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Chapter 19
Magnetism
Magnets
• In each magnet there are two poles present (the ends
where objects are most strongly attracted): north and
south
• Like (unlike) poles repel (attract) each other (similar to
electric charges)
• Magnetic poles cannot be isolated – if a permanent
magnet is cut in half, you will still have a north and a
south pole (unlike electric charges)
• There is some theoretical basis for monopoles, but
none have been detected
Magnetism
• An unmagnetized piece of iron can be magnetized by
stroking it with a magnet (like stroking an object to
charge an object)
• Magnetism can be induced – if a piece of iron, for
example, is placed near a strong permanent magnet, it
will become magnetized
• Soft magnetic materials (such as iron) are easily
magnetized and also tend to lose their magnetism
easily
• Hard magnetic materials (such as cobalt and nickel)
are difficult to magnetize and they tend to retain their
magnetism
Magnetic Fields
• The region of space surrounding a moving charge
includes a magnetic field (the charge will also be
surrounded by an electric field)
• A magnetic field surrounds a properly magnetized
magnetic material
• A magnetic field is a vector quantity symbolized by B
• Its direction is given by the direction a north pole of a
compass needle pointing in that location
• Magnetic field lines can be used to show how the field
lines, as traced out by a compass, would look
Magnetic Field Lines
• A compass can be used to show the direction of
the magnetic field lines
Magnetic Field Lines
• Iron filings can also be used
to show the pattern of the
magnetic field lines
• The direction of the field is
the direction a north pole
would point
• Unlike poles (compare to the
electric field produced by an
electric dipole)
Magnetic Field Lines
• Iron filings can also be used
to show the pattern of the
magnetic field lines
• The direction of the field is
the direction a north pole
would point
• Unlike poles (compare to the
electric field produced by an
electric dipole)
• Like poles (compare to the
electric field produced by like
charges)
Earth’s Magnetic Field
• The Earth’s geographic north (south) pole corresponds
to a magnetic south (north) pole – a north (south) pole
should be a “north- (south-) seeking” pole
• The Earth’s magnetic field
resembles that achieved by burying
a huge bar magnet deep in the
Earth’s interior
• The most likely source of the
Earth’s magnetic field – electric
currents in the liquid part of the
core
Earth’s Magnetic Field
• The magnetic and geographic poles are not in the
same exact location – magnetic declination is the
difference in directions to the geographic north pole
and the magnetic south pole
• The amount of declination varies
by location on the earth’s
surface
• The direction of the Earth’s
magnetic field reverses every
few million years (the origin of
these reversals is not
understood)
Earth’s Magnetic Field
• If a compass is free to rotate vertically as well as
horizontally, it points to the earth’s surface
• The angle between the horizontal and the direction of
the magnetic field is called the dip angle
• The farther north the device is moved, the farther from
horizontal the compass needle would be
• The compass needle would be horizontal at the
equator and the dip angle would be 0°
• The compass needle would point straight down at the
south magnetic pole and the dip angle would be 90°
Magnetic Fields
• When moving through a magnetic field, a charged
Nikola Tesla
particle experiences a magnetic force
1856 – 1943
• This force has a maximum (zero) value when the
charge moves perpendicularly to (along) the magnetic
field lines
• Magnetic field is defined in terms of the magnetic force
exerted on a test charge moving in the field with
velocity v
F
• The SI unit: Tesla (T)
N
T
Am
B
N
T
qv sin 
C  (m / s)
F  qvB sin 
Magnetic Fields
• Conventional laboratory magnets: ~ 2.5 T
• Superconducting magnets ~ 30 T
• Earth’s magnetic field ~ 5 x 10-5 T
Direction of Magnetic Force
• Experiments show that the direction of
the magnetic force is always
perpendicular to both v and B
• Fmax occurs when v is perpendicular to
B and F = 0 when v is parallel to B
F  qvB sin 
• Right Hand Rule #1 (for a + charge):
Place your fingers in the direction of v
and curl the fingers in the direction of B
– your thumb points in the direction of F
• If the charge is negative, the force
points in the opposite direction
Direction of Magnetic Force
• The blue x’s indicate the magnetic field when it is
directed into the page (the x represents the tail of the
arrow)
• Blue dots would be used to represent the field directed
out of the page (the • represents the head of the arrow)
Force on a Charged Particle in a
Magnetic Field
• Consider a particle moving in an
external magnetic field so that its
velocity is perpendicular to the field
• The force is always directed toward
the center of the circular path
• The magnetic force causes a
centripetal acceleration, changing
the direction of the velocity of the
particle
F  qvB sin 
mv
F  qvB 
r
mv
r
qB
2
Force on a Charged Particle in a
Magnetic Field
• This expression is known as the
cyclotron equation
• r is proportional to the momentum
of the particle and inversely
proportional to the magnetic field
• If the particle’s velocity is not
perpendicular to the field, the path
followed by the particle is a spiral
(helix)
mv
r
qB
Chapter 19
Problem 6
A proton moves perpendicularly to a uniform magnetic
field at 1.0 × 107 m/s and exhibits an acceleration of 2.0 ×
1013 m/s2 in the +x-direction when its velocity is in the +zdirection. Determine the magnitude and direction of the
field.
Chapter 19
Problem 40
A particle with charge q and kinetic energy KE travels in a
uniform magnetic field of magnitude B. If the particle
moves in a circular path of radius R, find expressions for
(a) its speed and (b) its mass.
Magnetic Force on a Current Carrying Wire
• The current is a collection of many charged particles in
motion
• The magnetic force is exerted on each moving charge
in the wire
• The total force is the sum of all the magnetic forces on
all the individual charges producing the current
• Therefore a force is exerted on a current-carrying wire
placed in a magnetic field:
F  qvd B sin  # carriers  qvd B sin   nAl
F  BIl sin 
Magnetic Force on a Current Carrying Wire
• The direction of the force is given by right hand rule #1,
placing your fingers in the direction of I instead of v
Chapter 19
Problem 21
Consider the system pictured in the figure. A 15-cm length of conductor of mass 15 g,
free to move vertically, is placed between two thin, vertical conductors, and a uniform
magnetic field acts perpendicular to the page. When a 5.0-A current is directed as
shown in the figure, the horizontal wire moves upward at constant velocity in the
presence of gravity. (a) What forces act on the horizontal wire, and under what
condition is the wire able to move upward at constant velocity? (b) Find the magnitude
and direction of the minimum magnetic field required to move the wire at constant
speed. (c) What happens if the magnetic field exceeds this minimum value? (The wire
slides without friction on the two vertical conductors.)
Torque on a Current Loop
F1  F2  BIb  max
a
a
 F1  F2
2
2
a
a
 BIb  BIb  BIab  BIA
2
2
  BIA sin 
  NBIA sin 
Torque on a Current Loop
• Applies to any shape loop
• Torque has a maximum value when  = 90°
• Torque is zero when the field is perpendicular to the
plane of the loop
  NBIA sin 
Magnetic Moment
• The vector
coil

is called the magnetic moment of the
• Its magnitude is given by
μ = IAN
• The vector always points perpendicular to the plane of
the loop(s)
• The equation for the magnetic torque can be written as
τ = BIAN sinθ = μB sinθ
• The angle is between the moment and the field
Chapter 19
Problem 30
A copper wire is 8.00 m long and has a cross-sectional area of 1.00 ×
10−4 m2. The wire forms a one-turn loop in the shape of square and is
then connected to a battery that applies a potential difference of 0.100
V. If the loop is placed in a uniform magnetic field of magnitude 0.400
T, what is the maximum torque that can act on it? The resistivity of
copper is 1.70 × 10−8 Ω · m.
Electric Motor
• An electric motor converts electrical energy to
mechanical energy (rotational kinetic energy)
• An electric motor consists of a rigid current-carrying
loop that rotates when placed in a magnetic field
• The torque acting on the loop
will tend to rotate the loop to
smaller values of θ until the
torque becomes 0 at θ = 0°
Electric Motor
• If the loop turns past this point and the current remains
in the same direction, the torque reverses and turns the
loop in the opposite direction
• To provide continuous rotation in one direction, the
current in the loop must periodically reverse
• In ac motors, this reversal
naturally occurs
• In dc motors, a split-ring
commutator and brushes are
used
Electric Motor
• Just as the loop becomes perpendicular to the magnetic
field and the torque becomes 0, inertia carries the loop
forward and the brushes cross the gaps in the ring,
causing the current loop to reverse its direction
• This provides more torque to
continue the rotation
• The process repeats itself
• Actual motors would contain
many current loops and
commutators
Magnetic Fields – Long Straight Wire
• A current-carrying wire produces a
magnetic field
• The compass needle points in the
direction of the magnetic field
produced by the current (tangential
to the circle)
• Right Hand Rule #2: Grasp the wire
in your right hand and point your
thumb in the direction of the current
• Your fingers will curl in the direction
of the field
Magnetic Fields – Long Straight Wire
• The magnitude of the field at a distance r from a wire
carrying a current of I is
o I
B
2 r
• µo = 4  x 10-7 T.m / A: permeability of free space
Ampère’s Law
• Ampère’s Circuital Law: a procedure for deriving the
relationship between the current in an arbitrarily shaped
wire and the magnetic field produced by the wire
• Choose an arbitrary closed path around the current and
sum all the products of B|| Δℓ around the closed path
 B|| Δℓ = µo I
Ampère’s Law for a Long Straight Wire
• Use a closed circular path
• The circumference of the circle is 2
 B|| Δℓ = µo I
B  Δℓ = B 2 r = µo I
o I
B
2 r
r
Chapter 19
Problem 51
A wire carries a 7.00-A current along the x-axis, and another wire
carries a 6.00-A current along the y-axis, as shown in the figure. What
is the magnetic field at point P, located at x = 4.00 m, y = 3.00 m?
Magnetic Force Between Two Parallel
Conductors
0 I 2
B2 
2d
0 I 2
F1  B2 I1l 
I1l
2d
F1  0 I1 I 2

l
2d
Magnetic Force Between Two Parallel
Conductors
• The force (per unit length ) on wire 1
due to the current in wire 1 and the
magnetic field produced by wire 2:
F
o I1 I2

2 d
• Parallel conductors carrying
currents in the same direction
attract each other
• Parallel conductors carrying
currents in the opposite directions
repel each other
Chapter 19
Problem 57
A wire with a weight per unit length of 0.080 N/m is
suspended directly above a second wire. The top wire
carries a current of 30.0 A and the bottom wire carries a
current of 60.0 A. Find the distance of separation between
the wires so that the top wire will be held in place by
magnetic repulsion.
Magnetic Field of a Current Loop
• The strength of a magnetic field produced by a wire
can be enhanced by forming the wire into a loop
• All the segments, Δx, contribute to the field, increasing
its strength
Magnetic Field of a Current Loop
• The magnitude of the magnetic field at the center of a
circular loop with a radius R and carrying current I is
B
o I
2R
• With N loops in the coil, this becomes
BN
o I
2R
Magnetic Field of a Solenoid
• If a long straight wire is bent into a coil of
several closely spaced loops, the
resulting device is called a solenoid
• It is also known as an electromagnet since
it acts like a magnet only when it carries a
current
• The field inside the solenoid is nearly
uniform and strong – the field lines are
nearly parallel, uniformly spaced, and
close together
• The exterior field is nonuniform, much
weaker, and in the opposite direction to
the field inside the solenoid
Magnetic Field of a Solenoid
• The field lines of the solenoid resemble
those of a bar magnet
• The magnitude of the field inside a
solenoid is approximately constant at all
points far from its ends
B = µo n I
• n = N / ℓ : the number of turns per unit
length
• The same result can be obtained by
applying Ampère’s Law to the solenoid
Magnetic Field of a Solenoid
• A cross-sectional view of a long tightly
wound solenoid
• If the solenoid is long compared to its
radius, we assume the field inside is
uniform and outside is zero
• Apply Ampère’s Law to the blue dashed
rectangle
B
l  Bl   0 NI
N
B   0 I   0 nI
l
Magnetic Effects of Electrons – Orbits
• An individual atom should act like a magnet because of
the motion of the electrons about the nucleus
• Each electron circles the atom once in about every 10-16
seconds; this would produce a current of 1.6 mA and a
magnetic field of about 20 T at the center of the circular
path
• However, the magnetic field produced by one electron
in an atom is often canceled by an oppositely revolving
electron in the same atom
• The net result is that the magnetic effect produced by
electrons orbiting the nucleus is either zero or very
small for most materials
Magnetic Effects of Electrons – Spins
• Electrons also have spin (it is a
quantum effect)
• The classical model is to consider
the electrons to spin like tops
• The field due to the spinning is
generally stronger than the field due
to the orbital motion
• Electrons usually pair up with their
spins opposite each other, so their
fields cancel each other, hence most
materials are not naturally magnetic
Magnetic Effects of Electrons – Domains
• In some materials – ferromagnetic – the spins do not
naturally cancel
• Large groups of atoms in which the spins are aligned
are called domains
• When an external field is applied, it causes the material
to become magnetized: the domains that are aligned
with the field tend to grow at the expense of the others
Domains and Permanent Magnets
• In hard magnetic materials, the domains remain
aligned after the external field is removed
• The result is a permanent magnet
• In soft magnetic materials, once the external field is
removed, thermal agitation causes the materials to
quickly return to an unmagnetized state
• With a core in a loop, the magnetic field is enhanced
since the domains in the core material align,
increasing the magnetic field
Answers to Even Numbered Problems
Chapter 19:
Problem 2
(a) toward the left; into the page; out of
the page; toward top of page; into the
page; out of the page
(b) the answers for part (b) are reversed
from those given in part (a)
Answers to Even Numbered Problems
Chapter 19:
Problem 26
4.33 × 10−3 N m
Answers to Even Numbered Problems
Chapter 19:
Problem 44
(a) toward the left
(b) out of the page
(c) lower left to upper right
Answers to Even Numbered Problems
Chapter 19:
Problem 48
(a) 40.0 μT into the page
(b) 5.00 μT out of the page
(c) 1.67 μT out of the page