Lecture Set 07—Magnetism
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Overview: Magnetism
Magnetism is closely related
to electricity:
●
Moving charges produce
magnetic fields
●
Magnetic fields can affect
moving charges
●
Magnetic fields can create
electric fields
Magnetic fields have as their
ultimate source charges in
motion (electric current)
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Fig. 7.1.1: A variety of
magnets.
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Part 1: Particle in a Magnetic Field
(Openstax 22.1-22.5)
Magnetic fields (B) are associated with moving charges, but
there are permanent magnets possible
Permanent magnets are associated
with poles—they have a north pole
and a south pole
●
The poles are the magnetic
equivalent of charges—same poles
repel, opposite poles attract
●
There
are
theoretically
no
magnetic monopoles
●
Fig. 7.1.2: A rail gun
uses magnetic fields to
launch a projectile at
very high speed
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Hard Magnetic Materials
Hard magnetic materials are used in
permanent magnets
●
Hard to magnetize, but then keep
their magnetization
●
These are “permanent” magnets
●
Provide permanent magnetic field
without electricity
●
Typically made from alloys of
cobalt-iron-nickel or aluminumnickel-cobalt (“alnico”), some rare
earth elements
●
Used in loudspeakers, magnetic
motors, computer hard drives
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Fig. 7.1.3: Field lines
from a permanent
magnet
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Soft Magnetic Materials
Soft magnetic materials can be
temporarily
magnetized
when
brought into contact with a magnet
●
They are easily magnetized but lose
their magnetization easily
●
Examples: iron, nickel, nickel-iron
alloys, ferrites (ferric oxide +
magnesium, nickel, etc)
●
Commonly used as magnetic
shields to prevent signals from
escaping from or being added to a
cable (e.g. a coaxial cable)
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Fig. 7.1.4: a) A coaxial cable
and b) a sketch of the
(external) magnetic field lines
approaching the shield
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Earth's Magnetic Field
The earth has a magnetic field which
can be treated as similar to the field
generated by a bar magnet:
Magnetic
south
corresponds
to
geographic north, and magnetic north
corresponds to geographic south
●
Result: a regular bar magnet's north pole
will point toward the north and a regular
bar magnet's south pole will point
towards the south
●
Magnetic field of the earth is likely
generated by the movement of charges in
the earth's liquid core
●
Direction reverses over time (~millions
of years)
●
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Fig. 7.1.5: The earth's
magnetic field (same
as the field produced
by a bar magnet)
along with geographic
poles (axis of rotation)
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Magnetic Fields
A charged particle interacts with magnetic fields if it is
moving:
●
A stationary charged particle is not affected by magnetic
fields
●
It does not generate magnetic fields either
Magnetic field can be defined in an analogous way to electric
field:
● Electric field defined via E = F /q.
E
●
Magnetic field defined via magnetic force and electric
charge, but also via particle velocity (magnitude and
direction):
FB
B≡
q v sin θ
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Units of Magnetic Field Magnitude
The SI unit for magnetic field strength (magnitude) in mks
units is the Tesla (T)
●
●
●
This is a very large field, as shown in Table 7.1.1.
Alternative unit is the cgs unit Gauss (G)
Equivalent to 1 T = 1 N/(C•m/s) and 1 T = 104 G.
Table 7.1.1: Some approximate magnetic field strengths; field strengths
at atomic nucleus are ~100 T, surface of a neutron are ~108 T.
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Magnetic Force
The magnetic force exerted on a moving
charge is thus:
⃗ B =q ⃗v × B
⃗
F
⇒
F B =qvB sin θ
Magnetic force is in a direction which
is perpendicular to both the magnetic
filed direction and the particle's
velocity
●
Force magnitude is maximized if the
velocity is perpendicular to the
magnetic field, and minimized (=0) if
the two are parallel
●
Opposite charges experience opposite
forces
●
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Fig. 7.1.6: The force
must be perpendicular
to the magnetic field
and to the particle's
velocity. Direction can
be found from the
right-hand rule.
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Right Hand Rule
The direction of the magnetic force acting on a moving positivelycharged particle is found using the right-hand rule
●
If the charge is negative, reverse the direction of the force
Fig. 7.1.7: Two
versions of the
right-hand rule
for determining
direction of the
magnetic force
(FB) given the
velocity (v) of a
positive charge
and magnetic
field (B).
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Longer Example—Proton in Earth's Magnetic
Field
LE7.1.1. A proton is traveling perpendicularly to the earth's
magnetic field near the earth's surface (strength: ~0.5 G). The
field points due north and the proton is moving straight
down.
a) What is the magnitude of the magnetic force if the proton's
speed is 1.00×104 m/s?
b) What is the magnitude of the acceleration of the proton?
c) In which direction will the magnetic force act?
d) What changes in your answers if the particle is an electron
instead?
N.B. A proton has mass 1.67×10-27 kg and charge +1.60×10-19 C,
whereas an electron has mass 9.11×10-31 kg and charge -1.60×10-19
C.
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Motion of a Charged Particle in a Uniform
Magnetic Field
The force acting on a moving
charged
particle
is
perpendicular
to
the
instantaneous velocity
This causes the particle to
“turn,” so the force is a type of
centripital force
●
In a constant, uniform field
this force results in the
particle's moving in a helix
●
“End-on” view is a circle, so
can consider in terms of
circular motion
●
If v ┴ B, then motion will be
circular only (not helical)
●
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Fig. 7.1.8: The motion of a
charged particle in a uniform
and constant magnetic field is
a helix
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Motion of a Charged Particle in a Uniform
Magnetic Field: Circular Component
Let us consider the circular
component of the particle's motion:
Magnetic force supplies the
centripital force, FC = FB.
●
Recall that centripital force is given
by FC = mac = m v2/r.
● Magnetic force is given by F =qvB.
B
●
Thus, the radius of the circular path
is
●
r=mv /qB
●
Period is given by:
T =2 π /ω=2 π m/qB
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Fig. 7.1.9: End-on view:
particle moves in a circle
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Nonuniform Magnetic Field: Magnetic Mirrors
The motion of a particle in a
nonuniform magnetic field is
more complex:
Magnetic mirror: particle reflects
from a region of high magnetic
field strength to a region of low
magnetic field strength
●
It's typically a combination of the
radial component of the field and
the azimuthal (circular) component
of the particle's velocity
●
Perpendicular force will be in the
opposite direction as the direction
along which the particle is
translating
z
●
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Fig. 7.1.10: A pair of magnetic
mirrors make a magnetic
bottle. Note that vz anf Fz are
in opposite directions when
moving from low field density
to high field density.
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Magnetic Bottles and the Van Allen Belt
Two opposing magnetic mirrors can
a)
be combined to create a magnetic
bottle:
●
Particle become trapped between the
two mirrors
●
b)
As each is mirror is approached, the
particle's translational (z-) velocity is
reversed
●
The earth forms a magnetic bottle
●
Charged particles ejected from the sun
(solar wind, Fig. 7.1.11a) get trapped
by the magnetic fields running
between the poles of the earth (Fig.
7.1.11b)
●
Fig. 7.1.11: Van Allen belts
These are the Van Allen belts
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Aurora Borealis (1)
The Aurora Borealis is is caused when charged ions from the
solar wind approach the earth near the poles:
●
Velocity component parallel to field lines carries the particles
towards poles (until reflected)
●
Perpendicular component confines the particles
Fig. 7.1.12: So much
space! In this photo of
the night sky over Ifjord,
Finland, we see stars,
the
Milky Way, the
Aurora Borealis, and
(look
closely)
some
shooting stars.
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Aurora Borealis (2)
Concentration of ions is greatest near poles:
●
Because the field lines are most dense near the poles, this
means that there will also be a greater concentration of
particles here
●
Hence, more collisions between charged particles and air
molecules
Fig. 7.1.12: So much
space! In this photo of
the night sky over Ifjord,
Finland, we see stars,
the
Milky Way, the
Aurora Borealis, and
(look
closely)
some
shooting stars.
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Aurora Borealis (3)
High particle concentration means more collisions:
●
Collisions between these charged particles and the air
molecules cause ionization (electrons ripped free)
●
Electrons re-combine with atoms in the air, releasing photons
(light)—hence the northern lights!
Fig. 7.1.12: So much
space! In this photo of
the night sky over Ifjord,
Finland, we see stars,
the
Milky Way, the
Aurora Borealis, and
(look
closely)
some
shooting stars.
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Lorentz Force
The combined force from an electric
field and a magnetic field is called
the Lorentz force (FL):
⃗ L=q [ ⃗
⃗ )]
F
E +(⃗v × B
Motion of a particle in such a
combination of fields is a
corkscrew motion.
●
However, the contribution to the
motion from the E-field causes the
particle to speed up (or slow down)
●
If particle speeds up, gyration
radius must increase
FL = FE + FB
FB
●
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+q
E F
E
B
θ
v
Fig. 7.1.13: Lorentz
force acting on a
charged particle
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Application: Cyclotron
The cyclotron is an early type of
particle accelerator invented by
E.O. Lawrence and M.S.
Livingston:
Accelerates charged particles to
very high speed
●
Used both E-field and B-Field
●
Charged particle moves in a spiral
through two semi-circular, DShaped
containers
(called
“dees”).
●
E-Field accelerates the particle as
it moves in the gap between the
two dees
●
B-Field confines the particle
●
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Fig. 7.1.14: Cyclotron design
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Application: Cyclotron (2)
There are two parameters of
interest for cyclotrons:
●
Cyclotron frequency, which is
also the frequency at which the
electric field should alternate
ωC =q B /m
●
The exiting energy for an ion:
K =mv 2 / 2=( q 2 B2 R 2 ) /(2 m)
R is the radius if the dees
Cyclotrons are often used in
hospitals to produce radioactive
substances for diagnosis and for
treatment (e.g. radiation therapy)
●
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Fig. 7.1.14: Cyclotron design
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Application: Velocity Selector
A velocity selector can be build by
making the E-Field and B-Field
perpendicular to each other:
●
Both are also perpendicular to the
initial particle velocity
●
Particles pass through a slit to set up
beam orientation
● F
= qv×B acts to the left in Fig.
B
7.1.15.
● F = qE acts to the right.
E
● If (and only if) F = F , there will be
E
B
no deflection
●
This happens when v = E/B
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Fig. 7.1.15: v selector
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Application: Mass Spectrometer
The velocity selector can be used to
make a mass spectrometer:
●
Particles pass through a second
pinhole near the end of the velocity
selector
●
The particle's initial velocity and
position are thus known
●
An additional B-field is set up after
the velocity selector
●
This deflects the particle with a
centripital acceleration a = v2/rm.
●
Ratio of mass to charge is thus:
m r B0 r B0 B
=
=
q
v
E
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⃗v
B
⃗
Ev
Fig. 7.1.16: Mass deflection
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Earth’s Magnetic Field Strength
Fig. 7.1.17: Magnetic field strength contour lines
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Earth’s Magnetic Field Strength
The earth’s magnetic field strength is not uniform everywhere at
the earth’s surface
●
The contour map to the
right has magnetic field
strengths near the surface
●
Units are in nT, so 50000
on this map is 50000 nT or
0.5 G.
●
The
maximum
field
strength is just greater than
0.65 G, and the minimum is
a bit less than 0.25 G
Fig. 7.1.17b: Closeup of the magnetic
contours for North America
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Summary of New Terms and Equations (1)
●
●
●
●
●
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Magnetic fields are caused by charged particles in
motion; they act upon other charged particle sin
motion.
The magnetic field strength is defined in terms of the
magnitude of the magnetic force (FB) per unit charge
(q) per speed (v)
The magnetic force (FB) is the electrical charge (q)
times velocity (v) crossed with the magnetic field (B)
The magnetic force must be perpendicular to both the
field and the charge velocity
Charged particles moving in a magnetic field will
experience a magnetic force which causes them to
take on a motion with a circular component; the
radius (r) and period (T) of the circular motion can be
determined from the field strength (B), the mass (m),
the charge (q), and the speed (v) of the particle
FB
B=
q v sin θ
⃗ B =q ⃗v × B
⃗
F
mv
r=
qB
2πm
T=
qB
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Summary of New Terms and Equations (2)
●
●
●
●
●
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Lorentz force (FL) is the total force experienced by
⃗ L=q E
⃗ + ( ⃗v × B
⃗)
a charged particle moving through an electric ( E) F
and magnetic (B) field.
Soft magnetic materials are easy to magnetize
when placed in an external magnetic field;
however, they lose this magnetization quickly once
the external magnetic field is removed
Hard magnetic materials are more difficult to
magnetize when placed in an external magnetic
field; however, they retain their magnetization
once the field is removed.
The earth's magnetic field is comparable to that of
a bar magnet with the south pole of the magnet at
the north pole of the earth
Opposite magnetic poles attract, and like poles
repel
[
]
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Lecture Set 07 Part 1 References (1)
Fig. 7.1.1: P. Urone, R. Hinrichs, K. Dirks, and M. Sharma, College Physics, OpenStax College, Fig. 22.3 (pp.
780).
Fig. 7.1.2: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole),
Fig. 19.1 (pp. 660).
Fig. 7.1.3: S. Hannahs, “Permanent Magnet,” National High Magnetic Field Laboratory, 23 September 2014,
web. Accessed 28 December 2015.
Fig. 7.1.4: a) “Coaxial Cable Cutaway,” Wikimedia Commons, 02 January 2012, web. Accessed 28 December
2015. b) “Magnetic Shielding,” Excel @ Physics, 2014, web. Accessed 28 December 2015.
Fig. 7.1.5: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice
Hall (Pearson), Fig. 27-5 (pp. 709).
Table 7.1.1: R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning
(Brooks and Cole), Table. 29.1 (pp. 873).
Fig. 7.1.6: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole),
Fig. 19.6 (pp. 665).
Fig. 7.1.7: R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning
(Brooks and Cole), Fig. 29.5 (pp. 872).
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Lecture Set 07 Part 1 References (2)
Fig. 7.1.8: R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning
(Brooks and Cole), Fig. 29.9 (pp. 876).
Fig. 7.1.9: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice
Hall (Pearson), Fig. 27-17 (pp. 715).
Fig. 7.1.10: “The Magnetic Bottle: How does a magnetic mirror reflect charged particles?” The XMM Satellite
Schoolpage, Astrophysics and Space Research Group, The University of Birmingham, web. Accessed
31 December 2015.
Fig. 7.1.11: a) E. Zolfagharifard, “Earth is being protected by a 'Star Trek-style invisible shield,': Scientists
probe mysterious barrier blocking 'killer electrons,” Daily Mail, 26 November 2014, web. Accessed 31
December 2015. Image copyright I. Cuming, Ikon Images, Corbis. b) R. Serway and J. Jewett, Physics
for Scientists and Engineers 9th Ed. (2014), Cengage Learning (Brooks and Cole), Fig. 29.12 (pp. 879).
Fig. 7.1.12: “Spectacular displays of the northern lights or aurora borealis in northern Norway,” Telegraph,
web. Accessed 31 December 2015. Photo by T. Eliassen, Barcroft Media.
Fig. 7.1.14: “Cyclotron,” Hyperphysics, Department of Physics and Astronomy, Georgia State University, web.
Accessed 31 December 2015.
Fig. 7.1.16:R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning
(Brooks and Cole), Fig. 29.13 (pp. 880).
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Lecture Set 07 Part 1 References (3)
Fig. 7.1.16: R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning
(Brooks and Cole), Fig. 29.14 (pp. 880).
Fig. 7.1.17: NOAA/NGDC and CIRES, “US/UK World Magnetic Model—Epoch 2015.0 Main Field Total
Intensity,” Wikipedia/Wikimedia Commons, 14 February 2015, web. Accessed 18 May 2018.
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Part 2: Current-Carrying Conductors
(Openstax 22.7-22.8)
Magnetic fields are generated by
moving charges, but they also
influence other moving charges:
●
This means that a current-carrying
conductor (e.g. a wire) can produce
a magnetic field
●
First observed by Hans Christian
Oersted
●
A current-carrying wire which is in
a magnetic field which it does not
produce will experience a force
from that field
●
If the wire make a loop, then it may
also experience a torque
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Fig. 7.2.1: a) Hans
Christian Oersted and b)
a close model of his
compass experiment
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Magnetic Force on a Wire
A current-carrying wire
contains
many
moving
charged particles:
●
Magnetic field can exert a
force on the charges in the
wire provided that current
is not parallel to the
magnetic field
●
Consequently the magnetic
field exerts a force on the
wire proportional to the
magnitude of charge and
the speed at which the
charges are moving
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Fig. 7.2.1: Magnetic field acts on a
current-carrying wire and causes
the wire to bend. The magnetic
field is into the page, the current is
0 (a), up (b) or down (c).
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Force on a Wire in an Uniform Magnetic Field
The total magnetic force magnitude is
therefore proportional to current:
●
The force is actually given by
F=qvdBnAcsLsinθ, where n is the density
of charge carriers and Acs is the crosssection area of the wire (see Fig. 7.2.2)
● The current I is given by I=nqv A .
d cs
●
Therefore, the force acting on a wire of
length L and current I in a magnetic field
of strength B (Fig. 7.2.3) is
⃗ B =I ⃗L× B
⃗
F
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Fig. 7.2.2
Fig. 7.2.3
F B =BIL sin θ
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Short Examples
SE7.2.1. A certain power line has a
current of 20.0 A and a length of 1.0
km. The power line (and hence the
current) is running from east-to-west.
What is the magnitude and direction
of the magnetic force acting on this
line?
SE7.7.2. If the linear density of the
Fig. 7.2.4: Power lines
above wires is 0.100 kg/m, how much which are running eastcurrent would the wires need for the west
magnetic force to have an equal
magnitude to gravity?
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Force on a Wire in a semi-Uniform
Magnetic Field
The previous expressions assume a
uniform magnetic field
●
Wire must also everywhere make
the same angle with the field
●
A more general case is given by
∑ F⃗ =∑ BILi sin(θi ) b⃗i
This works if, for example, the
wire makes a loop with discrete
sides (as in Fig. 7.2.5).
●
Force is applied to each side from
the magnetic field—to get the total
force, add up the forces for each
side!
●
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Fig. 7.2.5: Non-uniform B;
really, non-uniform L (or I).
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Application: Electromagnetic Pump for
Artificial Heart or Kidneys
One application of magnetic field force on a current-carrying
wire is the electromagnetic pump (Fig. 7.2.6):
Blood or other conducting fluids flows into the pump in one
direction
●
Current passes through the fluid perpendicularly to the direction in
which it is to be pumped
●
A constant magnetic field is
applied in a third direction,
perpendicular to both current and
velocity
●
Result: fluid is accelerated out of
pump
●
Does not risk damaging bloods
cells as a mechanical pump would Fig. 7.2.6: Electromagnetic
●
pump
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Longer Example—Electromagnetic Pump
LE7.2.2. Blood has a density of 1060 kg/m3. Suppose you are
designing an electromagnetic pump which has a cross-section
area of 1.00 cm2 and a pumping length of 2.00 cm. Blood will
slowly flow into the pump with vi=1.00 cm/s, but needs to
leave with a speed of vf=2.00 cm/s. You are able to put a
current of 1.00 mA through the blood without damaging it.
Estimate the magnetic field strength will you need to use for
this pump.
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Torque on a Current-Carrying Loop in a
Uniform Magnetic Field
When a current-carrying loop is
placed in a magnetic field:
Current direction is changing as
it moves around the loop.
●
Initially it may be in the +x
direction, and after traversing
180o worth of loop it will be in
the -x direction.
●
Thus if the magnetic field is
uniform, the force it exerts on
the loop will not have a uniform
direction
●
Result: magnetic field exerts a
net torque on the loop!
●
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Fig. 7.2.8: A current-carrying
loop may experience a net
torque and thus begin to spin
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Torque on a Current-Carrying Loop
Consider the current loop in a uniform B-field in Fig. 7.2.9:
●
Two sides are parallel to the B-field, so there is no force from the
magnetic field on these sides
●
Current in other two sides has a component which is
perpendicular to the magnetic field: thus there is a nonzero force
on each side of equal magnitudes
Fig. 7.2.9
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Torque on a Current-Carrying Loop (2)
Since current is in opposite directions, the force points in opposite
direction for these two sides, resulting in a torque:
●
⃗ ⇒ τ=r F sin θ
Torque is given by ⃗τ =⃗r × F
● The forces are given by F =F =BIb (Fig. 7.2.9a)
1
2
●
If the wire rotates about point O (Fig. 7.2.9b), then the lengths
will add to a. This is true for any axis of rotation!
Fig. 7.2.8
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Torque on a Current-Carrying Loop (3)
The total torque is therefore given by:
●
If the axis of rotation is some distance x from the end of the side
of length a, then the torque is given by
τ=τ1 + τ2=F 1 x+F 2 (a−x)=BIb x+BIb(a−x)=BIba
●
The product of the lengths b and a is the area Aloop.
Fig. 7.2.9
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Torque on a Current-Carrying Loop (4)
The magnetic field need not be perpendicular to one pair of
sides—it may make an oblique angle as in Fig. 7.2.9c.
●
τ=B I A loop sin θ
The total magnitude of torque is therefore
●
Finally, the loop may consist of N coils, in which case the
torque becomes
τ=B I A loop N sin θ
Fig. 7.2.9
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Magnetic Moment
The quantity NIA is the magnitude of the loop's magnetic
moment. The magnetic moment μB is a vector
●
It is always perpendicular to the direction of the plane of the
loop or loops
●
Direction defined by right hand rule: curl fingers in the direction
of the current in the loop, and the thumb of the right hand points
in the direction of the magnetic moment
●
The angle θ is therefore the angle between the magnetic field
and the magnetic moment
●
⃗ ⇒ τ=μ B sin θ=BIAN sin θ
The torque is therefore
⃗τ =μ
⃗×B
Although this discussion has been limited to a rectangular loop,
the loop's geometry is not actually important
●
⃗
U B =−μ
⃗⋅B
Magnetic potential energy is given by
●
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Rotation in the Magnetic Field
A net torque causes an angular
acceleration, so the loops will
rotate in the B-field:
●
The torque will as a
consequence
change
magnitude (Fig. 7.2.10a-b)
●
There will also be a pair of
equilibrium orientations (Fig.
7.2.10c) at which the torque is
0.
●
Finally, the torque will change
directions as the loop passes
through
the
equilibrium
positions (Fig. 7.2.10d)
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Short Examples
SE7.2.3. Calculate the initial magnetic moment of a circular
loop (single coil) of radius 1.00 m carrying a current of 10.0
mA in a magnetic field of strength 1.00 mT. Assume that the
field is pointing to the right and that the loop is lying flat on
the page with a clockwise current.
SE7.2.4. Calculate the initial torque for the above loop.
SE7.2.5. How do your answers change if the magnetic field is
point up out of the page instead?
G.P.II
L07P2
Application: DC Motor
If the current and magnetic field are both constant, then the loop
will behave like an oscillator:
● For a small maximum angle θ
, the loop will undergo simple
max
harmonic oscillations about it equilibrium
●
A motor requires that
the loop continues to
rotate in the same
direction
●
This is accomplished
by changing the current
direction every halfcycle
●
Accomplished
using
Fig. 7.2.11: Brushes with split-ring
brushes (Fig. 7.2.11)
commutator reverse current every 180o.
G.P.II
L07P2
Application: Galvanometer
In principle, we could also use an ammeter to measure the
induced current. What's the difference?
●
Galvanometer is used to measure relatively small currents;
ammeter is needed for larger currents (~1 mA or more)
●
Galvanometer can detect direction of current
●
Based on torque applied to a wire carrying current in a Bfield
●
Can convert a Galvanometer into an ammeter by connecting a
small resistor (“shunt resistance”) in parallel with it.
●
Can also convert to a voltmeter by connecting a relatively
large resistor in series with it.
G.P.II
L07P2
Application: Galvanometer
In principle, we could also use an ammeter to measure the
induced current. What's the difference?
●
Galvanometer is used to measure relatively small currents;
ammeter is needed for larger currents (~1 mA or more)
Galvanometer
can
detect direction of
current
●
Based on torque
applied to a wire
carrying current in a
B-field (Fig. 7.2.12)
●
Fig. 7.2.12: Basic schematic of a Galvanometer
G.P.II
L07P2
Application: Z Machine for Fusion Research
and X-Ray Generation (1)
One interesting application of
the Lorentz force is found in
Sandia National Laboratory's ZMachine (Fig. 7.2.13a):
●
A series of wires surrounds a
DT fuel pellet
●
A large current (Fig. 7.1.13 bd, orange lines) is driven
through the wires, which
vaporizes and then ionizes
them to make a plasma
●
The current continues through
the plasma (an excellent
conductor)
G.P.II
a)
b)
c)
d)
Fig. 7.1.13: a) The Z Machine,
and b)-d) the physics of z-pinch
L07P2
Application: Z Machine for Fusion Research
and X-Ray Generation (2)
The current runs in the Z- a)
direction:
●
Current generates a B-field
(Figs. 7.1.13a-c, blue)
●
Lorentz force compresses the
wires or plasma (Fig. 7.1.13c)
by applying a net “inward”
force on it.
●
You can check this by taking b)
c)
d)
the cross product: z×θ = -r
(cylindrical)
or
z×y=-x
(Cartesian) using RHR
●
This heats the plasma and
generates x-rays (Fig. 8.1.13d) Fig. 7.1.13: a) The Z Machine,
and b)-d) the physics of z-pinch
G.P.II
L07P2
Summary of New Terms and Equations
●
●
●
●
●
G.P.II
The force (FB) which a magnetic field (B) exerts on a
current-carrying wire is determined by the current (I)
in and length (L) of the wire
The magnetic torque on a current-carrying loop of
wire is given by the magnetic moment (μ) crossed
with the magnetic field (B)
The potential energy (UB) of a current-carrying loop
of wire is given by the magnetic moment ( μ) dotted
with the magnetic field (B)
The magnetic moment (μ) of a current-carrying loop
is determined by the current (I) in the loop, the
number of turns (N), and the enclosed area (A) of
each complete turn
Thus, a current-carrying loop of N turns and enclosed
area A per turn placed in a magnetic field will
experience a torque (τ)
⃗ B =I ⃗L× B
⃗
F
⃗
⃗τ =μ
⃗×B
⃗
U B =−μ
⃗⋅B
μ
A
⃗ =N I ⃗
τ=BIAN sin θ
L07P2
Lecture Set 07 Part 2 References (1)
Fig. 7.2.1: a) “Hans-Christian Oersted,” Indian Centre for Ocean Information Services (ESSO), web. Accessed
29 December 2015. b) A. Privat-Deschanel, “Elementary Treatise on Natural Philosophy, Part 3:
Electricity and Magnetism” (1876), D. Appleton and Co.,New York, Tr. J.D. Everett, Fig. 456 (pp.656).
Fig. 7.2.2: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole),
Fig. 19.11 (pp. 668).
Fig. 7.2.3: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole),
Fig. 19.12 (pp. 668).
Fig. 7.2.4: S. Salzberg, “Do High Voltage Power Lines Cause Cancer?”, Fighting Pseudoscience, Forbes, 01
September 2014, web. Accessed 28 December 2015.
Fig. 7.2.5: My own drawing. Copyright J. Sanders, 2016.
Fig. 7.2.6: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole),
Fig. 19.14 (pp. 669).
Fig. 7.2.7: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice
Hall (Pearson), Fig. 27-52 (pp. 731).
Fig. 7.2.8: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole),
Fig. 19.15 (pp. 670).
G.P.II
L07P2
Lecture Set 07 Part 2 References (1)
Fig. 7.2.9: P. Urone, R. Hinrichs, K. Dirks, and M. Sharma, College Physics, OpenStax College, Fig. 22.35
(pp. 797).
Fig. 7.2.10: P. Urone, R. Hinrichs, K. Dirks, and M. Sharma, College Physics, OpenStax College, Fig. 22.36
(pp. 798).
Fig. 7.2.11: P. Urone, R. Hinrichs, K. Dirks, and M. Sharma, College Physics, OpenStax College, Fig. 22.37
(pp. 798).
Fig. 7.2.12: R. Hamberly, “Laser Fusion: Lighting the Earth with Exploding Stars,” Stanford University, web.
Accessed 04 January 2016. Fig a) from Fig 12, Figs. b)-d) from Fig. 10, of this source.
Fig. 7.2.13: C. Freudenrich, “How Fusion Reactors Work—Fusion Reactors: Magnetic Confinement,”
HowStuffWorks.com, 11 August 2005, web. Accessed 04 January 2016.
G.P.II
L07P2
Part 3: Sources of Magnetic Fields
(Openstax 22.8-22.11, 22.6)
We have so far considered the
effects of magnetic fields on
moving charges
●
Now we turn to the sources of
magnetic fields, which are other
moving charges
●
Can be produced by a single
moving charge (does not effect
this charge)
●
Can be produced by a current,
e.g. in a wire
●
Can be produced a magnetized
material
G.P.II
Yo
dawg!
Fig. 7.3.1: I heard you like
magnets, so I put an
electromagnet inside a
horseshoe magnet….
L07P3
Magnetism Produced by Current
A current carrying wire produces a magnetic field:
●
First discovered by Hans Christian Oersted. Can use a compass
to map magnetic field (Fig. 7.3.2).
●
Direction can be found from the right-hand rule: point thumb
in direction of current, and grab wire; fingers point in direction
of B-field
Fig. 7.3.2: Using a compass to
map the magnetic field. When
there is no current (a), the
compasses all point northsouth, but when a current is
applied (b) the compasses
point in a circle around the
wire.
G.P.II
L07P3
Magnetism Produced by Current (2)
The strength (magnitude) of the
magnetic field can be measured:
●
A loop placed partially into the B-field
can be used to measure field strength
(Fig. 7.3.3).
●
Alternatively, iron filings can be
placed into the field to get relative
field strength
●
The magnetic field strength for a long
straight wire is thus
μ0 I
B=
2πr
Proportionality constant μ0=4π×10-7
T•m/A is permeability of free space
G.P.II
Fig. 7.3.3: A simple
device for measuring
magnetic field strength
L07P3
Magnetic Force Between Two Long
Parallel Wires
Consider two long, straight wires carrying
currents I1 and I2 as in Fig. 7.3.4:
●
Each current produces a B-Field (the field for
wire 1 wire shown, but not for wire 2)
●
Magnitude of field produced by wire 1 at
the location of wire 2 is B1 = μ0I1/(2πr) .
●
This B-Field applies a force to the current
(moving charges) in the other wire
●
Can thus find force per unit length, F/L
between the two wires
Fig.
7.3.4a
Fig.
7.3.4b
μ0 I 1 I 2
F
=I 2 B 1=
L
2π r
G.P.II
L07P3
Short Examples
SE7.3.1: A certain long wire carries a current
of 1.00 A. What is the magnitude of the
magnetic field 2.00 cm away from this wire?
SE7.3.2: Two parallel wires are placed a
distance of 2.00 cm apart. Each carries a
current of 1.0 A, and both are 1.0 m long.
What is the magnitude of the force between
them?
G.P.II
Fig.
7.3.4a
Fig.
7.3.4b
L07P3
Operational Definitions for Units:
Ampere and Coulomb
The force between two parallel
wires can be used to
conveniently define units of
current and charge:
●
Current: one Ampere is the
current needed to cause a force
of 2×10-7 N/m to exist between
two long parallel wires
carrying equal current and
separated by 1 meter
●
Charge: one Coulomb is the
charge passing through a cross
section per second of a
conduct carrying 1 A of steady
current
G.P.II
Fig. 7.3.5: a) Currents in same
direction pull wires together. b)
Currents in opposite directions
push wires apart
L07P3
Ampere's Law
The wire does not need to be straight to determine the
magnetic field which it produces:
●
Consider an arbitrary closed
loop around a current (Fig.
7.3.6)
●
The path consists of many
small segments of length Δl
●
The magnetic field has some
component Bǁ which is
parallel to the small segment
of the loop
●
The product of these can be
added for all segments
Fig. 7.3.6: An Amperian loop.
G.P.II
L07P3
Ampere's Law (2)
Ampere's Law—discovered by Andre-Marie Ampere:
●
States that the sum of these
products ΔlBǁ is equal to the
enclosed current Ienc times the
permeability of free space:
∑ B ∥ Δ L=μ0 I enc
⃗
Note that Δ L B ∥ =Δ ⃗l⋅B
●
Ampere's Law is only valid in
this form if the currents and field
are constant (not changing in
Fig. 7.3.6: An Amperian loop.
time)
●
G.P.II
L07P3
Right Hand Rules
The right-hand rule can be used to
determine the direction of the field:
●
With a straight wire
●
With an Amperian loop
Fig. 7.3.7: The right-hand rule applied to two geometries
G.P.II
L07P3
Longer Example—A Coaxial Cable
LE7.3.1. Consider a coaxial cable: it consists of an insulated
wire of radius r = 0.400 mm which carries a current of 4.00
A, surrounded by a cylindrical conductor of inner radius
2.40 mm and thickness 0.200 mm which carries a current of
4.00 A in the opposite direction. Determine the magnetic
fields magnitudes (and directions) at the following
locations:
a) At r = 0.100 mm (inside the wire)
b) At r = 0.500 mm (between the wire and the conductor)
c) At r = 2.50 mm (inside the conductor)
d) At r = 2.60 mm (immediately outside of the conductor)
Assume that the wire and the conductor each have a
uniform current density.
G.P.II
L07P3
Solenoid
A solenoid consist of a long currentcarrying wire which is twisted in the
shape of a helix:
The magnetic field inside a solenoid is
shown in Fig. 7.3.9.
●
Inside the solenoid, the field lines are
~uniform straight lines running parallel
to the solenoid's central axis
●
Outside, the lines resemble though of a
bar magnet
●
Tighter coiling means more uniform
(straight) interior fields
●
Ideal solenoid: turns are infinitesimally
close and solenoid's length >> radius of
Fig. 7.3.9: Solenoid B-field
solenoid's turns
●
G.P.II
L07P3
B-Fields in a Solenoid
Use Ampere's Law to determine the value
of the B-field inside the solenoid:
●
Take the Amperian loop to be the dashed
rectangle in Fig. 7.3.10.
●
Dashed circle gives field outside the
solenoid, which is very weak for an ideal
solenoid!
●
⃗⋅Δ ⃗L =BL
The closed loop gives ∑ B
● And the enclosed current is I
= NI,
enc
where N is the number of loops enclosed
●
Ampere's Law thus gives (for n = N/L)
B L=μ 0 N I
G.P.II
⇒
B=μ 0 n I
Fig. 7.3.10
L07P3
Torroid
A torroid is a solenoid which is wrapped in
a circle as in Fig. 7.3.11a:
●
We can use Ampere's law both inside and
outside of the torroid
●
Outside the torroid, we follow path 2
●
The total current enclosed is 0
●
Thus, the B-field outside the torroid is
also 0!
●
Inside the torroid, we follow path 1
● Enclosed current is I
=NI
enc
●
Path length is L = 2πr
●
Total field inside the torroid is thus
B=μ 0 N I /2 π r
G.P.II
Fig.
7.3.11
L07P3
Short Examples
SE7.3.3: A certain solenoid has a total
length of 1.00 m and 500 turns. The
wire of the solenoid carries a current
of 1.50 A. What is the magnitude of
the magnetic field produced inside this
solenoid?
SE7.3.4: A certain torroid has a total
diameter of 2.00 m and 1250 turns.
The wire of the torroid carries a
current of 0.50 A. What is the
magnitude of the magnetic field
produced inside of this torroid?
Fig. 7.3.9: Solenoid B-field
G.P.II
L07P3
Longer Examples
LE7.3.2: A certain solenoid has a total
length of 1.00m, an inner radius of
2.50 cm, and an outer radius of 2.60
cm. The wire of the solenoid carries a
current of 1.50 A, and it is wound such
that the space between consecutive
wires is equal to the wire's radiu. What
is the magnitude of the magnetic field
produced by this solenoid?
Fig. 7.3.9: Solenoid B-field
G.P.II
L07P3
Gauss's Law in Magnetism
Ampere's Law for B-Fields is the functional or
operational equivalent to Gauss's law for EFields
●
However, Gauss's Law can apply for any field,
including B-Fields (Fig. 7.3.12a)
●
Recall that the flux of a field is found using ( V
is a vector field)
⃗ A
⃗
ΦV ≡V⋅
a)
b)
Unit of magnetic flux is T•m2 = Wb (weber)
●
For magnetic field flux through a closed
surface, Gauss's law gives Φ B =0
●
The implication is that there are no isolated
magnetic monopoles in nature (Fig. 7.3.12b)
●
Fig. 7.3.12: The flux
G.P.II
L07P3
Magnetic Materials
The field B0 inside a a solenoid is
given by B0 = μ0nI if the solenoid is
filled with vacuum:
●
Can fill the space of the solenoid with
other material (like iron)
●
The total magnetic field of the
solenoid will be increased in this
case.
●
The total field can be written as:
⃗ =⃗
⃗M
B
B0 + B
●
G.P.II
BM is the field due to the magnetic
material
Fig. 7.3.13: The total Bfield in an iron-core toroid
as a function of Bo.
L07P3
Magnetic Domains: Understanding the
Enhancement of the B-Field (1)
Certain materials like iron (or
nickel, cobalt, and their alloy) can
be made into strong magnets
●
This enhances the externally
produced magnetic field of the
solenoid
●
Such materials consist of small
magnetic
domains,
each
behaving like a tiny bar magnet
with a north and south pole
●
Bar magnet is a magnetic
dipole, magnetic analogue to
the electric dipole
G.P.II
Fig.
7.3.14:
Magnetic
domains in an unmagnetized
and in a magnetized piece of
material. Arrow heads point
toward the north pole of the
magnetic domain.
L07P3
Magnetic Domains: Understanding the
Enhancement of the B-Field (2)
Domains' alignment determines
whether the whole material will act
as a magnet or not:
●
If the domains are oriented
randomly (Fig 7.3.14 left), the
material will be unmagnetized
●
Individual domains will cancel
with each other
●
If the individual domains are
oriented in a common direction,
the material will be magnetized
●
Individual domains will largely
add to each other to enhance the
field strength.
G.P.II
Fig.
7.3.14:
Magnetic
domains in an unmagnetized
and in a magnetized piece of
material. Arrow heads point
toward the north pole of the
magnetic domain.
L07P3
Magnetic Domains: Understanding the
Enhancement of the B-Field (3)
Unmagnetized
pieces
of
ferromagnetic material can become
magnetized by placing them in a
sufficiently strong B-field
Domains align with the existing
external field (Fig. 7.3.14 right)
●
This also explains why a permanent
magnet can pick up “unmagnetized”
pieces of iron
●
Domains in iron realign with
external field
●
Conversely, if a permanent magnet
is
jarred
sufficiently,
it
demagnetizes
●
G.P.II
Fig.
7.3.14:
Magnetic
domains in an unmagnetized
and in a magnetized piece of
material. Arrow heads point
toward the north pole of the
magnetic domain.
L07P3
The Hall Effect (1)
Consider a conductor carrying a current in a B-field (Fig.
7.3.15):
●
Magnetic field applies a force to the charge carriers in the
current
●
The
charge-carriers
will
deflect as a result of this
●
This ultimately means that net
charges
will
begin
to
accumulate along edges of the
conductor
●
As net charges accumulate, a
potential difference
builds
between the two opposite edges
Fig. 7.3.15: Hall effect acting
●
G.P.II
This is the Hall voltage ΔVH
on electrons in the conductor
L07P3
The Hall Effect (2)
The existence of a potential difference across a gap implies the
existence of an electric field, ΔVH = Ehd:
●
This E-field will apply a force on the charge carriers which
acts in the opposite direction as the B-Field's force
⃗ B=− F
⃗ EH ⇒ q ⃗v d × B
⃗ =q E
⃗H
F
Where vd is the charge-carrier
drift-velocity
●
Therefore,
the
potential
difference across the conductor
(between points C and D in Fig.
7.3.15) is
Δ V H =v d B d
●
G.P.II
Fig. 7.3.15: Hall effect acting
on electrons in the conductor
L07P3
Hall Probe
We can use the hall effect to probe magnetic field strength:
Need to measure the Hall voltage VH between points C and D
(Fig.7.3.16)
●
The drift speed can be expressed as
(Lecture Set 05)
●
I=n q v d A ⇒ v d =I /(nqA )
A is the cross-sectional area of the
conductor, n is the charge carrier
density
●
A = wd in Fig. 7.3.17.
●
Can measure the current I
●
The Hall voltage is thus
D
C
●
IB R H IB
Δ V H=
=
nqw
w
G.P.II
L
d
w
Fig. 7.3.16: Hall effect acting
on electrons in the conductor
L07P3
Longer Example: Hall Probe
LE7.3.3: A rectangular silver strip
is 2.5 cm long and 0.095 cm thick.
It carries a current of 4.0 A.
Determine the following:
D
C
a) Hall coefficient RH = 1/nq.
b) Hall voltage when this strip is
placed in a magnetic field of L
magnitude 0.75 T such that the Hall
voltage is developed between top and
w
bottom edges of the strip (e.g.
d
between the edges which are 2.5 cm
Fig. 7.3.16: Hall effect acting
apart).
c) What magnetic field strength would on electrons in the conductor
produce a Hall voltage of 100 nV?
G.P.II
L07P3
Summary of New Terms and Equations (1)
●
●
●
●
●
G.P.II
Magnetic fields are ultimately produced by moving
charges (e.g. electrical current)
The magnetic field strength (B) produced by a
current-carrying wire is given by the current (I) at a
distance r from the wire
The constant μ0 is called the permeability of free
space, and it is a fundamental constant of the
universe, μ0=4π×10-7 T•m/A.
The force per unit length of two current-carrying
wires is dependent on their separation distance (r)
and the currents in each wire ( I1 and I2 respectively)
Ampere's Law states that the sum of the products of
the magnetic field and the path length about any
closed loop is equal to the total current (I) enclosed
by the loop.
⃗ B =I ⃗L× B
⃗
F
μ0 I
B=
2πr
F μ0 I 1 I 2
=
L 2π r
∑ B⃗⋅Δ ⃗l =μ0 I enc
L07P3
Summary of New Terms and Equations (2)
●
●
G.P.II
Magnetic field produced by a solenoid is
dependent on the current (I) and the number of
turns per unit length (n) for the solenoid
Gauss's Law for magnetic field states the the flux
of the magnetic field through a closed surface is
zero, the implication of which is that there are no
magnetic monopoles.
B=μ 0 n I
⃗⋅⃗
ΦB≡ B
A =0
L07P3
Lecture Set 07 Part 3 References (1)
Fig. 7.3.1: M. Brain, “How Electric Motors Work: Electromagnets and Motors,” HowStuffWorks.com, 01 April
2000, web. Accessed 31 December 2015. Inset from Yo Dawg.com.
Fig. 7.3.2: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole),
Fig. 19.23 (pp. 676).
Fig. 7.3.3: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice
Hall (Pearson), Fig. 27-13 (pp. 713).
Fig. 7.3.4: P. Urone, R. Hinrichs, K. Dirks, and M. Sharma, College Physics, OpenStax College, Fig. 22.38
(pp. 799). and Fig. 22.42 (pp. 804)
Fig. 7.3.5: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice
Hall (Pearson), Fig. 28-6 (pp. 735).
Fig. 7.3.6: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole),
Fig. 19.25 (pp. 678).
Fig. 7.3.7: P. Urone, R. Hinrichs, K. Dirks, and M. Sharma, College Physics, OpenStax College, Fig. 22.38
(pp. 799). and Fig. 22.39 (pp. 800).
Fig. 7.3.9: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice
Hall (Pearson), Fig. 28-15 (pp. 741).
G.P.II
L07P3
Lecture Set 07 Part 3 References (2)
Fig. 7.3.10: R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning
(Brooks and Cole), Fig. 30.18 (pp. 916).
Fig. 7.3.11: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice
Hall (Pearson), Fig. 28-17 (pp. 742).
Fig. 7.3.12: a) R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage
Learning (Brooks and Cole), Fig. 30.19 (pp. 917). b) R. Serway and J. Jewett, Physics for Scientists and
Engineers 9th Ed. (2014), Cengage Learning (Brooks and Cole), Fig. 30.22 (pp. 918).
Fig. 7.3.13: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice
Hall (Pearson), Fig. 28-28 (pp. 748).
Fig. 7.3.14: A.D. Elster, “Ferromagnetism: What is Ferromagnetism,” Questions and Answers in MRI, 2014,
web. Accessed 05 January 2016.
Fig. 7.3.15: R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning
(Brooks and Cole), Fig. 29.26 (pp. 890).
Fig. 7.3.16: A. Ajaja, “The Hall Effect,” 2000, web. Accessed 06 January 2016.
G.P.II
L07P3
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