Chapter 29
Magnetic Fields
Introduction
• Knowledge of Magnetism and application
dates back to 13th Century BCE in China.
– Magnetic Compass Needle (of Arabic/Indian
invention)
• Greeks discovered magnetism ~800 BCE.
– Magnetite (Fe3O4)attracts iron.
• 1269, Pierre de Maricourt discovered that
magnets have two poles, N and S, which can
attract and repel one another.
Introduction
• 1600, William Gilbert suggested the Earth
itself is a magnet.
• 1750, experimenters collectively show the
attractive/repulsive forces follow the inverse
square law.
– Very similar behavior to electric charges
– A single magnetic pole (monopole) has not been
isolated.
Introduction
• 1819, Hans Christian Oersted, discovered the
link between electricity and magnetism.
– Current in wire caused deflection of a compass
needle.
• 1820’s, Michael Faraday and Joseph Henry
(independently) showed that a changing
magnetic field creates an electric field.
Introduction
• 1860’s, James Clerk Maxwell, theoretically
shows the reverse.
– A changing electric field causes a magnetic field.
• This Chapter will focus on the effects of
magnetic field on charges and current carrying
wires.
• We will identify sources of magnetic field in
Ch 30.
29.1 Magnetic Fields and Forces
• Remember that an electric field surrounds any
electric charge.
• A magnetic field also surrounds any “moving”
charge, and any permanent magnet.
29.1
• Magnetic field is a vector quantity.
– Represented by the symbol B
– The vector direction of B aligns with the direction
that the needle of a compass would point.
– Field’s can be represented with magnetic field
lines.
29.1
• A typical bar magnetic field lines.
– A compass can be used in the presence of a
magnet to trace the field lines.
29.1
• Iron filings are also useful (but messy) to
identify magnetic field patterns.
29.1
29.1
• We can define a magnetic field B, at some
point in space, in terms of the magnetic force
FB, that the field exerts on a charged particle
moving through the space with velocity v.
• For now we will assume there are no
electric/gravitational fields in the space.
29.1
• Experiments have shown that
– The magnitude of FB is proportional the charge q
and speed v of the test particle.
– The magnitude and direction of FB depend on the
velocity and the magnitude and direction of B
– When a charged particle moves parallel to the
magnetic field vector, the value of FB is zero.
29.1
• Experiments show cont’d:
– At any angle θ ≠ 0, FB acts perpendicular to both v
and B.
– The direction of magnetic force on a positive
charge is opposite to the force on a negative
charge.
– The magnitude of FB is proportional to sinθ
29.1
• We can summarize these observations with
the expression
FB  qv  B
29.1
• Remember the direction of the cross product
v x B is determined by the “Right Hand Rule”
– Point your fingers in the direction of v.
– “Curl” them in the direction of B.
– Thumb points in the direction of FB.
• Remember the magnitude of a cross product
FB  q vBsin 
29.1
29.1
• From this we see that FB = 0 when θ = 0, 180o
• Electric field Differences
– FE acts along field lines, FB perpendicular
– FE acts on any particle w/ charge, FB only acts on
“moving” charges.
– FE does work displacing a charged particle, FB does
no work, (perpendicular force)
29.1
• By rearranging the equation we determine
units for magnetic field should be
N
1T  1
C m s
• The SI unit for magnetic field is the tesla (T)
also given as
N
1T  1
A m
and
1 T  10 4 G
29.1
• Common B Fields
29.1
• Quick Quizzes p. 899
• Example 29.1
For Reference
29.2 Magnetic Force on a Current
Carrying Conductor
• If a Force acts on a single charge moving
through a B-Field, then it should also act on a
wire with current placed in a B-Field.
• We can demonstrate this
by looking at a wire suspended
between the poles of a magnet.
29.2
• When there is no current in the wire, there is
no force on the wire.
29.2
• When the current flows upward through the
wire, the force causes a deflection to the left.
29.2
• When the current flows down through the
wire, there is a deflection to right.
29.2
• To quantify these observations we will look at
a wire segment of length L, cross-sectional
area A, carrying a current I, in a uniform
magnetic field B.
29.2
• We see that the force acting on a single
moving charge (traditional current) follows the
equation
FB  qv d  B
• If we multiple this force by the number of
charges moving through the segment given as
nAL, we have the force on the whole segment.
FB  qv d  B nAL
29.2
• Now remember that
I  nqvd A
• So we can rewrite the expression as
FB  IL  B
FB  ILB sin 
• Where L is a vector that points in the direction
of the current and has a magnitude equal to
the length of the wire segment.
29.2
• This expression only applies for a straight wire
segment passing through a uniform B-Field.
29.2
• Now consider an uniformly shaped, but
arbitrarily bent wire, in a B-Field.
29.2
• The force on any small segment will be
dFB  Ids  B
• We integrate from end points a and b to find
the total force on the wire…
b
FB  I  ds  B
a
b

FB  I   ds   B
 a 
29.2
b
• The quantity a ds represents the vector sum
of each small segment, which will equal the
vector length from point a to point b, L’
29.2
• So we can conclude that the magnetic force
on a curved current carrying wire, in a uniform
B-field is equal to that on a straight wire
connecting the end points and carrying the
same current.
FB  IL'B
29.2
• We also note that if our conductor is a closed
loop, we take the integral over the entire loop
 
FB  I  ds  B
• But since the vector sum of a closed loop is
zero
FB  0
29.2
• Quick Quizzes p. 903
• Example 29.2
29.3 Torque on a Current Loop in a
Uniform Magnetic Field
• Consider the rectangular loop carrying current
I in the figure below.
• Which sides have the
magnetic force acting on
them?
• Sides 2 and 4
• What directions do the
forces act?
29.3
• We see that the magnitude of the forces are
equal,
F2  F4  IaB
• Since the forces act on opposite sides in
opposing direction, they have
equivalent torque around
the central axis.
29.3
• The maximum torque is given as
 max  F2  b2   F4  b2   IabB
• This can be simplified to
 max  IAB
29.3
• If the loop is angled in the B-Field as shown
• The lever arm for each torque is
b2 sin 
• So the overall magnitude
of the torque is
  IAB sin 
29.3
• This can easily be expressed in vector notation
as
  IA B
• The direction of vector A points perpendicular
to the area of the loop following the RH Rule.
• Curl your fingers around the loop in the
direction of the current. Thumb Points out A
direction.
29.3
• Often the product IA is referred to as the
“magnetic dipole moment, μ” or “magnetic
moment” of the loop.
μ  IA
• The magnetic moment has units of A.m2 and
points in the same direction as A.
  μB
29.3
• For a coil of wire, with many turns
  Nμ loop  B  μ coil  B
• The potential energy associated with a loop of
wire is given as
U  μ  B
29.3
• From the expression, Umin = -μB, when μ and
B point in the same direction and Umax = +μB,
when μ and B point in opposite directions.
• Quick Quizzes p 906
• Example 29.3, 29.4
29.4 The Motion of a Charged Particle
in a Uniform Magnetic Field
• Remember from 29.1, the magnetic force that
acts on a particle is always perpendicular to
the velocity (and therefore does no work).
• If a positively charged particle moves with
velocity v, perpendicular to a magnetic field B,
what shape will its path take?
29.4
• Circular Path
29.4
• Using what we know about circular motion,
and centripetal acceleration, we can find the
radius of the circular path.
 F  ma
c
v2
FB  qvB  m
r
r
mv
qB
29.4
• We can define the angular speed (also called
the cyclotron frequency)
v qB
 
r m
• The period of the cycle is given as
T
2r 2 2m


v

qB
29.4
• If the velocity is not perpendicular to B, but
some arbitrary angle, the perpendicular
component causes circular motion, but the
parallel component induces no force.
• The path is helical.
29.4
• In non-uniform fields the motion is complex.
Particles can be come trapped oscillating back
and forth.
29.4
• Trapping effect is often referred to as a
magnetic bottle.
• This effect is shown in the Van Allen radiation
belts, surrounding the earth, and is
responsible for the Auroras Borealis and
Australis .
29.4
• Van Allen Belts
29.4
• Quick Quizzes p. 908
• Examples 29.6, 29.7
29.5 Applications of Charged Particles
moving in Magnetic Fields
• Charged particles moving in both Electric and
Magnetic Fields
• The net force on the particle is generally sum
of the Electric and Magnetic Forces
F  qE  qv  B
29.5
• Velocity Selector– Used to guarantee that only particles of a specific
velocity enter an experiment.
– Consists of uniform and perpendicular electric and
magnetic fields.
29.5
• As in the picture, the positively charged
particle experiences an Electric force down,
and magnetic force up.
• Only when the velocity is correct will the
forces balance (net force zero) and the particle
will pass straight through.
• Particles with incorrect speeds will deflect
either up or down.
29.5
• Setting equal to zero
F  qE  qv  B  0
• V is given as the ratio of E to B
E
v
B
29.5
• Mass Spectrometer- a method for isolating
ions of different masses/charges.
• Particles begin by passing through a velocity
selector, then pass though an additional
magnetic field.
29.5
• The greater the mass of the particle, the wider
the radius of arc it will travel, creating a
“spectrum” of masses on the detection
screen.
• Positively charged particles will deflect to one
side, negatively charged particles will deflect
to the other.
29.5
• We can determine the ratio of m/q of
unknown particles by rearranging the radius
equation
m rBo

q
v
• And we know the velocity from the selector so
m rBo B

q
E
29.5
• Using this information we can determine the
mass ratios of various isotopes of a given ion.
• Can be used for collecting fairly pure samples
of specific isotopes it does not scale well to
industrial levels.
• Quick Quiz p. 912
29.5
• A variation of the concept was completed by
J.J. Thomson in 1897 to verify the ratio of
e/me for electrons, helping to confirm their
existence.
29.5
• Cyclotron- a device used to accelerate charged
particle to high speeds often to bombard
atomic nuclei to produce various nuclear
reactions.
• Depends on both
electric and magnetic
fields.
29.5
• The source P, emits charged particles.
• The magnetic field causes the particles to
travel a circular path, with a period of T.
• The circular path is divided into two
semicircular sections “dees”, with an
alternating potential difference across the
gap.
29.5
• Every time the charged particle crosses the
gap it accelerates, and the potential difference
flips by the time it comes halfway around.
• Each gap acceleration widens the circular path
due to the increased velocity. The K gained
across the gap is equal to qΔV.
29.5
• The kinetic energy of the particles upon
exiting the cyclotron is given as
2 2 2
q
B R
1
K  2 mv 
2m
2
• And is often measured in eV (electron volts),
keV and MeV. (1 eV = 1.6 x 10-19 J)
• With an upper limit of 20 MeV before the
effects of relativity come into play.
29.6 The Hall Effect
• When a current carrying conductor is placed
in a magnetic field, a potential difference
across the conductor is created.
• This comes from the deflection of charge
carriers due to the magnetic force.
• The potential difference is known as the Hall
Voltage, and helps determine the sign of the
charge carriers, and is also a method for
measuring magnetic fields.
29.6
• An easy way to demonstrate the Hall Effect is
to look at a flat rectangular shaped conductor.
• The perpendicular magnetic field creates a
magnetic force that causes the
charge carriers to deflect to
the top of the conductor.
29.6
FB  qv d  B
• The deflection of carriers to the top, leaves a
deficit of carriers on the bottom side, resulting
in an electric field (Hall Field) and Voltage
across the conductor.
29.6
• Eventually equilibrium is reached and there is
no further deflection. The upward magnetic
force is balanced by the resulting downward
Electric Force.
• A sensitive voltmeter can measure the
potential difference and its polarity gives the
sign of the charge carrier.
29.6
29.6
• At equilibrium
qvd B  qEH
• So the Hall Field is
EH  vd B
• And the Hall Voltage is
VH  EH d  vd Bd
29.6
• Since the drift velocity is directly related to
charge density and current
vd 
I
nqA
• So the Hall Voltage can be given as
IBd
VH 
nqA
29.6
• One last adjustment, A/d represents the
thickness of the conductor, t
IB
VH 
nqt
• Everything in this expression can be measured
directly, except for 1/nq which is named the
Hall Coefficient, RH.
RH IB
VH 
t
29.6
• Since we can measure everything else,
determining the Hall Coefficient easily gives
the sign and density of charge carriers.
• It allows us to confirm that many conducting
metals give up one electron per atom.
• Certain metals and semiconductors give up
much fewer.
29.6
• Medical Application- because of the ions
carried by in the blood stream, the Hall effect
is used by electromagnetic blood flowmeters.
• The diameter of the artery is measured, a
magnetic field is applied, then the Hall voltage
is measured. From this vd can be determined.
• Example 29.8 p. 916