Chapter 29
Magnetic Fields
A Partial History of Magnetism

13th century BC

Chinese used a compass



Uses a magnetic needle
Probably an invention of Arabic or Indian origin
800 BC

Greeks

Discovered magnetite (Fe3O4) attracts iron
A Partial History of Magnetism

Hans Christian Oersted
 Danish Physicist
 Discovered the
relationship between
electricity and
magnetism in 1819


during a lecture
demonstration!
An electric current in a
wire deflected a nearby
compass needle
Demo Ec4:
Oersted’s Experiment



When a current flows through the
conductor the compass needle
deflects due to the magnetic field
set up by the current.
Reversing the direction of the
current reverses the direction of
the magnetic field.
Oersted in 1819 revolutionised the
study of electricity & magnetism,
sparking scientists to investigate
the subject. He himself couldn’t
set up experiments and had to rely
on assistants. He got left behind!
Demo Ec1: Lines of force
in a magnetic field

Iron filings sprinkled
onto perspex sheet
on top of bar
magnet
Magnetic Poles

Every magnet, regardless of its shape,
has two poles


Called north and south poles
Poles exert forces on one another
 Similar to the way electric charges exert
forces on each other

Like poles repel


N-N or S-S
Unlike poles attract

N-S
Magnetic Poles, cont.


The poles received their names due to the way
a magnet behaves in the Earth’s magnetic field
If a bar magnet is suspended so that it can
move freely, it will rotate

Magnetic north pole points toward the Earth’s north
geographic pole
 i.e. Earth’s north geographic pole is a magnetic
south pole
 Similarly, the Earth’s south geographic pole is a
magnetic north pole
Magnetic Poles, final


The force between two poles varies as
the inverse square of the distance
between them
A single magnetic pole has never been
isolated


i.e. magnetic poles always found in pairs
There is some theoretical basis for the
existence of monopoles – single poles
Magnetic Fields
(magnetic induction)

A vector quantity



Symbol B
Direction given by that which the north
pole of a compass needle points
Magnetic field lines trace this direction
in space
MFA02VD1: Compass needle
shows direction of magnetic
field around a bar magnet
Magnetic Field Lines
for a Bar Magnet


Compass can be
used to trace the
field lines
The lines outside
the magnet point
from the North pole
to the South pole
MFM02VD1: Magnetic Field
lines around bar magnets
Magnetic Field Lines,
Bar Magnet


Iron filings are used
to show the pattern
of the magnetic field
lines
The direction of the
field is the direction
a north pole would
point
Magnetic Field Lines,
Unlike Poles


Iron filings are used to
show the pattern of
the electric field lines
The direction of the
field is the direction a
north pole would point
 c.f. electric field
produced by an
electric dipole
Magnetic Field Lines,
Like Poles


Iron filings are used to
show the pattern of the
electric field lines
The direction of the
field is the direction a
north pole would point
 c.f. electric field
produced by like
charges
Magnetic Fields and Forces

The magnetic field, B, at some point in
space can be defined in terms of the
magnetic force, FB


Note: B is sometimes known as the
Magnetic Induction
A magnetic force will be exerted on a
charged particle moving in a magnetic
field
FB on a Charge Moving
in a Magnetic Field


Magnitude proportional to charge and speed
of the particle
Direction depends on the velocity of the
particle and the direction of the magnetic field


It is perpendicular to both
FB = q v x B




FB is the magnetic force
q is the charge
v is the velocity of the moving charge
B is the magnetic field
Direction


FB perpendicular to plane formed by v & B
Oppositely directed forces are exerted on
charges of different signs

cause the particles to move in opposite directions
Direction given by
Right-Hand Rule

Fingers point in the
direction of v



(for positive charge;
opposite direction if
negative)
Curl fingers in the
direction of B
Then thumb points in the
direction of v x B; i.e. the
direction of FB
The Magnitude of F

The magnitude of the magnetic force on
a charged particle is FB = |q| vB sin 


 is the angle between v and B
FB is zero when v and B are parallel or
antiparallel


 = 0° or 180°
FB is a maximum when v and B are
perpendicular

 = 90°
Example
Proton moves at v = 8x106 m/s in x-direction.
Enters region where B = 2.5 T directed at 60°
to x-axis in xy-plane. What is the initial force
on the proton?
Right Hand rule puts F in the z-direction.
F  qvB sin(  )
 (1 . 6  10
 2.77  10
 19
C)(8
 12
 10
N . Tiny!
6
m/s)(2.5 T)sin(60
)
Differences Between Electric
and Magnetic Fields

Direction of force



Electric force acts along direction of electric field
Magnetic force acts perpendicular to magnetic
field
Motion


Electric force acts on a charged particle
regardless of whether the particle is moving
Magnetic force acts on a charged particle only
when the particle is in motion
Differences Between Electric
and Magnetic Fields

Work


Electric force does work in displacing a
charged particle
Magnetic force associated with a steady
magnetic field does no work when a
particle is displaced
 Force is perpendicular to the
displacement, so F.ds=0
Work in Magnetic Fields


The kinetic energy of a charged particle
moving through a magnetic field cannot
be altered by the magnetic field alone
The field can only alter the direction of
the particle, not its speed

Units of Magnetic Field

The SI unit of magnetic field is the tesla (T)

Since FB = |q| vB sin 
1T 1

N
C  m/s
1
N
A m
The cgs unit is a gauss (G)

1 T = 104 G
Typical Magnetic Field Values
Quick Quiz 29.1
The north-pole end of a bar magnet is held near a positively
charged piece of plastic. The plastic is
(a) attracted
(b) repelled
(c) unaffected by the magnet
Quick Quiz 29.1
Answer: (c). The magnetic force exerted by a magnetic field
on a charge is proportional to the charge’s velocity relative
to the field. If the charge is stationary, as in this situation,
there is no magnetic force.
Quick Quiz 29.2
A charged particle moves with velocity v in a magnetic field
B. The magnetic force on the particle is a maximum when v
is:
(a) parallel to B
(b) perpendicular to B
(c) zero
Quick Quiz 29.2
Answer: (b). The maximum value of sin θ occurs for θ =
90°.
Quick Quiz 29.3
An electron moves in the plane of this paper toward the top
of the page. A magnetic field is also in the plane of the page
and directed toward the right. The direction of the magnetic
force on the electron is
(a) toward the top of the page
(b) toward the bottom of the page
(c) toward the left edge of the page
(d) toward the right edge of the page
(e) out of the page
(f) into the page
Quick Quiz 29.3
Answer: (e). Out of the page. The right-hand rule gives the
direction. Be sure to account for the negative charge on the
electron.
Magnetic Force on a Current
Carrying Conductor

A force is exerted on a current-carrying
wire placed in a magnetic field


The current is a collection of many charged
particles in motion
The direction of the force is given by the
right-hand rule
Ec9: Magnetic force on
current carrying conductor



Flexible wire placed
between poles of a
strong magnet. When
current is passed
through it, the wire
dramatically jumps out.
Can demonstrate F is
perpendicular to L and
B by suspending wire in
different orientations.
See also corridor
display.
Note on Notation


The dots indicate the
direction is out of the page
 The dots represent the
tips of the arrows
coming toward you
The crosses indicate the
direction is into the page
 The crosses represent
the feathered tails of the
arrows
MFA03AN2: Force on a
current-carrying conductor
Force on a Wire


When no current
flows there is no
force
Therefore, the wire
remains vertical
Force on a Wire (2)



With B into the page, and
Current up the page, then
Force is to the left

Apply the right hand rule
Force on a Wire, equation

Magnetic force is exerted
on each moving charge in
the wire


F = q vd x B
Total force is the product
of force on one charge
times the number of
charges

F = (q vd x B)nAL

n is the number of charges
per unit volume
Force on a Wire, (4)

In terms of the current, this becomes
F=ILxB

L is a vector that points in the direction of
the current (i.e. of vD)


Magnitude is the length L of the segment
I is the current = nqAvD
(think about the units to see this)

B is the magnetic field
Force on a Wire of
Arbitrary Shape


Consider a small
segment of the wire,
ds
The force exerted
on this segment is
F = I ds x B

The total force is
F  I

b
a
ds  B
Force on a Wire, Case 1


Suppose that B is uniform.
b
Then F  I
ds  B

a
becomes F=I L’xB


L’ is the vector sum of all the
length elements from a to b
Thus, the magnetic force
on a curved currentcarrying wire in a uniform
field is equal to that on a
straight wire connecting
the end points and carrying
the same current
Force on a Wire, Case 2


An arbitrary shaped closed
loop carrying current I in a
uniform magnetic field
FB  I




d s  B  0 since

ds  0
The length elements form a
closed loop, so the vector sum is
zero
Thus, the net magnetic force
acting on any closed current
loop in a uniform magnetic
field is zero
Quick Quiz 29.4
The four wires shown below all carry the same current from point A to
point B through the same magnetic field. In all four parts of the figure, the
points A and B are 10 cm apart. Which of the following choices ranks the
wires according to the magnitude
of the magnetic force exerted on
them, from greatest to least?
(a) a, b, c
(b) a, c, b
(c) a, c, d
(d) a, d, c
(d) b, c, d
(e) c, b, d
Quick Quiz 29.4
Answer: (c). The order is (a), (b) = (c), (d). The magnitude
of the force depends on the value of sin θ. The maximum
force occurs when the wire is perpendicular to the field (a),
and there is zero force when the wire is parallel (d). Choices
(b) and (c) represent the same force because a straight wire
between A and B will have the same force on it as the curved
wire for a uniform magnetic field.
Quick Quiz 29.5
A wire carries current in the plane of this paper towards the
top of the page. The wire experiences a magnetic force
toward the right edge of the page. The direction of the
magnetic field causing this force is
(a) in the plane of the page and toward the left edge
(b) in the plane of the page and toward the bottom edge
(c) out of the page
(d) into the page
Quick Quiz 29.5
Answer: (c). Use the right-hand rule to determine the
direction of the magnetic field.
Charged Particle in a
Magnetic Field



Consider a particle moving
in an external magnetic
field with its velocity
perpendicular to the field
The force is always
directed toward the centre
of the circular path
The magnetic force causes
a centripetal acceleration,
changing the direction of
the velocity of the particle
Force on a Charged Particle

Equating the magnetic and centripetal
forces:
F  qvB 


mv
2
r
Solving gives r = mv/qB

r is proportional to the momentum of the
particle and inversely proportional to the
charge and to the magnetic field
More About Motion of
Charged Particle

The angular speed of the particle is
 

 
v
r

qB
m
The angular speed,  is also referred to as
the cyclotron frequency
The period of the motion is
T 
2r
v

2


2m
qB
Van Allen Radiation Belts



The Van Allen radiation
belts consist of charged
particles surrounding the
Earth in doughnut-shaped
regions
“Cosmic ray” particles from
the Sun are trapped by the
Earth’s magnetic field
The particles spiral from
pole to pole

Can result in Auroras
The Earth’s Magnetosphere
Charged particles trapped in the Van Allen belts and create aurorae
Aurorae: MFM11VD1
Typically 100–400 km above the Earth
Ec5: Deflection of electron beam
by magnetic & electric fields



Use either a cathode
ray tube or electron
beam deflection tube.
Possible to determine
e/m by size of the
deflection.
Mark North pole of
magnet to show that
F = q v x B.
MFM03AN1: Lorentz force in
electric & magnetic fields
Charged Particles Moving in
Electric and Magnetic Fields



In many applications, charged particles
will move in the presence of both
magnetic and electric fields
In that case, the total force is the sum of
the forces due to the individual fields
In general: F = qE + qv x B
Hall Effect (not examinable)



When a current carrying conductor is placed
in a magnetic field, a potential difference is
generated in a direction perpendicular to both
the current and the magnetic field
This phenomena is known as the Hall effect
It arises from the deflection of charge carriers
to one side of the conductor as a result of the
magnetic forces they experience
Hall Voltage

Observing the Hall
effect

The Hall voltage is
measured between
points a and c
Hall Effect



When the charge carriers are negative, the upper
edge of the conductor becomes negatively charged
When the charge carriers are positive, the upper edge
becomes positively charged
Sign of Hall voltage, VH, gives the sign of the
charges

Hall Voltage


In equilibrium qEH = qvdB
VH = EHd = vd B d


d is the width of the conductor
vd is the drift velocity

Since


So can be found if B and d are known and VH measured
vd 
I
nqA
, then
VH 
IBd
nqA

IB
nqy

R H IB
y
Where RH = 1 / nq is called the Hall coefficient and y is the
thickness of the conductor, with area A = yd
A properly calibrated conductor can be used to
measure the magnitude of an unknown magnetic field
Quick Quiz 29.8
A charged particle is moving perpendicular to a magnetic
field in a circle with a radius r. An identical particle enters
the field, with v perpendicular to B, but with a higher speed
v than the first particle. Compared to the radius of the circle
for the first particle, the radius of the circle for the second
particle is
(a) smaller
(b) larger
(c) equal in size
Quick Quiz 29.8
Answer: (b). The magnetic force on the particle increases in
proportion to v, but the centripetal acceleration increases
according to the square of v. The result is a larger radius, as
we can see from r = mv/qB.
Quick Quiz 29.9
A charged particle is moving perpendicular to a magnetic
field in a circle with a radius r. The magnitude of the
magnetic field is increased. Compared to the initial radius of
the circular path, the radius of the new path is
(a) smaller
(b) larger
(c) equal in size
Quick Quiz 29.9
Answer: (a). The magnetic force on the particle increases in
proportion to B. The result is a smaller radius, as we also see
from r = mv/qB.
End of Chapter
Quick Quiz 29.6
Rank the magnitudes of the torques acting on the rectangular
loops shown in the figure below, from highest to lowest. (All the
loops are identical and carry the same current.)
(a) a, b, c
(b) b, c, a
(c) c, b, a
(d) a, c, b.
(e) All loops experience zero torque.
Quick Quiz 29.6
Answer: (c). Because all loops enclose the same area and
carry the same current, the magnitude of μ is the same for
all. For part (c) in the image, μ points upward and is
perpendicular to the magnetic field and τ = μB, the
maximum torque possible. For the loop in (a), μ points along
the direction of B and the torque is zero. For (b), the torque
is intermediate between zero and the maximum value.
Quick Quiz 29.7
Rank the magnitudes of the net forces acting on the rectangular
loops shown in this figure, from highest to lowest. (All the loops
are identical and carry the same current.)
(a) a, b, c
(b) b, c, a
(c) c, b, a
(d) b, a, c
(e) All loops experience zero net force.
Quick Quiz 29.7
Answer: (e). Because the magnetic field is uniform, there is
zero net force on all three loops.
Quick Quiz 29.10a
Three types of particles enter a mass spectrometer like the one shown
in your book as Figure 29.24. The figure below shows where the
particles strike the detector array. Rank the particles that arrive at a,
b, and c by speed.
(a) a, b, c
(c) c, b, a
(b) b, c, a
(d) All their speeds are equal.
Quick Quiz 29.10a
Answer: (d). The velocity selector ensures that all three
types of particles have the same speed.
Quick Quiz 29.10b
Rank the particles that arrive at a, b, and c by m/q ratio.
(a) a, b, c
(b) b, c, a
(c) c, b, a
(d) All their m/q ratios are equal.
Quick Quiz 29.10b
Answer: (c). We cannot determine individual masses or
charges, but we can rank the particles by m/q ratio. Equation
29.18 indicates that those particles traveling through the
circle of greatest radius have the greatest m/q ratio.