is said to exist. To define current more precisely,
suppose that the charges are moving perpendicular to a surface of area
A, as shown in Figure 51. This area could be the cross-sectional area of
a wire, for example.
• The electric current intensity I is the rate at which charge flows
through this surface. If ∆Q is the amount of charge that passes through

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Figure 51 Charges in
motion through an area A.
The time rate at which
charge flows through the
area is defined as the
current intensity I. The
direction of the current is
the direction in which
positive charges flow
when free to do so.


this area in a time interval ∆t, the average current intensity Iave is equal to the charge that passes
through A per unit time:
Iave = ∆Q/∆t

(81)

• If the rate at which charge flows varies in time, then the current varies in time; we define the
instantaneous current intensity I as the differential limit of average current:

I = lim

∆t → 0

∆Q dQ
=
∆t
dt

(82)

I = Q/t

(82’)

• If I is constant, (82) becomes

where Q is the quantity of electric charge passing through the cross-sectional area in the time t.
.
• The SI unit of electric current intensity is the ampère (A): 1 A = 1 C/1 s. That is, 1 A of current
is equivalent to 1 C of charge passing through the surface area in 1 s.
• If the ends of a conducting wire are connected to form a loop, all points on the loop are at the
same electric potential, and hence the electric field is zero within and at the surface of the
conductor. Because the electric field is zero, there is no net transport of charge through the wire,
and therefore there is no electric current.
• If the ends of the conducting wire are connected to a battery, all points on the loop are not at
the same potential. The battery sets up a potential difference between the ends of the loop,
creating an electric field within the wire. The electric field exerts forces on the electrons in the
wire, causing them to move around the loop and thus creating an electric current. It is common
to refer to a moving charge (positive or negative) as a mobile charge carrier. For example, the
mobile charge carriers in a metal are electrons.
♦ Electric current direction
• The charges passing through the surface, as shown in Figure 51, can be positive or negative, or
both. It is conventional to assign the electric current direction the same direction as the flow of
positive charge. In electrical conductors, such as copper or aluminum, the electric

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current is due to the motion of negatively charged electrons. Therefore, when we speak of
electric current in an ordinary conductor, the direction of the current is opposite to that of the
flow of electrons. However, if we are considering a beam of positively charged protons in an
accelerator, the current is in the direction of motion of the protons. In some cases - such as those
involving gases and electrolytes, for instance - the electric current is the result of the flow of
both positive and negative charges.
• An electric current can be represented by an arrow. The sense of the electric current arrow is
defined as follows:
If the current is due to the motion of positive charges, the current arrow is parallel to the
charge velocity.
If the current is due to the motion of negative charges, the current arrow is antiparallel
to the charge velocity.
Example: During 4 minutes a 5.0 A current is set up in a metal wire, how many (a)
coulombs and (b) electrons pass through any cross section across the wire’s width? ANS:
(a) Q = It = 1.2x103 C (b) N = Q/e = 7.5 x 10 21
Solution
(a) From (82’) we have Q = It; plugging numbers leads to Q = It = 1.2x103 C.
(b) N = Q/|qe| = Q/e = 7.5 x 1021 (e = 1.60 x 10 -19 C)
4.2.2. Electric current density
• Electric current density J is a vector quantity whose magnitude is the ratio of the magnitude
of electric current flowing in a conductor to the cross-sectional area perpendicular to the
current flow and whose direction points in the direction of the current.

• In other words, J is a vector quantity, and the scalar product of which with the cross-sectional
area vector A is equal to the electric current intensity. By magnitude it is the electric current
intensity divided by the cross-sectional area.

If the electric current density is constant then

I= J.A

(scalar product of J and A ). If the current density is not constant, then
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(83)


I=

∫ J .dA

(84)

where the current is in fact the integral of the dot product of the
current density vector J and the differential surface element dA
of the conductor’s cross-sectional area.
• The SI unit of J is the ampère per square meter (A/m2).
Figure 52 Depicting the
electric current density.

• Electric current density is important to the design of electrical
and electronic systems. For example, in the domain of electrical
wiring (isolated copper), maximum current density can vary
from 4 A/mm2 for a wire isolated to 6 A/mm2 for a wire in free

air.

If J is constant over the whole cross section of the wire and perpendicular to the cross sectional area, Equation 84 becomes
I = JA

(84’)

where A is the area of the wire’s cross section. In this case, J is parallel to the arrow showing
the direction of the electric current.
Example: The electric current density in a cylindrical wire of radius R = 2 mm is
uniform across a cross section of the wire and is J = 4.0x10 5 A/m2. Find the intensity of the
electric current through the wire.
Solution
J is constant, using (84’) we have I = JA with A = πR2
leads to I = 5.029 A.

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I = J(πR2). Plugging numbers


4.3 Magnetic interaction - Ampère’s law
4.3.1. Magnetic interaction
♦ Between two permanent magnets

There are no individual
magnetic poles
(or magnetic charges).
Electric charges can
be separated, but magnetic poles always

come in pairs
- one north and one south.
Unlike poles (N and S)
attract and like
poles (N and N,
or S and S) repel.
These bar magnets will remain
"permanent"
until something
happens to eliminate
the alignment of
atomic magnets
in the bar of
iron, nickel,
or cobalt.
Figure 53 Magnetic interaction between two bar

magnets (permanent magnets).
♦ Between an electric current and a compass

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The connection between electric current and
magnetic field was first observed when the
presence of a current in a wire near a magnetic
compass affected the direction of the compass
needle. We now know that electric current

gives rise to magnetic fields, just as electric
charge gives rise to electric fields.

Figure 54 Compass near an electric current-

carrying wire.

♦ Magnetic force acting on a moving charge
• A charged particle q when moving with velocity v in a magnetic field B experiences a
magnetic force F .
• Experiments on various charged particles moving in a magnetic field give the following
results:

• The magnitude F of the magnetic force exerted on the particle is proportional to the
charge magnitude |q| and to the speed v of the particle.
• The magnitude and direction of F depend on the velocity v of the particle and
on the magnitude and direction of the magnetic field B .

• When a charged particle moves parallel to the magnetic field vector, the magnetic

force acting on the particle is zero.
• When the particle’s velocity vector v makes any angle φ ≠ 0 with the magnetic
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field B , the magnetic force F acts in a direction perpendicular to both v and B ; that
is, F is perpendicular to the plane formed by v and B (see Figure 55).
• Mathematically, the magnetic force F is given by

F = qv x B

(85)

where the direction of F is in the direction of v x B if q is positive, which, by definition of the
cross product, is perpendicular to both v and B .
• We can regard Equation 85 as an operational definition of the magnetic field at some point in
space.
• The magnitude of the magnetic force is

F = |q|vB sinφ

(86)

where φ is the smaller angle between v and B . From this expression, we see that F is zero when
v is parallel or antiparallel to B (φ = 0 or 180°) and maximum, Fmax = |q|vB, when v is

perpendicular to B (φ = 90°).

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The direction of the cross product of the
two vectors can be obtained by using a
right-hand rule:

The index finger of the right hand
points in the direction of the first vector

( v ) in the cross product;
then adjust your wrist so that
you can bend the rest fingers toward the
direction of the second vector ( B );
extend the thumb to get the
direction of the magnetic force.

Figure 55 Magnetic force acting on a moving charge.

♦ MOTION OF A CHARGED PARTICLE IN A UNIFORM MAGNETIC FIELD
• We previously found that the magnetic force acting on a charged particle moving in a
magnetic field is perpendicular to the velocity of the particle, and consequently the work done
on the particle by the magnetic force is zero.
• Let us now consider the special case of a positively charged particle moving in a uniform
magnetic field with the initial velocity vector perpendicular to the field vector B . Let us assume
that the direction of the magnetic field is into the page. Figure 56 shows that the particle moves
in a circle in a plane perpendicular to the magnetic field.
• The particle moves in this way because the magnetic force F is at right angles to both v and

B has a constant magnitude |q|vB (sinφ = 1). As the force deflects the particle, the directions of
v and F change continuously, as shown in Figure 56.

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• Because F always points toward the center of the circle, it changes only the direction of v
and not its magnitude. As Figure 56 illustrates, the rotation is counterclockwise for a positive
charge. If q were negative, the rotation would be clockwise.

• Consequently, a charged particle
moving in a plane perpendicular to a
constant magnetic field will move in a
circular orbit with the magnetic force
playing the role of centripetal force. The
direction of the force is given by the
right-hand rule.
• Equating the centripetal force with the
magnetic force and solving for R, the
radius of the circular path, we get
mv2/R = |q|vB and

R = mv/|q|B

(87)

Figure 56 Motion of a charged particle in a constant

(uniform) magnetic field.

Example: (a) A proton is moving in a circular orbit of radius 14 cm in a uniform 0.35-T
magnetic field perpendicular to the velocity of the proton. Find the linear speed of the proton.
(Ans. v = 4.7 x 106 m/s)
(b) If an electron moves in a direction perpendicular to the same magnetic field with
this same linear speed, what is the radius of its circular orbit? (Ans. R = 7.6 x 10-5 m)
Solution
(a) Because magnetic field is perpendicular to the velocity of the proton, we can use (87)
v = R|q|B/m; plugging numbers leads to v = 4.7 x 106 m/s (qp = 1.60 x 10 -19 C and
mp = 1.67 x 10 -27 kg).

(b) Again using (87), we have R = 7.6 x 10-5 m (qe = –1.60 x 10-19 C and me = 9.1 x 10-31
kg).
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♦ MAGNETIC FORCE ACTING ON A CURRENT - CARRYING CONDUCTOR
• If a magnetic force is exerted on a single charged particle when the particle moves in a
magnetic field, it follows that a current-carrying wire also experiences a force when placed in a
magnetic field. This follows from the fact that the electric current is a collection of many
charged particles in motion; hence, the resultant force exerted by the field on the wire is the
vector sum of the individual forces exerted on all charged particles making up the current.

• Similar to the force on a moving
charge in a B field, a conductor of
length l carrying an electric current of
I in a B field experiences a magnetic
force given by:
F = Il xB

(88)

where I = J . A , according to
Equation 83.
• From (88), we have the formula for
the magnitude of the force given by
F = BIlsinø (88’)
where ø is the smaller angle

between the magnetic field and the
electric current direction.

Figure 57 Magnetic force on a current-carrying

conductor placed in a uniform magnetic field.
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♦ Magnetic force between two parallel electric current - carrying wires

Figure 58 Magnetic interaction between two parallel electric
current - carrying wires.
• Consider two long, straight, parallel wires separated by a distance a and carrying currents I1
and I2 in the same direction, as illustrated by Figure 58. We can determine the force exerted on
one wire due to the magnetic field set up by the other wire. Wire 1, which carries a current I1,
creates a magnetic field B1 at the location of wire 2. The direction of B1 is perpendicular to wire
2, as shown in Figure 58. According to Equation 88, the magnetic force on a length l of wire 2
is F21 = I 2l x B1 . Because l is perpendicular to B1 in this situation, the magnitude of F21 is
F21 = I2 l B1. Since the magnitude of B1 is given by B1 =

F12 = F21 = I2 l (

µ0 I1
, we have
2 a

à0 I1

àII
)= 0 1 2l
2 a
2 a

(89)

ã The direction of F21 is toward wire 1 because l x B1 is in that direction. If the field set up at
wire 1 by wire 2 is calculated, the force F12 acting on wire 1 is found to be equal in magnitude
and opposite in direction to F21 . This is what we expect because Newton’s third law must be
obeyed.
• When the currents are in opposite directions (that is, when one of the currents is reversed in
Fig 58), the forces are reversed and the wires repel each other. Hence, we find that parallel
straight conductors carrying currents in the same direction attract each other, and parallel
straight conductors carrying currents in opposite directions repel each other.
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• Because the magnitudes of the forces are the same on both wires, we denote the magnitude of
the magnetic force between the wires as simply FB. We can then rewrite this magnitude in terms
of the magnetic force per unit length:
FB µ0 I1 I 2
=
l
2π a

(90)

• The SI unit of FB is the newton (N), and that of FB/l is the newton per meter (N/m).

Example: A horizontal conductor is carrying 5.0 A of current to the east. A magnetic field
of 0.20 T pointing straight up cuts across 1.5 m of the conductor. Determine the force acting
on the conductor (magnitude and direction).
Solution
Because magnetic field is perpendicular to the velocity of the proton, we have
ø = 90º and from (88’)
F = BIlsinø = (0.20 T)(5.0 A)(1.5 m)sin90º = 1.5 N.
Using the three-finger rule, we find that the force points south.

4.3.2. Ampère’s law
• The magnetic field in space around an electric current is proportional to the intensity of the

electric current which serves as its source, just as the electric field in space is proportional to the
charge which serves as its source. Ampère’s law states that for any closed loop path, the sum of
the length elements times the magnetic field in the direction of the length element is equal to
the permeability times the intensity of the electric current enclosed in the loop (as expressed by
Equation 91).

(91)

• Oersted’s 1819 discovery about deflected compass needles demonstrates that a currentcarrying conductor produces a magnetic field. Figure 59a shows how this effect can be
Physic 1 Module 4: Electricity and magnetism

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demonstrated in the classroom. Several compass needles are placed in a horizontal plane near a
long vertical wire. When no current is present in the wire, all the needles point in the same
direction (that of the Earth’s magnetic field), as expected.
• When the wire carries a strong, steady current, all needles deflect in a direction tangent to a

circle, as shown in Figure 59b. These observations demonstrate that the direction of the

magnetic field produced by the current in the wire is consistent with the right-hand rule
described in Figure 30.3 (see Halliday’s book, page 941).
• When the current is reversed, the needles in Figure 59b also reverse. Because the compass
needles point in the direction of B , we conclude that the lines of B form circles around the
wire, as discussed in the preceding section. By symmetry, the magnitude of B is the same
everywhere on a circular path centered on the wire and lying in a plane perpendicular to the
wire. By varying the current intensity and distance a from the wire, we find that B is

proportional to the current intensity and inversely proportional to the distance from the wire, as
described by the following equation:

B=

µ0 I
2π a

(92)

• Now let us evaluate the dot product B . d s for a small length element ds on the circular path
defined by the compass needles (see Figure 59b) and sum the products for all elements over the
closed circular path. Along this path, the vectors d s and B are parallel at each point (see Fig.
59b), so B . d s = Bds. Furthermore, the magnitude B is constant on this circle and is given by
Equation 92. Therefore, the sum of the products B . d s over the closed path, which is equivalent
to the line integral of B . d s , is

µ0 I

∫ B.ds = B ∫ ds = 2πa (2πa) = µ I

0

where

(93)

∫ ds = 2πa is the circumference of the circular path. Although this result was calculated

for the special case of a circular path surrounding a wire, it holds for a closed path of any shape
surrounding an electric current that exists in an unbroken circuit.

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• As a result, in general Ampère’s law can be stated as follow:
The line integral of B . d s around any closed path equals µI, where I is the total
intensity of all continuous currents passing through any surface bounded by the closed path.

∫ B.ds = µI

(93’)

where µ is the magnetic permeability of the medium.
• Ampère’s law describes the creation of magnetic fields by all continuous current
configurations, but at our mathematical level it is useful only for calculating the magnetic field
of current configurations having a high degree of symmetry.

Figure 59 (a) When no electric current is present in the wire,

all compass needles point in the same direction (toward the
Earth’s north pole).
(b) When the wire carries a strong current, the
compass needles deflect in a direction tangent to a circle, which
is the direction of the magnetic field created by the current.

♦ Applications of Ampère’s law
1. Magnetic field created by an infinitely long straight wire carrying an electric
current

• The magnetic field lines around a long wire which carries an electric current form concentric
circles around the wire. The direction of the magnetic field is perpendicular to the wire and is in

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the direction the fingers of your right hand would curl if you wrapped them around the wire with
your thumb in the direction of the current (see Figure 60).
• The magnitude of the magnetic field vector B produced by a current-carrying straight wire
depends on the intensity of the current and is inversely proportional to the distance from the
wire, as given by Equation 92.
Magnetic field created by an infinitely long straight wire carrying an
electric current

• The magnetic field of an infinitely long
straight wire can be obtained by applying
Ampere's law. The expression for the magnitude
of magnetic field vector is

(94)
where r is the distance from the point of interest
to the wire and µ0 the magnetic permeability of
free space.

Figure 60 Depicting the magnetic field created by an infinitely long straight
wire carrying an electric current.

Example: Find the electric current intensity in a straight conducting wire (placed in the
air) that produces a magnetic field of 5.00 x 10-7 T at a distance of 10.0 cm from the
wire.
Solution

Because the wire is placed in the air, we can use (94)

I = B2πr/µ . Plugging numbers
0

leads to I = 5.00 x 10-7 T x [2π x 0.100 m]/4πx10-7 T.m/A = 2.5 x 102 A.
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2. Magnetic field created by a long straight coil of wire (solenoid) carrying an
electric current

• A long straight coil of wire can be used to generate a nearly uniform magnetic field similar to
that of a bar magnet. Such coils, called solenoids, have an enormous number of practical

applications. The field can be greatly strengthened by the addition of an iron core. Such cores
are typical in electromagnets.

• In Equation 95 for the magnetic field B inside a solenoid carrying an electric current, n is the
number of turns per unit length, sometimes called the "turn density". The expression is an
idealization to an infinitely long solenoid, but provides a good approximation to the magnetic
field created by a long solenoid carrying an electric current.

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Solenoid field from Ampère’s law

• Taking a rectangular path about which to evaluate ∫ B.ds such that the length of the
side parallel to the solenoid field is L gives a contribution BL inside the coil. The
field is essentially perpendicular to the other sides of the path, giving negligible
contribution. If the end is taken so far from the coil that the field is negligible, then
the length inside the coil is the dominant contribution.
• This admittedly idealized case
for Ampère’s law gives

(95)

• This turns out to be a good a
proximation for the solenoid
field, particularly in the case of
an iron core solenoid.
Figure 61 Magnetic field created by a long straight coil of wire (solenoid)

carrying an electric current.

Example: An air-core solenoid is 100 cm long and has 3000 turns of copper wire. It

carries an electric current of 4 A. Find the magnetic field inside the solenoid.
Solution

The core is air

µ = µ0. Using (95), we have B = µnI = µ0 (N/L)I. Plugging numbers

leads to B = 1.51x10-2 T. Recalling

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3. Magnetic field created by a toroid carrying an electric current

• A device called a toroid (see Figure 62) is often used to create a magnetic field with almost
uniform magnitude in some enclosed area. The device consists of a conducting wire wrapped
around a ring (a torus) made of a nonconducting material. For a toroid having N closely spaced
turns of wire, we calculate the magnetic field in the region occupied by the torus, a distance r
from the center.
• To calculate this field, we must evaluate ∫ B.ds over the circle of radius r, as shown in Figure
62. By symmetry, we see that the field has a constant magnitude on this circle and is tangent to
it, so B . d s = Bds. Furthermore, note that the closed circular path surrounds N loops of wire,
each of which carries a current I. Therefore, the right side of Equation 93 is µ0NI in this case.

ã Ampốres law applied to the circle gives

B=

à0 NI
2 r

(96)

ã This result shows that B varies as 1/r and hence is nonuniform in the region occupied by the
torus. However, if r is very large compared with the cross-sectional radius of the torus, then the
field is approximately uniform inside the torus.
• For an ideal toroid, in which the turns are closely spaced, the external magnetic field is zero.
This can be seen by noting that the net current passing through any circular path lying outside
the toroid (including the region of the “hole in the doughnut”) is zero. Therefore, from Ampère’s
law we find that B = 0 in the regions exterior to the torus.

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