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Physics 12 Lesson Notes: Natural Magnetism and Electromagnetism
Magnets
When a bar magnet is dipped into iron filings, the filings are attracted to it, accumulating
most noticeably around regions at each end of the magnet the poles. When the bar magnet is
allowed to rotate freely the pole that tends to seek the northerly direction is called the northseeking pole, or simply, the N-pole. The other is called the south-seeking pole, or S-pole.
By placing two bar magnets first with similar poles together, then with opposite poles
together, you can demonstrate the law of magnetic poles (Figure 1):
Magnetic Fields
Since an iron filing experiences a force when placed near a magnet, then, by definition,
a magnet is surrounded by a magnetic force field. This field is often detected by its effect on a
small test compass (magnetized needle). It is visually depicted by drawing magnetic field lines
that show the direction in which the N-pole of the test compass points at all locations in the field.
Experimentally, the lines in a magnetic field can easily be traced by sprinkling iron filings on a
sheet of paper placed in the field. The filings behave like many tiny compasses and line up in the
direction of the field at all points. They produce a picture of the magnetic field, as shown in
Figure 2.
Since iron filings have no marked north or S-poles, they reveal only the pattern of the
magnetic field lines, not their direction (Figure 3). The relative strength of the magnetic field is
indicated by the spacing of adjacent field lines: where lines are close together, the magnetic field
is strong.
(a) Similar poles
other.
face each
(b) Opposite poles face each
other.
The magnetic field at any point is a vector quantity, represented by the symbol B. The
magnitude B is given by the magnitude of the torque (or turning action) on a small test compass
not aligned with the direction of the field. We will make a more precise definition of B later in
this chapter, when we examine electromagnetism.
The characteristics of magnetic field lines are summarized below.
1. The spacing of the lines indicates the relative strength of the force. The closer
together the lines are, the greater the force.
2. Outside a magnet, the lines are concentrated at the poles. They are closest
within the magnet itself.
3. By convention, the lines proceed from S to N inside a magnet and from N to
S outside a magnet, forming closed loops. (A plotting compass indicates these directions.)
4. The lines do not cross one another.
Note that the magnetic field around a bar
magnet is three-dimensional in the diagram shown;
it does not exist just in the horizontal plane.
Earth’s Magnetic Field
A pivoted magnet will rotate and point north south because of its interaction with the
magnetic field of Earth. As early as the 16th century, Sir William Gilbert, the distinguished
English physicist, had devised a model to describe Earth s magnetism. He determined that
Earth s magnetic field resembled the field of a large bar magnet, inclined at a slight angle to
Earth s axis, with its S-pole in the northern hemisphere. Figure 4(a) shows this field and the bar
magnet that was thought, in Gilbert s time, to be responsible for it.
(a) The magnetic field of Earth
closely resembles the field of a large
bar magnet.
(b) Lines of magnetic declination in
Canada
A compass points toward Earth s magnetic S-pole, rather than toward its geographic north
pole (the north end of Earth s axis of rotation). The angle, or magnetic declination, between
magnetic north and geographic north varies from position to position on the surface of Earth
(Figure 4(b)). In navigating by compass, the angle of declination for a particular location must
be known so that true north can be determined.
In addition, Earth s magnetic field is three-dimensional, with both a horizontal and a
vertical component. A magnetic compass on a horizontal surface reveals only the horizontal
component. The angle between Earth s magnetic field, at any point, and the horizontal is called
the magnetic inclination, or dip, and is measured with a magnetic dipping needle (Figure 5).
A dipping needle is a compass pivoted
at its centre of gravity and free to rotate in a vertical plane. When
aligned with a horizontal compass pointing north, it points in the direction of
Earth s magnetic field. The angle of inclination is then read directly from the
attached protractor.
Inclination and declination charts must be revised from time to time because Earth s
magnetic field is slowly changing. It is believed that these changes result from the rotation of the
magnetic field about Earth s axis; one complete rotation takes about 1000 years (Figure 6).
The Domain Theory of Magnetism
Although not normally magnetized, some ferromagnetic materials, such as iron,
nickel, cobalt, and gadolinium, may become magnetized under certain circumstances. How they
are able to acquire magnetic properties may be explained by the domain theory of magnetism.
Ferromagnetic substances are composed of a large number of tiny regions called
magnetic domains. Each domain behaves like a tiny bar magnet, with its own N- and S-poles. When
a specimen of the material is unmagnetized, these millions of domains are oriented at random, with
their magnetic effects canceling each other out, as in Figure 7.
However, if a piece of ferromagnetic material is placed in a sufficiently strong
magnetic field, some domains rotate to align with the external field, while others, already aligned,
tend to increase in size at the expense of neighboring nonaligned domains (Figure 8). The net
result is a preferred orientation of the domains (in the same direction as the external field),
causing the material to behave like a magnet. When the external field is removed, this orientation
will either remain for a long time or disappear almost immediately, depending on the material.
When magnets are made in this way, they are known as induced magnets.
Figure 7
The atomic dipoles are lined up in
each domain. The domains point in
random directions. The magnetic
material is unmagnetized.
Figure 8
The atomic dipoles (not the domains) turn so that all
domains point in the direction of the magnetizing field.
The magnetic material is fully magnetized.
The domain model provides a simple explanation for many properties of induced magnets:
1. A needle is magnetized by rubbing it in one direction with a strong permanent magnet. This
aligns the domains with the field of the permanent magnet.
2. When a bar magnet is broken in two, two smaller magnets result, each with its own N- and Spoles. It is impossible to produce an isolated N- or S-pole by breaking a bar magnet.
3. Induced magnets made of soft iron demagnetize as soon as the external field is removed.
Examples include temporary magnets such as lifting electromagnets. In contrast, hard steel or
alloys remain magnetized indefinitely. These include permanent magnets such as magnetic
door catches. Impurities in the alloys seem to lock the aligned domains in place and
prevent them from relaxing to their random orientation.
4. Heating or dropping a magnet can cause it to lose its magnetization, jostling the domains
sufficiently to allow them to move and resume their random orientation. Each ferromagnetic
material has a critical temperature above which it becomes demagnetized and remains
demagnetized even upon cooling.
5. A strong external magnetic field can reverse the magnetism in a bar magnet, causing the
former south-seeking pole to become north-seeking. This occurs when the domains reverse
their direction of orientation by 180° due to the influence of the strong external field in the
opposite direction.
6. Ships hulls, columns and beams in buildings, and many other steel structures are often found
to be magnetized by the combined effects of Earth s magnetic field and the vibrations
imposed during construction. The effect is similar to stroking a needle with a strong magnet,
in that the domains within the metals are caused to line up with Earth s magnetic field.
Vibrations during construction aid in the realignment of the domains.
Example 1) Two iron nails are held to a magnet, as shown in Figure. Predict what will happen
when the nails are released. If possible, verify your prediction experimentally.
Example 2) In the diagrams below, each circle represents a compass. Show the direction of the
needle in each compass.
Example 3) In the diagrams below, draw the magnetic field lines between the ends of three bar
magnets if
(a) all the S-poles are close together
(b) one N-pole and two S-poles are close together
Small pieces of iron rubbed in one direction with lodestone become magnetized. Even
bringing a piece of iron near a magnet causes the iron to be magnetized. Nickel and cobalt, and
any alloy containing nickel, cobalt, or iron, behave in the same way. These substances are called
ferromagnetic, and you can induce them to become magnetized by placing them in a magnetic
field.
Oersted’s Discovery
For centuries, people believed that electricity and magnetism were somehow related, but
no one could prove a connecting link between them. Then, in 1819, the Danish physicist Hans
Christian Oersted (1777 1851) discovered the connection by accident while lecturing on electric
circuits at the University of Copenhagen.Oersted noticed that a compass needle placed just below
a wire carrying a current would take up a position nearly perpendicular to the wire while the
current was flowing (Figure 1). When the direction of the current was reversed, the compass
needle again set itself at right angles to the wire, but with its ends reversed. The effect lasted only
while the current flowed. Much to his own surprise, Oersted had discovered the basic principle
of electromagnetism.
Magnetic Field of a Straight Conductor
When an electric current flows through a long, straight conductor, the resulting
magnetic field consists of field lines that are concentric circles, centred on the conductor
(Figure 9). You can remember the direction of these field lines (as indicated by the N-pole of a
small test compass) if you use the right-hand rule for a straight conductor:
(a) Iron filings reveal the circular pattern
of the magnetic field around a conductor
with a current.
(b) If the right thumb points in the
direction of the current, then
the fingers curl around the wire
in the direction of the magnetic
field lines.
Models of the magnetic field of a straight conductor
(a) Imagine the X as being the tail of an arrow moving away from you.
(b) Imagine the dot as being the tip of an arrow facing you.
Understanding Concept.
1) Refer to diagram on the right and fill in the needles by showing
which direction are they pointing if current is applied in the
circuit?
2) Figure shows three current-carrying conductors with their
magnetic fields. Indicate the direction of electric current in
each wire.
3) Figure shows three conductors with the direction of the electric current. Draw magnetic field
lines around each, indicating polarities where applicable.
4) Choose the diagram from below that best illustrates the strength of the magnetic field
surrounding a conductor. Explain your answer.
Determine the magnetic field around a current carrying straight wire
To determine the magnitude of the magnetic field strength, we use the formula
Where B is the magnetic field strength generated by the current.
μo is a constant, called the permeability of free space,
T m
μo = 4 π x 10 7
A
I is the current through the wire
R is the distance to the conductor.
Example 1)What is the magnetic field strength 15 cm from a straight conductor with a current of
60 A flowing through it?
Example 2) What current is flowing through a straight wire if the magnetic field strength 35 cm
from the wire is 6.4 x 10 6 T?
Example 3) What is the magnetic field strength at a point midway between two long parallel
wires, 1.0 m apart, carrying currents of 10 A and 20 A respectively, if the currents are
flowing in the same direction?
Example 4) What is the magnetic field strength at a point midway between two long parallel
wires, 1.0 m apart, carrying currents of 10 A and 20 A respectively, if the currents are
flowing in the opposite direction?
Understanding the concepts.
1) What is the magnetic field strength 20 cm from a ling straight conductor with a current of 60
A flowing through it? 6.0 x 10 5 T
2) What current is flowing through a straight wire if the magnetic field strength 10 cm from the
wire is 2.4 x 10 5 T? 12 A
3) At what distance from a straight conductor, with a current of 200 A flowing through it, is the
magnetic field intensity 8.0 x 10 4 T? 5.0 x 10 2 m.
4) What is the magnetic field strength at a point midway between two long parallel wires, 1.0 m
apart, carrying currents of 10 A and 20 A respectively, if the currents are
a) in opposite directions? 1.2 x 10 5 T
b) in the same direction? 4.0 x 10 6T
5) A long, solid, copper rod has a circular cross-section of diameter 10 cm. It carries a current of
1000A, uniformly distributed across its area. Calculate the magnetic field strength at these
four positions: (Tricky!!!)
a) at the centre of the rod 0T
b) 2.5 cm from the centre
2.0 x 10 3 T
c) 5.0 cm from the centre
4.0 x 10 3 T
d) 7.5 cm from the centre
2.7 x 10 3 T
Magnetic Field of a Current Loop
When a straight wire is formed into a circular loop, its magnetic field will appear as
shown in Figure 10. Note that the field lines inside the loop are closer together, indicating a
stronger magnetic field than on the outside of the loop.
Magnetic Field of a Coil or Solenoid
A solenoid is a long conductor wound into a coil of many loops. The magnetic field
of a solenoid (Figure 11) is the sum of the magnetic fields of all of its loops. The field inside the
coil can consequently be very strong. If the coil is tightly wound, the field lines are nearly
straight and very close together (Figure 12).
Figure 11
(a) A solenoid
(b) Iron filings reveal the field
lines
in and around a solenoid.
Figure 12
(a) When the solenoid is loosely wound, field lines within the coil are curved.
(b) The field becomes stronger and straighter inside the coil when the coil is wound
tighter. The right-hand rule for solenoids (a corollary of the right-hand rule for a
straight conductor) gives the direction of the field inside the coil.
A solenoid has a magnetic field very similar to the field of a bar magnet, with the
convenient additional feature that the field can be switched off and on. To remember the
direction of the magnetic field of a solenoid, we apply a special right-hand rule for a solenoid:
Note that the right-hand rule for a solenoid is consistent with the right-hand rule
for a straight conductor if we point our thumb along, or tangent to, the curved wire
of the coil.
Determine the magnetic field at the centre of a current carrying circular loops
If the current carrying conductor is bent into a circular loop, the magnetic field at the
centre of the loop is found using the formula:
,
Example 1) A circular coil has 10 loops of wire with a diameter of 12 cm. If the current flowing
through these loops is 5.0 A, what is the magnetic field strength at the centre of this coil?
Example 2) What is the magnitude of the magnetic field in the core of a solenoid 5.0 cm long,
with 300 turns and a current of 8.0 A?
Example 3) An air core solenoid is 25 cm long and has 1000 loops. If it has a current of 9.0 A,
what is the magnetic field in this solenoid?
Understanding Concepts.
1) A circular coil has a diameter of 9.0 cm and 12 loops. If the current flowing through the coil is
15 A, what is the magnetic field strength at the centre of this coil? 2.5 x 10 3 T
2) A 25.0 cm solenoid has 1800 loops and a diameter of 3.0 cm. Calculate the magnitude of the
magnetic field in the air core of the solenoid when a current of 1.25 A is flowing. 1.13 x 10 2 T
3) A circular coil has 9 loops and a current of 8.0 A flowing through it. If the magnetic field at
the centre of this coil is 1.1 x 10 3 T, what is its diameter? 0.082 m
4) A circular coil with 18 loops of wire and has a diameter of 12 cm. If the magnetic field at the
center of this coil is 6.2 x 10 4 T, what is the current flowing through the coil? 3.3 A
5) An air core solenoid is 25 cm long and carries a current of 0.72 A. If the magnetic field in the
core is 2.1 x 10 3 T, how many turns does this solenoid have? 580
6) An air core solenoid is 30.0 cm long and has 775 turns. If the magnetic field in the core is
0.100 T, what is the current flowing through this solenoid? 31 A
7) Two long fixed parallel wires are 7.2 cm apart and carry currents of 25 A and 15 A in the
opposite direction. What is the magnitude of the magnetic field midway between the two
wires? 2.2 x 10 4 T
8) A 14-gauge copper wire has a current of 12 A. How many turns would have to be wound on a
coil 15 cm long to produce a magnetic field of strength 5.0 x 10 2 T?
5.0 x 102
9) Calculate the magnitude of the magnetic field strength in the core of a coil 10.0 cm long, with
420 turns and a current of 6.0 A.
3.2 x 10 2 T
10) A coil 8.0 cm long, with 400 turns, produces a magnetic field of magnitude 1.4 x 10 2 T in
its core. Calculate the current in the coil.
2.2 A
Physics 12
Electromagnetism.
Lesson Notes:
Magnetic Force on Moving Charge
Before we start this lesson, let us review the interactions of magnetic fields.
What happens in the case of a pair of parallel current carrying conductors?
Case I) In the same direction.
Case II) In the opposite direction.
From the above diagrams, we have observed that the magnetic field s set up
plays a big role in the outcome of either force of repulsion or attraction. We recognize that the
bigger the magnetic field exists between the wires, the bigger the magnetic force occurs.
To determine the magnitude of these attractive or repulsive forces between the
conductors, we use the following formula:
F
Where = force per unit length,
L
I1 and I2 = the currents in the conductors.
R = the distance between the conductors
Example 1) What is the magnetic force between two wires carrying currents of 5.0 A and 8.0 A
in opposite directions? If the wires are 15 cm apart and 45 cm long.
Example 2) Two long parallel wires carrying currents of 10.0 A and 15.0 A in the same direction.
If the force per meter on each wire is 4.8 x 10 4 N/m, how far apart are the wires?
Behavior of a current carrying conductor in Magnetic Field.
In 1821, following Oersted s discovery of electromagnetism, English physicist Michael
Faraday (1791 1867) set out to prove that, as a wire carrying electric current could cause a
magnetized compass needle to move, so in reverse a magnet could cause a current-carrying wire
to move. Suspending a piece of wire inside a magnetic field which came from a fixed magnet,
Faraday connected the wire to a battery, and the wire began to deflect.
Faraday determined that the magnetic field of a permanent magnet can exert a force on
the charges in a current-carrying conductor. Figure 1 shows how the direction of this force is
related to the magnetic field of the conductor and to the external magnetic field.
To the left of the conductor, the field lines point in the same direction and tend to
reinforce one another, producing a strong magnetic field. To the right, the fields are opposed and,
as a result, tend to cancel one another, producing a weaker field. This difference in field strength
results in a force to the right on the conductor. If either the external field or the direction of the
electric current were reversed, the force would act in the opposite direction. A more detailed
investigation would show that the actual magnitude of the force depends on the magnitude of
both the current and the magnetic field. These effects are summarized in the motor principle.
Motor Principle
A current-carrying conductor that cuts across external magnetic
field lines experiences a force perpendicular to both the magnetic field
and the direction of electric current. The magnitude of this force depends
on the magnitude of both the external field and the current, as well as the
angle between the conductor and the magnetic field it cuts across.
The magnitude of the magnetic force on a current carrying conductor can
be calculated using:
Fm = B I L sin θ
where Fm = magnetic force
B = magnetic field strength
I = current through the conductor
L = length of conductor
sin θ = angle at which the conductor passes through
the magnetic field
Example 1) A conductor 3.2 x 10 1 m is placed in a magnetic field of 2.10 x 10 1 T. Assuming
the conductor is perpendicular to the magnetic field, and the magnetic force acting on the
conductor is 4.00 x 10 2 N, what is the current flowing through the conductor?
Example 2) Calculate the magnitude and the direction of the magnetic force on a conductor
which is 5.5 cm long with a current of 4.0 A heading North. The conductor is placed
perpendicular to the magnetic field with strength of 3.5 x 10 2 T from West to East.
Example 3) Calculate the magnitude and the direction of the magnetic force on a conductor
which is 8.0 cm long with a current of 7.5 A. The conductor is placed at an angle 30° to the
magnetic field with strength of 3.5 x 10 2 T from East to West.
Moving charges in Magnetic Fields.
It is not only current carrying conductors that are deflected by magnetic
fields. Moving charges (electrons, protons, and alpha particles) can also be
deflected by magnetic fields. These particles are deflected in the same way as
current carrying conductors. However, to determine the direction of the deflection,
you use your left hand rule for negative charge (electrons) and right hand rule for
all positive charges like you did with the current carrying wire in magnetic field.
The magnitude of the deflecting magnetic forces on moving charged particles can
be calculated using:
Fm = q v B sin θ
where Fm = magnetic force
B = magnetic field strength
q = charge of the particle
v = speed of the particle
sin θ = angle at which the particle passes through the
magnetic field
Example 4) Calculate the magnitude of the magnetic force on an electron traveling at a speed of
3.6 x 104 m/s perpendicular through a magnetic field 4.20 T. 2.42 x 10 14 N
Example 5) A proton traveling at an angle 35° to the vertical line at a speed of 2.1 x 10 5 m/s
through a horizontal magnetic field experiences a magnetic force of 7.8 x 10 14 N.
What is the magnitude of the magnetic field? 2.83 T
Figure 5 shows a positively charged particle in a magnetic field perpendicular to its velocity. (If
the particle were negative, its trajectory would curve the other way, in a clockwise circle.) If the
magnetic force is the sole force acting on the particle, it is equal to the net force on the particle
and is always perpendicular to its velocity. This is the condition for uniform circular motion; in
fact, if the field is strong enough and the particle doesn t lose any energy, it will move in a
complete circle as shown.
Figure 5
(a) A positive charge moving at
constant speed through a uniform
magnetic field follows a curved path.
(b) Ideally, a charged particle will
move in a circle because the
magnetic force is perpendicular
to the velocity at all times.
We represent these magnetic fields in two-dimensional diagrams by drawing Xs for field lines
directed into and perpendicular to the page and dots for field lines pointing out of and
perpendicular to the page. If the velocity is perpendicular to the magnetic field lines, then both
the velocity and the magnetic force are parallel to the page, as in Figure 6.
Figure 6
In this case, the particle is negatively charged, with the magnetic
field directed into the page, perpendicular to the velocity. To
determine the direction of the magnetic force, point your thumb in
the opposite direction of the velocity because the charge is
negative.
Example 1)
An electron accelerates from rest in a horizontally directed electric field
through a potential difference of 46 V. The electron then leaves the
electric field, entering a magnetic field of magnitude 0.20 T directed into
the page (Figure 7).
(a) Calculate the initial speed of the electron upon entering the
magnetic field. 4.0 x 106 m/s
(b) Calculate the magnitude and direction of the magnetic force on the
electron.
1.3 x 10 13 N.
(c) Calculate the radius of the electron s circular path. 1.1 x 10 4 m
Example 2) Calculate the mass of chlorine-35 ions, of charge 1.60 x 10 19 C, accelerated into a
mass spectrometer through a potential difference of 2.50 x 102 V into a uniform 1.00-T
magnetic field. The radius of the curved path is 1.35 cm. 5.83 x 10 26 kg.
Understanding Concept.
1) Determine the magnitude and direction of the magnetic force on a proton moving horizontally
northward at 8.6 x 104 m/s, as it enters a magnetic field of 1.2 T directed vertically upward.
(The mass of a proton is 1.67x 10 27 kg.)
1.7 x 10 14 N [E]
2) An electron moving through a uniform magnetic field with a velocity of 2.0 x 106 m/s [up]
experiences a maximum magnetic force of 5.1x 10 14 N [left]. Calculate the magnitude and
direction of the magnetic field.
0.16 T [horizontal, toward observer]
3) Calculate the radius of the path taken by an α particle (He2+ ion, of charge 3.2 x 10 19 C and
mass 6.7x 10 27 kg) injected at a speed of 1.5 x 107 m/s into a uniform magnetic field of 2.4 T,
at right angles to the field.
0.13 m
4) Calculate the speed of a proton, moving in a circular path of radius 8.0 cm, in a plane
perpendicular to a uniform 1.5-T magnetic field. What voltage would be required to
accelerate the proton from rest, in a vacuum, to this speed? (m proton = 1.67 x 10 27 kg)
1.1 x 107 m/s; 6.9 x 105 V
5) An airplane flying through Earth s magnetic field at a speed of 2.0 x 102 m/s acquires a charge
of 1.0 x 102 C. Calculate the maximum magnitude of the magnetic force on it in a region
1.0 N
where the magnitude of Earth s magnetic field is 5.0 x 10 5 T.