Magnetism

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Chapter 19

• Magnetism

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Magnets have two ends poles called north and
south. Like poles repel unlike poles attract.
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If you cut a magnet in half, you dont get a
north pole and a south pole you get two smaller
magnets.
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Types of Magnetic Materials

• Soft magnetic materials, such as iron, are easily
  magnetized
• They also tend to lose their magnetism easily
• Hard magnetic materials, such as cobalt and
  nickel, are difficult to magnetize
• They tend to retain their magnetism

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Sources of 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

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Magnetic Fields

• A vector quantity
• Symbolized by
• Direction is given by the direction a north pole
  of a compass needle points in that location
• Magnetic field lines can be used to show how the
  field lines, as traced out by a compass, would
  look

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Magnetic Field Lines, sketch

• A compass can be used to show the direction of
  the magnetic field lines (a)
• A sketch of the magnetic field lines (b)

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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

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Magnetic Field Lines, Unlike Poles

• 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
• Compare to the magnetic field produced by an
  electric dipole

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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
• Compare to the electric field produced by like
  charges

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Earths Magnetic Field

• The Earths geographic north pole corresponds to
  a magnetic south pole
• The Earths geographic south pole corresponds to
  a magnetic north pole
• Strictly speaking, a north pole should be a
  north-seeking pole and a south pole a
  south-seeking pole

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Earths Magnetic Field

• The Earths magnetic field resembles that
  achieved by burying a huge bar magnet deep in the
  Earths interior

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More About the Earths Magnetic Poles

• The magnetic and geographic poles are not in the
  same exact location
• The difference between true north, at the
  geographic north pole, and magnetic north is
  called the magnetic declination
• The amount of declination varies by location on
  the earths surface

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Earths Magnetic Declination
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Source of the Earths Magnetic Field

• There cannot be large masses of permanently
  magnetized materials since the high temperatures
  of the core prevent materials from retaining
  permanent magnetization
• The most likely source of the Earths magnetic
  field is believed to be electric currents in the
  liquid part of the core

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Reversals of the Earths Magnetic Field

• The direction of the Earths magnetic field
  reverses every few million years
• Evidence of these reversals are found in basalts
  resulting from volcanic activity
• The origin of the reversals is not understood

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Magnetic Fields

• When moving through a magnetic field, a charged
  particle experiences a magnetic force
• This force has a maximum value when the charge
  moves perpendicularly to the magnetic field lines
• This force is zero when the charge moves along
  the field lines

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Magnetic Fields, cont

• One can define a magnetic field in terms of the
  magnetic force exerted on a test charge moving in
  the field with velocity
• Similar to the way electric fields are defined

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Units of Magnetic Field

• The SI unit of magnetic field is the Tesla (T)
• Wb is a Weber
• The cgs unit is a Gauss (G)
• 1 T 104 G

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A Few Typical B Values

• Conventional laboratory magnets
• 25000 G or 2.5 T
• Superconducting magnets
• 300000 G or 30 T
• Earths magnetic field
• 0.5 G or 5 x 10-5 T

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Finding the Direction of Magnetic Force

• Experiments show that the direction of the
  magnetic force is always perpendicular to both
  and
• Fmax occurs when is perpendicular to
• F 0 when is parallel to

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Right Hand Rule 1

• Place your fingers in the direction of
• Curl the fingers in the direction of the magnetic
  field,
• Your thumb points in the direction of the force,
  , on a positive charge
• If the charge is negative, the force is opposite
  that determined by the right hand rule

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1) out of the page 2) into the page 3)
downwards 4) to the right 5) to the left
A positive charge enters a uniform magnetic
field as shown. What is the direction of the
magnetic force?
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A positive charge enters a uniform magnetic
field as shown. What is the direction of the
magnetic force?
1) out of the page 2) into the page 3)
downwards 4) to the right 5) to the left
Using the right-hand rule, you can see that
the magnetic force is directed to the left.
Remember that the magnetic force must be
perpendicular to BOTH the B field and the
velocity.
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1) out of the page 2) into the page 3)
downwards 4) upwards 5) to the left
A positive charge enters a uniform magnetic
field as shown. What is the direction of the
magnetic force?
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A positive charge enters a uniform magnetic
field as shown. What is the direction of the
magnetic force?
1) out of the page 2) into the page 3)
downwards 4) upwards 5) to the left
Using the right-hand rule, you can see that
the magnetic force is directed upwards. Remember
that the magnetic force must be perpendicular to
BOTH the B field and the velocity.
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1) out of the page 2) into the page 3)
zero 4) to the right 5) to the left
A positive charge enters a uniform magnetic
field as shown. What is the direction of the
magnetic force?
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A positive charge enters a uniform magnetic
field as shown. What is the direction of the
magnetic force?
1) out of the page 2) into the page 3)
zero 4) to the right 5) to the left
Using the right-hand rule, you can see that
the magnetic force is directed into the page.
Remember that the magnetic force must be
perpendicular to BOTH the B field and the
velocity.
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1) out of the page 2) into the page 3)
zero 4) to the right 5) to the left
A positive charge enters a uniform magnetic
field as shown. What is the direction of the
magnetic force?
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A positive charge enters a uniform magnetic
field as shown. What is the direction of the
magnetic force?
1) out of the page 2) into the page 3)
zero 4) to the right 5) to the left
The charge is moving parallel to the magnetic
field, so it does not experience any magnetic
force. Remember that the magnetic force is given
by F v B sin(q) .
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Quick Quiz
A charged particle moves in a straight line
through a region of space. Which of the following
answers must be true? (Assume any other fields
are negligible.) The magnetic field (a) has a
magnitude of zero (b) has a zero component
perpendicular to the particles velocity (c) has
a zero component parallel to the particles
velocity in that region.
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Answer
(b). The force that a magnetic field exerts on a
charged particle moving through it is given by
, where is the component
of the field perpendicular to the particles
velocity. Since the particle moves in a straight
line, the magnetic force (and hence ,
since ) must be zero.
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Quick Quiz
The north-pole end of a bar magnet is held near a
stationary positively charged piece of plastic.
Is the plastic (a) attracted, (b) repelled, or
(c) unaffected by the magnet?
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Answer
(c). The magnetic force exerted by a magnetic
field on a charge is proportional to the charges
velocity relative to the field. If the charge is
stationary, as in this situation, there is no
magnetic force.
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• A beam of atoms enters a magnetic field
  region. What path will the atoms follow?

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• A beam of atoms enters a magnetic field
  region. What path will the atoms follow?

Atoms are neutral objects whose net charge is
zero. Thus they do not experience a magnetic
force.
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Example 1
An electron gun fires electrons into a magnetic
field directed straight downward. Find the
direction of the force exerted by the field on an
electron for each of the following directions of
the electrons velocity (a) horizontal and due
north (b) horizontal and 30 west of north (c)
due north, but at 30 below the horizontal (d)
straight upward. (Remember that an electron has a
negative charge.)
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Example 2
(a) Find the direction of the force on a proton
(a positively charged particle) moving through
the magnetic fields in Figure P19.2, as shown.
(b) Repeat part (a), assuming the moving particle
is an electron.
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Example 3
An electron is accelerated through 2 400 V from
rest and then enters a region where there is a
uniform 1.70-T magnetic field. What are (a) the
maximum and (b) the minimum magnitudes of the
magnetic force acting on this electron?
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Practice 1
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
z-direction. Determine the magnitude and
direction of the field.
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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 right hand
  rule 1

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Force on a Wire

• The blue xs indicate the magnetic field 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
• In this case, there is no current, so there is no
  force

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Force on a Wire,cont

• B is into the page
• The current is up the page
• The force is to the left

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Force on a Wire,final

• B is into the page
• The current is down the page
• The force is to the right

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Force on a Wire, equation

• 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
• F B I l sin ?
• ? is the angle between and the direction of I
• The direction is found by the right hand rule,
  placing your fingers in the direction of I
  instead of

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1) left 2) right 3) zero 4) into the
page 5) out of the page

• A horizontal wire carries a current and is in a
  vertical magnetic field. What is the direction
  of the force on the wire?

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• A horizontal wire carries a current and is in
  a vertical magnetic field. What is the direction
  of the force on the wire?

1) left 2) right 3) zero 4) into the
page 5) out of the page
Using the right-hand rule, we see that the
magnetic force must point out of the page. Since
F must be perpendicular to both I and B, you
should realize that F cannot be in the plane of
the page at all.
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Torque on a Current Loop

• Applies to any shape loop
• N is the number of turns in the coil
• Torque has a maximum value of NBIA
• When q 90
• Torque is zero when the field is parallel to the
  plane of the loop

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Magnetic Moment

• The vector is called the magnetic moment of
  the coil
• Its magnitude is given by m IAN
• The vector always points perpendicular to the
  plane of the loop(s)
• The angle is between the moment and the field
• The equation for the magnetic torque can be
  written as t mB sinq

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Electric Motor

• An electric motor converts electrical energy to
  mechanical energy
• The mechanical energy is in the form of
  rotational kinetic energy
• An electric motor consists of a rigid
  current-carrying loop that rotates when placed in
  a magnetic field

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A galvanometer takes advantage of the torque on a
current loop to measure current.
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Quick Quiz
A square and a circular loop with the same area
lie in the xy-plane, where there is a uniform
magnetic field pointing at some angle ? with
respect to the positive z-direction. Each loop
carries the same current, in the same direction.
Which magnetic torque is larger? (a) the torque
on the square loop (b) the torque on the circular
loop (c) the torques are the same (d) more
information is needed
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Answer
(c). The torque that a planar current loop will
experience when it is in a magnetic field is
given by . Note that this
torque depends on the strength of the field, the
current in the coil, the area enclosed by the
coil, and the orientation of the plane of the
coil relative to the direction of the field.
However, it does not depend on the shape of the
loop.
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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

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Force on a Charged Particle

• Equating the magnetic and centripetal forces
• Solving for r
• r is proportional to the momentum of the particle
  and inversely proportional to the magnetic field
• Sometimes called the cyclotron equation

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Particle Moving in an External Magnetic Field

• If the particles velocity is not perpendicular
  to the field, the path followed by the particle
  is a spiral
• The spiral path is called a helix

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Quick Quiz
As a charged particle moves freely in a circular
path in the presence of a constant magnetic field
applied perpendicular to the particles velocity,
its kinetic energy (a) remains constant, (b)
increases, or (c) decreases.
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Answer
(a). The magnetic force acting on the particle is
always perpendicular to the velocity of the
particle, and hence to the displacement the
particle is undergoing. Under these conditions,
the force does no work on the particle and the
particles kinetic energy remains constant.
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Hans Christian Oersted

• 1777 1851
• Best known for observing that a compass needle
  deflects when placed near a wire carrying a
  current
• First evidence of a connection between electric
  and magnetic phenomena

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Magnetic Fields Long Straight Wire

• A current-carrying wire produces a magnetic field
• The compass needle deflects in directions tangent
  to the circle
• The compass needle points in the direction of the
  magnetic field produced by the current

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Direction of the Field of a Long Straight Wire

• Right Hand Rule 2
• Grasp the wire in your right hand
• Point your thumb in the direction of the current
• Your fingers will curl in the direction of the
  field

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Magnitude of the Field of a Long Straight Wire

• The magnitude of the field at a distance r from a
  wire carrying a current of I is
• µo 4 ? x 10-7 T.m / A
• µo is called the permeability of free space

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Ampères Law

• André-Marie Ampère found a procedure for deriving
  the relationship between the current in an
  arbitrarily shaped wire and the magnetic field
  produced by the wire
• Ampères Circuital Law
• ?B ?l µo I
• Sum over the closed path

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Ampères Law, cont

• Choose an arbitrary closed path around the
  current
• Sum all the products of B ?l around the closed
  path

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Ampères Law to Find B for a Long Straight Wire

• Use a closed circular path
• The circumference of the circle is 2 ? r
• This is identical to the result previously
  obtained

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Example 4
In 1962, measurements of the magnetic field of a
large tornado were made at the Geophysical
Observatory in Tulsa, Oklahoma. If the magnitude
of the tornados field was B 1.50 10-8 T
pointing north when the tornado was 9.00 km east
of the observatory, what current was carried up
or down the funnel of the tornado? Model the
vortex as a long, straight wire carrying a
current.
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Example 5
The two wires in Figure P19.40 carry currents of
3.00 A and 5.00 A in the direction indicated. (a)
Find the direction and magnitude of the magnetic
field at a point midway between the wires. (b)
Find the magnitude and direction of the magnetic
field at point P, located 20.0 cm above the wire
carrying the 5.00-A current.
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Practice 2
A long, straight wire lies on a horizontal table
and carries a current of 1.20 µA. In a vacuum, a
proton moves parallel to the wire (opposite the
direction of the current) with a constant
velocity of 2.30 104 m/s at a constant distance
d above the wire. Determine the value of d. (You
may ignore the magnetic field due to Earth.)
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André-Marie Ampère

• 1775 1836
• Credited with the discovery of electromagnetism
• Relationship between electric currents and
  magnetic fields
• Mathematical genius evident by age 12

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Magnetic Force Between Two Parallel Conductors

• The force on wire 1 is due to the current in wire
  1 and the magnetic field produced by wire 2
• The force per unit length is

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Force Between Two Conductors, cont

• Parallel conductors carrying currents in the same
  direction attract each other
• Parallel conductors carrying currents in the
  opposite directions repel each other

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Parallel currents attract antiparallel currents
repel.
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Quick Quiz
If, in Figure 19.28, I1 2 A and I2 6 A, which
of the following is true? (a) F1 3F2 (b) F1
F2 or (c) F1 F2/3
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Answer
(b). The two forces are an action-reaction pair.
They act on different wires and have equal
magnitudes but opposite directions.
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1) toward each other 2) away from each
other 3) there is no force

• Two straight wires run parallel to each other,
  each carrying a current in the direction shown
  below. The two wires experience a force in which
  direction?

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• Two straight wires run parallel to each
  other, each carrying a current in the direction
  shown below. The two wires experience a force in
  which direction?

1) toward each other 2) away from each
other 3) there is no force
The current in each wire produces a magnetic
field that is felt by the current of the other
wire. Using the right-hand rule, we find that
each wire experiences a force toward the other
wire (i.e., an attractive force) when the
currents are parallel (as shown).
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Example 6
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.
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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, ?x, contribute to the field,
  increasing its strength

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Magnetic Field of a Current Loop Total Field
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Magnetic Field of a Current Loop Equation

• The magnitude of the magnetic field at the center
  of a circular loop with a radius R and carrying
  current I is
• With N loops in the coil, this becomes

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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

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Magnetic Field of a Solenoid, 2

• The field lines inside the solenoid are nearly
  parallel, uniformly spaced, and close together
• This indicates that the field inside the solenoid
  is nearly uniform and strong
• The exterior field is nonuniform, much weaker,
  and in the opposite direction to the field inside
  the solenoid

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Magnetic Field in a Solenoid, 3

• The field lines of the solenoid resemble those of
  a bar magnet

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Magnetic Field in a Solenoid, Magnitude

• The magnitude of the field inside a solenoid is
  constant at all points far from its ends
• B µo n I
• n is the number of turns per unit length
• n N / l
• The same result can be obtained by applying
  Ampères Law to the solenoid

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If a piece of iron is inserted in the solenoid,
the magnetic field greatly increases. Such
electromagnets have many practical applications.
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Example 7
What current is required in the windings of a
long solenoid that has 1 000 turns uniformly
distributed over a length of 0.400 m in order to
produce a magnetic field of magnitude 1.00 10-4
T at the center of the solenoid?
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Practice 3
It is desired to construct a solenoid that will
have a resistance of 5.00 O (at 20C) and produce
a magnetic field of 4.00 10-2 T at its center
when it carries a current of 4.00 A. The solenoid
is to be constructed from copper wire having a
diameter of 0.500 mm. If the radius of the
solenoid is to be 1.00 cm, determine (a) the
number of turns of wire needed and (b) the length
the solenoid should have.
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Mass Spectrometer
A mass spectrometer measures the masses of atoms.
If a charged particle is moving through
perpendicular electric and magnetic fields, there
is a particular speed at which it will not be
deflected
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All the atoms reaching the second magnetic field
will have the same speed their radius of
curvature will depend on their mass.
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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

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Magnetic Effects of Electrons Orbits, cont

• The net result is that the magnetic effect
  produced by electrons orbiting the nucleus is
  either zero or very small for most materials

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Magnetic Effects of Electrons Spins

• Electrons also have spin
• The classical model is to consider the electrons
  to spin like tops
• It is actually a quantum effect

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Magnetic Effects of Electrons Spins, cont

• 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
• That is why most materials are not naturally
  magnetic

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Magnetic Effects of Electrons Domains

• In some materials, the spins do not naturally
  cancel
• Such materials are called ferromagnetic
• Large groups of atoms in which the spins are
  aligned are called domains
• When an external field is applied, the domains
  that are aligned with the field tend to grow at
  the expense of the others
• This causes the material to become magnetized

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Domains, cont

• Random alignment, a, shows an unmagnetized
  material
• When an external field is applied, the domains
  aligned with B grow, b

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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