MINISTRY OF EDUCATION AND TRAINING
NONG LAM UNIVERSITY
FACULTY OF FOOD SCIENCE AND TECHNOLOGY
Course: Physics 1
Module 1: Electricity and Magnetism
Instructor: Dr. Son Thanh Nguyen
Academic year: 2008-2009
Contents
Module 4: Electricity and magnetism
4.1. Electromagnetic concepts and law of conservation of electric charge
1. Electromagnetic concepts
2. Law of conservation of electric charge
4.2. Electric current
1. Electric current
2. Current density
4.3. Magnetic interaction - Ampère’s law
1. Magnetic interaction
2. Ampère’s law for the magnetic field
4.4. Magnetic intensity
1. Magnetic intensity
2. Relationship between magnetic intensity and magnetic induction
4.5. Electromagnetic induction
1. Magnetic flux
2. Faraday’s law of induction
4.6. Magnetic energy.
1. Energy stored in a magnetic field
2. Magnetic energy density
Physic 1 Module 4: Electricity and magnetism
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4.1. Electromagnetic concepts and law of conservation of electric charge
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.
• The magnetic field forms one aspect of electromagnetism. 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, the electric and magnetic fields make up the
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 our 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 fields is called
magnetostatics, or static magnetic fields. 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
charges, like magnetic poles repel each other, while unlike poles attract.
• 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.
Physic 1 Module 4: Electricity and magnetism
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• Electric fields come from 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 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 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 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.
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 on the electron or the proton.
These things have the same size charge, but the sign is different. A proton has a charge of +e,
while an electron has a charge of -e. The amount of electric charge 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
Physic 1 Module 4: Electricity and magnetism
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(79)
q is the symbol used to represent charge, while n is a positive or negative integer (n = 0, ±1, ±2,
±3, …), and e is the electronic charge, 1.60 x 10-19 coulombs.
• Table of elementary particle masses and charges:
♦ The law of conservation of 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, charge conservation is the principle that electric charge 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 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 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.
• 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, plastic 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
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.
Physic 1 Module 4: Electricity and magnetism
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• 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
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 same 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, qin 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.
• The SI unit of electric charge is the coulomb (C).
4.2. Electric current
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.
Physic 1 Module 4: Electricity and magnetism
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• Whenever there is a net flow of charge through some region, a 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 ΔQ is the amount of charge that passes
through this area in a time interval Δt, the average current
intensity IAV is equal to the charge that passes through A per unit
time:
IAV = ΔQ/ Δt
(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
Δt → 0
ΔQ dQ
=
Δt
dt
(82)
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.
• 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 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 a 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.
♦ 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 current direction the same direction as the flow of
positive charge. In electrical conductors, such as copper or aluminum, the current is due to the
motion of negatively charged electrons. Therefore, when we speak of current in an ordinary
conductor, the direction of the current is opposite to that of 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 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 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.
Physic 1 Module 4: Electricity and magnetism
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If the current is due to the motion of negative charges, the current arrow is antiparallel to
the charge velocity.
2. Electric current density
G
• 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.
G
• In other words, J is a vector quantity, and the scalar product of which with the cross-sectional
G
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 current density is constant then
G G
I= J.A
(83)
G
G
(scalar product of J and A ).
If the current density is not constant, then
I=
G G
J
∫ .dA
(84)
where the current is in fact the integral of the dot product of the
G
G
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 from free air to 6 A/mm2 for a
wire at free air.
Example: During 4.0 minutes a 5.0-A current is set up in a wire,
find (a) charge quantity in coulombs and (b) number of electrons
passing through any cross section across the wire’s width.
(Ans. (a) 1.2 x 103 C; (b) 7.5 x 1021)
Physic 1 Module 4: Electricity and magnetism
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4.3. Magnetic interaction - Ampère’s law
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.
Opposite 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).
Physic 1 Module 4: Electricity and magnetism
9
♦ Between an electric currennt and a compass
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 current gives rise
to magnetic fields, just as electric charge gave
rise to electric fields.
Figure 54: Compass near a current-carrying
wire.
♦ Magnetic force acting on a moving charge
G
G
• A charged particle q when moving with velocity v in a magnetic field B experiences a
G
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.
G
G
• The magnitude and direction of F depend onGthe 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.
G
• When the particle’sGvelocity vector v makes any angle φ ≠ 0 with theG magnetic
G
G
field B , the magnetic force F acts in a direction perpendicular to both v and B ; that
G
G
G
is, F is perpendicular to the plane formed by v and B (see Figure 55).
Physic 1 Module 4: Electricity and magnetism
10
G
• Mathematiclly the force F is given by
G
G G
F = qv x B
(85)
G
G G
where the direction of F is in the direction of v x B if q is positive, which by definition of the
G
G
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)
G
G
where φ is the smaller angle between v and B . From this expression, we see that F is zero when
G
G
G
v is parallel or antiparallel to B (φ = 0 or 180°) and maximum, Fmax = |q|vB, when v is
G
perpendicular to B (φ = 90°).
The direction of the cross product can
be obtained by using a right-hand rule:
the index finger of the right hand points
G
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
G
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.
Physic 1 Module 4: Electricity and magnetism
11
• Let us now consider the special case of a positively charged particle moving in a uniform
magnetic field with the initial velocity vector of the particle perpendicular to the field. 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.
G
G
•G The particle moves in this way because the magnetic force F is at right angles to both v and
B and hasG a constant magnitude qvB (sinφ = 1). As the force deflects the particle, the directions
G
of v and F change continuously, as shown in Figure 56.
G
G
• 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
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
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)
♦ 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
Physic 1 Module 4: Electricity and magnetism
12
magnetic field. This follows from the fact that the 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 the charged particles making up the current.
• Similar to the force on a moving
G
charge in a B field, we have
for a conductor of length l carrying a
G
current of intensity I in a B field the
force experienced by the conductor:
G G
G
F = Il xB
(88)
G G
where I = J . A , according to
equation (83).
Figure 57: Magnetic force on a moving charge in a
current-carrying conductor.
♦ Magnetic force between two parallel current carrying wires
Figure 58: Magnetic interaction between two parallel current
carrying wires.
Physic 1 Module 4: Electricity and magnetism
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• 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,
G
G
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
G G
G
G
G
G
is F21 = I 2l x B1 . Because l is perpendicular to B1 in this situation, the magnitude of F21 is
G
μI
F21 = I2 l B1. Since the magnitude of B1 is given by B1 = 0 1 , we have
2π a
F21 = I2 l (
μ0 I1
μII
)= 0 1 2l
2π a
2π a
(89)
G G
G
• The direction of F21 is toward wire 1 because l x B1 is in that direction. If the field set up at
G
wire 1 by wire 2 is calculated, the force F12 acting on wire 1 is found to be equal in magnitude
G
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 56), 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.
• 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 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).
2. Ampère’s law
• The magnetic field in space around an electric current is proportional to 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 electric current enclosed in the loop (as expressed by equation 91).
(91)
Physic 1 Module 4: Electricity and magnetism
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• 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
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, the needles all deflect in a direction tangent to
the 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
G
G
needles point in the direction of B , we conclude that the lines of B form circles around the
G
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)
G G
• 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
G
G
closed circular path. Along this path, the vectors d s and B are parallel at each point (see Fig.
G G
59b), so B . d s = B ds. Furthermore, the magnitude B is constant on this circle and is given by
G G
equation (92). Therefore, the sum of the products B . d s over the closed path, which is
G G
equivalent to the line integral of B . d s , is
G G
μ0 I
v∫ B.ds = B v∫ ds = 2π a (2π a) = μ I
where
0
(93)
v∫ ds = 2π a 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 a current that exists in an unbroken circuit.
• As a result, the general case, known as Ampère’s law, can be also stated as follows:
G G
The line integral of B . d s around any closed path equals μ0I, where I is the total
continuous current passing through any surface bounded by the closed path.
G G
B
v∫ .ds = μ0 I
(94)
• 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.
Physic 1 Module 4: Electricity and magnetism
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Figure 59: (a) When no 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 the 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
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 58).
G
• The magnitude of the magnetic field vector B produced by a current-carrying straight wire
depends on the intensity of the current. It is also inversely proportional to the distance from the
wire, as given by equation (92).
Physic 1 Module 4: Electricity and magnetism
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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 magnetic field vector is
where r is the distance from the point of
interest to the wire. and μ0 the permeability of
free space
Figure 60: Depicting the magnetic field created by an infinitely long straight
wire carrying an electric current.
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.
Physic 1 Module 4: Electricity and magnetism
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G
• 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 "turns density". The expression is an
idealization to an infinite length solenoid, but provides a good approximation to the field of a
long solenoid.
Solenoid field from Ampère’s law
• Taking a rectangular path about which to evaluate Ampere's law 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 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 approximation 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.
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
G G
v∫ B.ds over the circle of radius r, as shown in Figure
62. By symmetry, we see that the magnitude of the field is constant on this circle and tangent to
G G
it, so B . d s = B ds. Furthermore, note that the circular closed 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.
• Ampère’s law applied to the circle gives
Physic 1 Module 4: Electricity and magnetism
18
B=
μ0 NI
2π r
(96)
• 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.
• Finding the magnetic field inside a
toroid is a good example of the power
of Ampère’s law. The current enclosed
by the dashed line is just the number of
loops times the current in each loop.
Ampere’s law then gives the magnetic
field by
(96)
• The toroid is a useful device used in
everything from tape heads to
tokamaks.
Figure 62: Magnetic field created by a toroid
carrying an electric current.
Magnetic field created by a toroid carrying an electric current = permeability x turn density x
current.
4.4. Magnetic field intensity or magnetic field strength
G
G
G
• There are two vectors namely B and H characterizing a magnetic field. The vector field B is
known among electrical engineers as magnetic flux density or magnetic induction, or simply
G
magnetic field, as used by physicists. The vector field H is known among electrical engineers as
the magnetic field intensity or magnetic field strength and is also known among physicists as
auxiliary magnetic field or magnetizing field.
Physic 1 Module 4: Electricity and magnetism
19
G
• The magnetic field B has the SI unit of teslas (T), equivalent to webers per square meter
G
(Wb/m²). The vector field H is measured in amperes per meter (A/m) in the SI units. An older
unit of magnetic field strength is the oersted: 1 A/m = 0.01257 oersted.
• The magnetic fields generated by currents and calculated from Ampere's law are characterized
G
by the magnetic field B measured in teslas. However, when the generated fields pass through
magnetic materials which themselves contribute internal magnetic fields, ambiguities can arise
about what part of the field comes from the external currents and what comes from the material
itself. It has been common practice to define another magnetic field quantity, usually called the
G
"magnetic field strength" and designated by H .
• The commonly used form for the relationship between B and H is
B = μH
(97)
where μ is the permeability of the medium and given by
μ = Kmμ0
(98)
μ0 being the magnetic permeability of free space and Km the relative permeability of the
material. If the material does not respond to the external magnetic field by producing any
magnetization, then Km = 1.
• For paramagnetic (μ > μ0) and diamagnetic (μ < μ0) materials, the relative permeability is very
close to 1. For ferromagnetic materials, μ is much greater than μ0.
4.5. Electromagnetic induction
1. Magnetic flux
• The magnetic flux, ΦB, through an element of area perpendicular to the direction of magnetic
field is given by the product of the magnetic field and the area element. More generally,
magnetic flux is defined by a scalar product of the magnetic field vector and the area element
vector. The SI unit of magnetic flux is the weber (Wb).
• The magnetic flux through a surface is proportional to the number of magnetic field lines that
pass through the surface. This is the net number, i.e., the
number passing through in one direction minus the number
passing through in the opposite direction.
• As illustrated by Figure 63, we divide the surface that has
the loop as its border into small elements of area dA. For
each element we calculate the differential magnetic flux of
G
the magnetic field B through it:
Figure 63: Depicting of the
magnetic.
Physic 1 Module 4: Electricity and magnetism
G G
dΦB = B.d A = B.dA.cosφ
20
(99)
G
G
where φ is the angle between the normal vector n̂ ( d A = n̂ dA) and the magnetic field vector B
at the position of the element.
• We then integrate all the terms
G G
ΦB = ∫ B.dA.cosφ = ∫ B.dA
(100)
2. Faraday’s law of induction
♦ Faraday's experiments
• These experiments helped formulate what is known as "Faraday's law of induction."
• The circuit shown in the left panel of Figure 64 consists of a wire loop connected to a
sensitive ammeter (known as a "galvanometer"). If we approach the loop with a permanent
magnet, we see a current being registered by the galvanometer. The results can be summarized
as follows:
i. A current appears only if there is relative motion between the magnet and the loop.
ii. Faster motion results in a larger current intensity.
iii. If we reverse the direction of motion or the polarity of the magnet, the current
reverses sign and flows in the
opposite direction.
• The current generated is known
as "induced current"; the
electromotive force (emf) that
appears is known as "induced
emf"; the whole effect is called
"induction."
Figure 64: Faraday’s experiments of induction;
(Left) A permanent magnet approaching a loop.
(Right) Switching the current in one loop induces
a current in another loop.
• In the right panel of Figure 64,
we show a second type of
experiment in which current is
induced in loop 2 when the switch
S in loop 1 is either closed or
opened. When the current in loop
1 is constant, no induced current
is observed in loop 2.
• We see that the magnetic field in an induction experiment can be generated either by a
permanent magnet or by an electric current in a coil.
• Faraday summarized the results of his experiments in what is known as Faraday's law of
induction.
An emf is induced in a loop when the number of magnetic field lines (or magnetic
flux) that pass through the loop is changing.
Physic 1 Module 4: Electricity and magnetism
21
• We can also express Faraday's law of induction in the following form:
The magnitude of the emf induced in a conductive loop is equal to the rate at which
the magnetic flux ΦB through the loop changes with time.
• The corresponding formula is
ε=-
dΦB
dt
(101)
where ε is the induced emf.
If the circuit is a coil consisting of N loops of the same area and if ΦB is the flux through
one loop, an emf is induced in every loop; thus, the total induced emf in the coil is given by the
expression
dΦB
ε = -N
(102)
dt
• The negative sign in equations (101) and (102) is of important physical significance, as
described later.
• The SI unif of emf is the volt (V).
♦ Methods for changing the magnetic flux ΦB through a loop
• We see that the magnetic flux ΦB can be changed and an emf is then induced in a circuit in
several ways:
G
• The magnitude of B can change with time.
• The area enclosed by the loop can change with time.
G
• The angle φ between the magnetic field vector B and the normal vector n̂ to the loop
can change with time.
• Any combination of the above can be used.
♦ Lenz’s law
• Faraday’s law of induction (equation 101 or equation 102) indicates that the induced emf and
the change in flux have opposite algebraic signs. This has a very real physical interpretation that
has come to be known as Lenz’s law:
The polarity of the induced emf is such that it tends to produce a current that
creates a magnetic flux to oppose the change in magnetic flux through the area
enclosed by the current loop.
• That is, the induced current tends to keep the original magnetic flux through the circuit from
changing. This law is actually a consequence of the law of
conservation of energy.
• We now concentrate on the negative sign in the equation that
expresses Faraday's law. The direction of the flow of induced
current in a loop is accurately predicted by what is known as
Lenz's law (or Lenz's rule).
Physic 1 Module 4: Electricity and magnetism
22
Figure 65: Depicting Lenz’s law.
• To understand Lenz’s law, we consider an example as shown in Figure 65. In the figure we
show a bar magnet approaching a loop. The induced current flows in the direction indicated
because this current generates an induced magnetic field that has the field lines pointing from
left to right. The loop is then equivalent to a magnet whose north pole faces the corresponding
north pole of the bar magnet that is approaching the loop. The loop then repels the approaching
magnet and thus opposes the change in the original magnetic flux that generated the induced
current.
Example: A coil consists of 200 turns of wire having a total resistance of 2.0 Ω. Each
turn is a square of side 18 cm, and a uniform magnetic field directed perpendicular to the plane
of the coil is turned on. If the field changes linearly from 0 to 0.50 T in 0.80 s,
(a) what is the magnitude of the induced emf in the coil while the field is changing? and
(b) what is the magnitude (intensity) of the induced current in the coil while the field is
changing?
(Ans. (a) |ε| = 4.1 V; (b) I = |ε|/R = 2.05 A)
♦ MOTIONAL ELECTROMOTIVE FORCE
• In examples illustrated by Figure 63, we considered cases in which an emf is induced in a
stationary circuit placed in a magnetic field when the field changes with time. In this section we
describe what is called motional electromotive force, which is the emf induced in a straight
conductor moving through a constant magnetic field.
• Consider a loop of width l shown in Figure 66. Part of the loop is located in a region where a
uniform magnetic field exists. The loop is being pulled outside the magnetic field region with
constant speed v. The magnetic flux through the loop is
ΦB = Blx. This flux decreases with time; according to
Faraday’s law, there is an induced emf given by
ε=-
dΦB
dx
= -Bl = -Blv
dt
dt
(103)
• Because the resistance of the circuit
is R, the intensity (magnitude) of the
induced current in the loop is
Figure 66: Depicting the
motional electromotive force.
I = |ε| = Blv/R
(104)
♦ Self – induction
• If we change the current i through an inductor whose inductance is L,
this causes a change in the magnetic flux ΦB = Li through the inductor
itself. Using Faraday's law we can determine the resulting emf known as
self-induced emf εL
Physic 1 Module 4: Electricity and magnetism
23
Figure 67: Depicting the
self-induction.
dΦB
di
= -L
dt
dt
εL = -
(105)
We have assumed that L is constant.
• If the inductor is an ideal solenoid of cross-sectional area A with N turns, its inductance is
given by
L = μ0(N2/l)A = μ0n2Al
(106)
where μ0 is the permeability of free space, and n = N/l is the number of turns per unit length or
the turn density of the solenoid.
• The permeability may be changed by putting a soft iron core into the solenoid, greatly
increasing the inductance of the solenoid. In this case we must replace μ0 by μ = Kmμ0 where Km
is the relative permeability of the core; for iron Km is much greater than 1.
• The SI unit of L is the henry (H).
Example: (a) Calculate the inductance of an air-core solenoid containing 300 turns if the
length of the solenoid is 25.0 cm and its cross-sectional area is 4.0 cm2.
(b) Calculate the self-induced emf in the solenoid if the current through it is
decreasing at the rate of 50.0 A/s. (Ans. (a) 0.181 mH; (b) 9.05 mV)
4.6. Magnetic energy
1. Energy stored in a magnetic field
♦ RL circuit
Figure 68: A series RL circuit.
As the current intensity
increases toward its maximum
value, an emf that opposes the
increasing current is induced
in the inductor.
• Consider a series RL circuit as shown in Figure 68. When the
switch S is closed, the current immediately starts to increase.
The induced emf (or back emf) in the inductor is large, as the
current is changing rapidly. As time goes on, the current
increases more slowly, and the potential difference across the
inductor decreases.
• It takes energy to establish a current in an inductor; this
energy is carried by the magnetic field inside the inductor.
• Considering the emf needed to establish a particular current
and the power involved, we find:
• As the current intensity through the coil increases, the magnetic field of the coil also
increases and electrical energy is stored in the coil as a magnetic field. The magnetic energy UB
stored in the coil is given by
UB =
1 2
LI
2
Physic 1 Module 4: Electricity and magnetism
24
(107)
• In capacitors we found that energy is stored in the electric field between their plates. In
inductors, energy is similarly stored, only now in the magnetic field. Just as with capacitors,
where the electric field is created by a charge on the capacitor and electric energy is stored
inside the capacitors, we now have a magnetic field created when there is a current through the
inductor. Thus, just as with the capacitors, the magnetic energy is stored inside the inductor.
• Again, although we introduce the magnetic field energy when talking about energy in
inductors, it is a generic concept – whenever a magnetic field is created, it takes energy to do so,
and that energy is stored in the field itself.
• The SI unit of magnetic energy is the joule (J).
2. Magnetic energy density
• For simplicity, consider an ideal solenoid whose inductance is given by
L = μo(N2/l)A = μon2Al
• The magnetic field inside a solenoid is given by B = μonI. As a result I = B/μon
• Substituting the expressions for L and for I into equation (107) leads to
UB =
B2
Al
2 μ0
(108)
• Because Al = V is the volume of the solenoid, the energy stored per unit volume in the
magnetic field or the magnetic energy density, uB = UB/V, inside the inductor is
(109)
• Although this expression was derived for the special case of a solenoid, it is valid for any
region of space in which a magnetic field exists regardless of its source. From equation (109),
we see that magnetic energy density is proportional to the square of the square of the field
magnitude.
• The SI unit of magnetic energy density is the joule per cubic meter (J/m3).
Example: The earth’s magnetic field in a certain region has the magnitude 6.0 x 10-5 T.
Find the magnetic energy density in this region. (Ans. 1.4 x 10-3 J/m3)
Physic 1 Module 4: Electricity and magnetism
25
REFERENCES
1) Halliday, David; Resnick, Robert; Walker, Jearl. (1999) Fundamentals of Physics 7th ed.
John Wiley & Sons, Inc.
2) Feynman, Richard; Leighton, Robert; Sands, Matthew. (1989) Feynman Lectures on Physics.
Addison-Wesley Publishing Company.
3) Serway, Raymond; Faughn, Jerry. (2003) College Physics 7th ed. Thompson, Brooks/Cole.
4) Sears, Francis; Zemansky Mark; Young, Hugh. (1991) College Physics 7th ed. AddisonWesley Publishing Company.
5) Beiser, Arthur. (1992) Physics 5th ed. Addison-Wesley Publishing Company.
6) Jones, Edwin; Childers, Richard. (1992) Contemporary College Physics 7th ed. AddisonWesley Publishing Company.
7) Alonso, Marcelo; Finn, Edward. (1972) Physics 7th ed. Addison-Wesley Publishing
Company.
8) Michels, Walter; Correll, Malcom; Patterson, A. L. (1968) Foundations of Physics 7th ed.
Addison-Wesley Publishing Company.
9) Hecht, Eugene. (1987) Optics 2th ed. Addison-Wesley Publishing Company.
10) Eisberg, R. M. (1961) Modern Physics, John Wiley & Sons, Inc.
11) WEBSITES
http://ocw.mit.edu/OcwWeb/Physics/8-02TSpring-2005/LectureNotes/index.htm
http://physics.bu.edu/~duffy/PY106/Charge.html
http://science.jrank.org/pages/1729/Conservation-Laws-Conservation-electric-charge.html
http://web.pdx.edu/~bseipel/ch31.pdf
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html#c1
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcon.html#c1
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