Magnetism

1
Chapter 19

• Magnetism

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Importance of Magnetism

• In Terms of applications one of the most
  important fields Generators, transformers,
  motors, loudspeakers, sensors, magnetic tapes,
  hard disks. Levitation, MRI, cyclotrons,
  synchrotrons.
• Still very active field of experimental and
  theoretical research.

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Magnets

• Poles of a magnet are the ends where objects are
  most strongly attracted
• Two poles, called north and south
• Like poles repel each other and unlike poles
  attract each other
• Similar to electric charges
• Magnetic poles cannot be isolated
• If a permanent magnetic is cut in half
  repeatedly, you will still have a north and a
  south pole
• This differs from electric charges
• There is some theoretical basis for monopoles,
  but none have been detected

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More About Magnetism

• An unmagnetized piece of iron can be magnetized
  by stroking it with a magnet
• Somewhat 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

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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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Assortment of Commercial Magnets
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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 electric 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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Dip Angle of Earths Magnetic Field

• If a compass is free to rotate vertically as well
  as horizontally, it points to the earths 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

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More About the Earths Magnetic Poles

• The dip angle of 90 is found at a point just
  north of Hudson Bay in Canada
• This is considered to be the location of the
  south magnetic pole
• 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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Magnetic declination map from 1860
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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

Vector cross product
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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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Cork-screw rule

• Alternatively
• Turn v in the direction of B on the shortest way.
  The direction of the force is given by the
  direction in which the cork-screw proceeds.

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Example 19.1 A proton traveling in EarthMagnetic
field

• A proton moves with a speed of 105 m/s through
  Earthmagnetic field, which has a value of 55 µT.
  When the proton moves northward, no magnetic
  force acts on it. (a) What is the direction of
  the magnetic field, and (b) what is the strength
  of the magnetic force when the proton moves
  eastward? (c) Calculate the gravitational force
  on the proton and compare it with the magnetic
  force. Compare it also with the electric force if
  there were an electric field with magnitude E
  1.5 102 N/C at that location, a common value at
  Earthsurface (mproton 1.6710-27 kg).

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Example 19.1 A proton traveling in EarthMagnetic
field Solution

• Since no force acts on the proton when it goes
  north, the magnetic field direction must be
  either 0 or 180. When the particle travels
  east, the magnetic force is directed upward. If
  the open flat hands points to the east and the
  thumb to the right (upward from the earth) the
  curled fingers point to the north, i.e. the
  magnetic field points north.
• Magnitude of the field

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Example 19.1 A proton traveling in EarthMagnetic
field Solution 2
(c) Gravitational force on the proton
Electric force for E 1.5102 N/C
Compared with magnetic force
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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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Demonstration of the force of a magnetic field on
a current carrying wire
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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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Application Loudspeaker
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Application Electromagnetic Pump for Blood
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Example 19.3 A Current-Carrying Wire in Earths
Magnetic field

• A wire carries a current of 22 A from west to
  east. Assume that at this location the magnetic
  field of Earth is horizontal and directed from
  south to north and that is has a magnitude of
  0.510-4 T. (a) Find the magnitude and direction
  of the magnetic force on a 36-m length of cable.
  (b) calculate the gravitational force on the same
  length of wire if its made of copper and has a
  cross-sectional area of 2.510-6 m2.

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Example 19.3 A Current-Carrying Wire in Earths
Magnetic field Solutions
Magnetic force on the wire
The force is directed upwards
Gravitational force on the wire
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Torque on a Rectangular Loop in a Magnetic Field
Top view of a rectangular loop in a uniform
magnetic field B
Side view of the rectangular loop and magnetic
forces
Forces on the sides of length b F1 F2 BIb
Maximum Torque tmax F1a/2 F2a/2 BIab
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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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Quick Quiz 19.3

• A square and a circular loop with the same area
  lie in the xy-plane, where there is a uniform
  magnetic Field B 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

Answer (c)
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Example 19.4 The Torque on a Circular Loop in a
Magnetic Field

• A circular wire loop of radius 1 m is placed in a
  magnetic field of 0.5 T. The normal of the loop
  makes an angle of 30 with the magnetic field.
  The current in the loop is 2 A in the direction
  shown. (a) Find the magnetic moment of the loop
  and the magnitude of the torque at this instant.
  (b) The same current is carried by the
  rectangular 2-m by 3-m coil with three loops
  shown in Fig. (b). Find the magnetic moment of
  the coil and the torque acting on the coil at
  that instant.

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Example 19.4 The Torque on a Circular Loop in a
Magnetic Field

• Solutions
• (a) Magnetic moment circular loop
• µ IAN 2 p11 6.28 Am2
• Torque
• t µBsin? 6.280.5sin 30 1.57 Nm
• (b) Magnetic moment rectang. loop

µ IAN 2233 36 Am2
Torque t µBsin? 360.5sin30 9 Nm
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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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Electric Motor, 2

• The torque acting on the loop will tend to rotate
  the loop to smaller values of ? until the torque
  becomes 0 at ? 0
• 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

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Electric Motor, 3

• 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
• Actual motors would contain many current loops
  and commutators

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Electric Motor, final

• 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

51
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

52
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
• Cyclotron frequency ? qB/m ? measure-ment of
  q/m (also in solids)

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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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Applying Physics 19.3 Trapping Charges
Consider separately the components of the
particle velocity parallel vp and perpendicular
vs to the field lines. There is no magnetic force
on the particle associated with the velocity
component parallel to the field lines ? ?vp
0 vs will experience a magnetic force
perpendicular to it and to B. The particle
follows a circular arc. It exits the field on the
other side of the circle. To remain trapped, the
particle should either lose kinetic energy due to
collisions, or it should enter the field deeper,
so that the magnetic field extends below the part
shown.
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QUICK Quiz 19.4

• 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?

Answer (a), the absolute value of v remains
constant, only its direction changes!
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The Mass Spectrometer
Two singly ionized atoms move out of a slit at
point S and into a magnetic field of 0.1 T
pointing into the page. Each has a speed of 106
m/s. The first atom is hydrogen (one proton), the
second atom is deuterium (an isotope with one
proton and one neutron). Find their separation
near P.
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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

General form of Amperes law
? Field inside outside a wire, coaxial cable
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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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Example 19.7 The magnetic field of a long wire
A long, straight wire carries a current of 5 A.
At one instant, a proton, at 4 m from the wire
travels at 1.5103 m/s parallel to the wire and
in the same direction as the current. Find (a)
the magnitude and direction of the magnetic field
created by the wire. (b) Find the magnitude and
direction of the magnetic force the wire
magnetic field exerts on the proton.
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Example 19.7 The magnetic field of a long wire
Solutions
(a) Magnitude of the magnetic field
The current points into the page (rule 2) (b)
Magnetic force on the proton
The force points to the left (rule 1)
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Magnetic Force Between Two Parallel Conductors

• The force on wire 1 is due to the field B2
  produced by the current I2 in wire 2

• The force is

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Magnetic Force Between Two Parallel Conductors,
Cont.

• The force per unit length is then

• Definition of the Ampere
• If two long, parallel wires 1m apart carry the
  same current, and the magnetic force per unit
  length on each wire is 210-7 N/m, then the
  current is defined to be 1 A.

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Quick Quiz 19.5
If the currents I1 2A and I2 6 A, which of
the following is true? (a) F1 3 F2 (b) F1 F2
or (c) F1 F2/3
Answer (b), since F I1I2
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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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Example 19.8 Levitating a wire
Two wires, each having a weight per unit length
of 10-4 N/m, are parallel with one directly above
the other. Assume that the wires carry currents
that are equal in magnitude and opposite in
direction. If the wires are 10 cm apart, and the
sum of gravitational force and magnetic force on
the upper wire is zero, calculate the current in
the wires.
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Example 19.8 Levitating a wire Solution
For I1 I2 I follows
Solving for I2
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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 loop of 4 straight wires
What is the field in the center of 4 long wires
forming a square, with a circle of radius R
inscribed within it?
This is the factor 1.27 larger than for the
inscribed loop. Although the loop is closer to
the center, there are contributions from the
wires beyond the square.
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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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Magnetic Field in a Solenoid from Ampères Law

• A cross-sectional view of a tightly wound
  solenoid
• If the solenoid is long compared to its radius,
  we assume the field inside is uniform and outside
  is zero
• Apply Ampères Law to the blue dashed rectangle

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Magnetic Field in a Solenoid from Ampères Law
The sum over B?l has only contributions from the
path 1. Path 2 and 4 are perpendicular to B and
path 2 is outside the solenoid, where B 0.
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Example 19.9 The magnetic Field inside a
Solenoid
A certain solenoid consists of 100 turns of wire
and has a length of 10 cm. (a) Find the magnitude
of the field inside the solenoid when it carries
a current of 0.5 A. (b) What is the momentum of a
proton orbiting inside the solenoid in a circle
with a radius of 0.02 m? The axis of the solenoid
is perpendicular to the plane of the orbit. (c)
approximately how much wire would be needed to
build this solenoid? Assume the solenoids radius
is 5 cm.
84
Example 19.9 The magnetic Field inside a
Solenoid Solutions
(a) Magnetic field
(b) Momentum of a proton orbiting near the
center of the solenoid
(c) Approximate length of the wire needed to
build the solenoid
85
Television Screen Using Electro-magnets to
Deflect Electrons
86
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

87
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

88
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

89
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

90
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

91
Domains, cont

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

92
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