1. Magnetic Forces and Magnetic Fields
Chapter 22
Magnetic Forces
and
Magnetic Fields

2. A Brief History of Magnetism
13th century BC
Chinese used a compass
Uses a magnetic needle
Probably an invention of Arab or Indian origin
800 BC
Greeks
Discovered magnetite attracts pieces of iron

3. A Brief History of Magnetism, 2
1269
Pierre de Maricourt found that the direction of a needle near a spherical natural magnet formed lines that encircled the sphere
The lines also passed through two points diametrically opposed to each other
He called the points poles

4. A Brief History of Magnetism, 3
1600
William Gilbert
Expanded experiments with magnetism to a variety of materials
Suggested the earth itself was a large permanent magnet
1750
John Michell
Magnetic poles exert attractive or repulsive forces on each other
These forces vary as the inverse square of the separation

5. A Brief History of Magnetism, 4
1819
Hans Christian Oersted
Pictured, 1777 – 1851
Discovered the relationship between electricity and magnetism
An electric current in a wire deflected a nearby compass needle
André-Marie Ampère
Deduced quantitative laws of magnetic forces between current-carrying conductors
Suggested electric current loops of molecular size are responsible for all magnetic phenomena

6. A Brief History of Magnetism, final
Faraday and Henry
Further connections between electricity and magnetism
A changing magnetic field creates an electric field
Maxwell
A changing electric field produces a magnetic field

7. Electric and Magnetic Fields
An electric field surrounds any stationary electric charge
The region of space surrounding a moving charge includes a magnetic field
In addition to the electric field
A magnetic field also surrounds any material with permanent magnetism
Both fields are vector fields

8. Magnetic Poles Every magnet, regardless of its shape, has two poles
Called north and south poles
Poles exert forces on one another
Similar to the way electric charges exert forces on each other
Like poles repel each other
N-N or S-S
Unlike poles attract each other
N-S

9. Magnetic Poles, cont
The poles received their names due to the way a magnet behaves in the Earth’s magnetic field
If a bar magnet is suspended so that it can move freely, it will rotate
The magnetic north pole points toward the earth’s north geographic pole
This means the earth’s north geographic pole is a magnetic south pole
Similarly, the earth’s south geographic pole is a magnetic north pole

10. Magnetic Poles, final
The force between two poles varies as the inverse square of the distance between them
A single magnetic pole has never been isolated
In other words, magnetic poles are always found in pairs
There is some theoretical basis for the existence of monopoles – single poles

11. 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

12. Magnetic Field Lines, Bar Magnet Example
The compass can be used to trace the field lines
The lines outside the magnet point from the North pole to the South pole

13. 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

14. 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

15. Magnetic Field Lines, Like 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 like charges

16. Definition of Magnetic Field
The magnetic field at some point in space can be defined in terms of the magnetic force,
The magnetic force will be exerted on a charged particle moving with a velocity,

17. Characteristics of the Magnetic Force
The magnitude of the force exerted on the particle is proportional to the charge, q, and to the speed, v, of the particle
When a charged particle moves parallel to the magnetic field vector, the magnetic force acting on the particle is zero

18. Characteristics of the Magnetic Force, cont
When the particle’s velocity vector makes any angle q ¹ 0 with the field, the magnetic force acts in a direction perpendicular to both the speed and the field
The magnetic force is perpendicular to the plane formed by

19. Characteristics of the Magnetic Force, final
The force exerted on a negative charge is directed opposite to the force on a positive charge moving in the same direction
If the velocity vector makes an angle q with the magnetic field, the magnitude of the force is proportional to sin q

20. More About Direction
The force is perpendicular to both the field and the velocity
Oppositely directed forces exerted on oppositely charged particles will cause the particles to move in opposite directions

21. Force on a Charge Moving in a Magnetic Field, Formula
The characteristics can be summarized in a vector equation
is the magnetic force
q is the charge
is the velocity of the moving charge
is the magnetic field

22. Units of Magnetic Field
The SI unit of magnetic field is the Tesla (T)
The cgs unit is a Gauss (G)
1 T = 104 G

23. Directions – Right Hand Rule #1
Depends on the right-hand rule for cross products
The fingers point in the direction of the velocity
The palm faces the field
Curl your fingers in the direction of field
The thumb points in the direction of the cross product, which is the direction of force
For a positive charge, opposite the direction for a negative charge

24. Direction – Right Hand Rule #2
Alternative to Rule #1
Thumb is the direction of the velocity
Fingers are in the direction of the field
Palm is in the direction of force
On a positive particle
Force on a negative charge is opposite
You can think of this as your hand pushing the particle

25. More About Magnitude of the Force
The magnitude of the magnetic force on a charged particle is FB = |q| v B sin q
q is the angle between the velocity and the field
The force is zero when the velocity and the field are parallel or antiparallel
q = 0 or 180o
The force is a maximum when the velocity and the field are perpendicular
q = 90o

26. Differences Between Electric and Magnetic Fields
Direction of force
The electric force acts parallel or antiparallel to the electric field
The magnetic force acts perpendicular to the magnetic field
Motion
The electric force acts on a charged particle regardless of its velocity
The magnetic force acts on a charged particle only when the particle is in motion and the force is proportional to the velocity

27. More Differences Between Electric and Magnetic Fields
Work
The electric force does work in displacing a charged particle
The magnetic force associated with a steady magnetic field does no work when a particle is displaced
This is because the force is perpendicular to the displacement

28. Work in Fields, cont
The kinetic energy of a charged particle moving through a constant magnetic field cannot be altered by the magnetic field alone
When a charged particle moves with a velocity through a magnetic field, the field can alter the direction of the velocity, but not the speed or the kinetic energy

29. Notation Note The dots indicate the direction is out of the page
The dots represent the tips of the arrows coming toward you
The crosses indicate the direction is into the page
The crosses represent the feathered tails of the arrows

30. Charged Particle in a Magnetic Field
Consider a particle moving in an external magnetic field with its velocity 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

31. Force on a Charged Particle
Using Newton’s Second Law, you can equate the magnetic and centripetal forces:
Solving for r:
r is proportional to the linear momentum of the particle and inversely proportional to the magnetic field and the charge

32. More About Motion of Charged Particle
The angular speed of the particle is
The angular speed, w, is also referred to as the cyclotron frequency
The period of the motion is

33. Motion of a Particle, General
If a charged particle moves in a magnetic field at some arbitrary angle with respect to the field, its path is a helix
Same equations apply, with

34. Bending of an Electron Beam
Electrons are accelerated from rest through a potential difference
Conservation of Energy will give v
Other parameters can be found

35. Charged Particle Moving in Electric and Magnetic Fields
In many applications, the charged particle will move in the presence of both magnetic and electric fields
In that case, the total force is the sum of the forces due to the individual fields
In general:
This force is called the Lorenz force
It is the vector sum of the electric force and the magnetic force

36. Velocity Selector
Used when all the particles need to move with the same velocity
A uniform electric field is perpendicular to a uniform magnetic field

37. Velocity Selector, cont
When the force due to the electric field is equal but opposite to the force due to the magnetic field, the particle moves in a straight line
This occurs for velocities of value v = E / B

38. Velocity Selector, final
Only those particles with the given speed will pass through the two fields undeflected
The magnetic force exerted on particles moving at speed greater than this is stronger than the electric field and the particles will be deflected upward
Those moving more slowly will be deflected downward

39. Mass Spectrometer
A mass spectrometer separates ions according to their mass-to-charge ratio
A beam of ions passes through a velocity selector and enters a second magnetic field

40. Mass Spectrometer, cont
After entering the second magnetic field, the ions move in a semicircle of radius r before striking a detector at P
If the ions are positively charged, they deflect upward
If the ions are negatively charged, they deflect downward
This version is known as the Bainbridge Mass Spectrometer

41. Mass Spectrometer, final
Analyzing the forces on the particles in the mass spectrometer gives
Typically, ions with the same charge are used and the mass is measured

42. Thomson’s e/m Experiment
Electrons are accelerated from the cathode
They are deflected by electric and magnetic fields
The beam of electrons strikes a fluorescent screen

43. Thomson’s e/m Experiment, cont
Thomson’s variation found e/me by measuring the deflection of the beam and the fields
This experiment was crucial in the discovery of the electron

44. Cyclotron
A cyclotron is a device that can accelerate charged particles to very high speeds
The energetic particles produced are used to bombard atomic nuclei and thereby produce reactions
These reactions can be analyzed by researchers

45. Cyclotron, 2 D1 and D2 are called dees because of their shape
A high frequency alternating potential is applied to the dees
A uniform magnetic field is perpendicular to them

46. Cyclotron, 3
A positive ion is released near the center and moves in a semicircular path
The potential difference is adjusted so that the polarity of the dees is reversed in the same time interval as the particle travels around one dee
This ensures the kinetic energy of the particle increases each trip

47. Cyclotron, final
The cyclotron’s operation is based on the fact that T is independent of the speed of the particles and of the radius of their path
When the energy of the ions in a cyclotron exceeds about 20 MeV, relativistic effects come into play

48. First Cyclotron Invented by E. O. Lawrence and M. S. Livingston
Invented in 1934

49. Force on a Current-Carrying Conductor
A current carrying conductor experiences a force when placed in an external magnetic field
The current represents a collection of many charged particles in motion
The resultant magnetic force on the wire is due to the sum of the magnetic forces on the charged particles

50. Force on a Wire
In this case, there is no current, so there is no force
Therefore, the wire remains vertical

51. Force on a Wire,cont The magnetic field is into the page
The current is upward, along the page
The force is to the left

52. Force on a Wire, final The field is into the page
The current is downward along the page
The force is to the right

53. Force on a Wire, equation
The magnetic force is exerted on each moving charge in the wire
The total force is the product of the force on one charge and the number of charges

54. Force on a Wire, cont
In terms of the current, this becomes
l is a vector that points in the direction of the current
Its magnitude is the length of the segment
This applies only to a straight segment of wire in a uniform external magnetic field

55. Force on a Wire, Arbitrary Shape
Consider a small segment of the wire,
The force exerted on this segment is
The total force is

56. Torque on a Current Loop
The rectangular loop carries a current I in a uniform magnetic field
No magnetic force acts on sides  & 
The wires are parallel to the field and cross product is zero

57. Torque on a Current Loop, 2
There is a force on sides  & 
These sides are perpendicular to the field
The magnitude of the magnetic force on these sides will be:
F2 = F4 = I a B
The direction of F2 is out of the page
The direction of F4 is into the page

58. Torque on a Current Loop, 3
The forces are equal and in opposite directions, but not along the same line of action
The forces produce a torque around point O

59. Torque on a Current Loop, Equation
The maximum torque is found by:
The area enclosed by the loop is ab, so tmax = I A B
This maximum value occurs only when the field is parallel to the plane of the loop

60. Torque on a Current Loop, General
Assume the magnetic field makes an angle of q<90o with a line perpendicular to the plane of the loop
The net torque about point O will be t = I A B sin q

61. Torque on a Current Loop, Summary
The torque has a maximum value when the field is perpendicular to the normal to the plane of the loop
The torque is zero when the field is parallel to the normal to the plane of the loop
where A is perpendicular to the plane of the loop and has a magnitude equal to the area of the loop

62. Direction of A
The right-hand rule can be used to determine the direction of
Curl your fingers in the direction of the current in the loop
Your thumb points in the direction of

63. Magnetic Dipole Moment
The product I is defined as the magnetic dipole moment, of the loop
Often called the magnetic moment
SI units: A m2
Torque in terms of magnetic moment:

64. Biot-Savart Law – Introduction
Biot and Savart conducted experiments on the force exerted by an electric current on a nearby magnet
They arrived at a mathematical expression that gives an expression for the magnetic field at some point in space due to a current

65. Biot-Savart Law – Set-Up
The magnetic field is
at some point P
The length element is
The wire is carrying a steady current of I

66. Biot-Savart Law – Observations
The vector is perpendicular to both ds and to the unit vector directed from
toward P
The magnitude of is inversely proportional to r2, where r is the distance from to P

67. Biot-Savart Law – Observations, cont
The magnitude of is proportional to the current and to the magnitude ds of the length element ds
The magnitude of is proportional to sin q, where q is the angle between the vectors and

68. Biot-Savart Law, Equation
The observations are summarized in the mathematical equation called Biot-Savart Law:
The Biot-Savart law gives the magnetic field only for a small length of the conductor

69. Permeability of Free Space
The constant mo is called the permeability of free space
mo = 4 p x 10-7 T. m / A
The Biot-Savart Law can be written as

70. Total Magnetic Field
To find the total field, you need to sum up the contributions from all the current elements
You need to evaluate the field by integrating over the entire current distribution
The magnitude of the field will be

71. B Compared to E Distance
The magnitude of the magnetic field varies as the inverse square of the distance from the source
The electric field due to a point charge also varies as the inverse square of the distance from the charge

72. B Compared to E, 2 Direction
The electric field created by a point charge is radial in direction
The magnetic field created by a current element is perpendicular to both the length element and the unit vector

73. B Compared to E, 3
Source
An electric field is established by an isolated electric charge
The current element that produces a magnetic field must be part of an extended current distribution
Therefore you must integrate over the entire current distribution

74. B for a Long, Straight Conductor, Direction
The magnetic field lines are circles concentric with the wire
The field lines lie in planes perpendicular to to wire
The magnitude of the field is constant on any circle of radius a
The right hand rule for determining the direction of the field is shown

75. B for a Circular Current Loop
The loop has a radius of R and carries a steady current of I
Find at point P

76. Field at the Center of a Loop
Consider the field at the center of the current loop
At this special point, x = 0
Then,

77. Magnetic Field Lines for a Loop
Figure a shows the magnetic field lines surrounding a current loop
Figure b shows the field lines in the iron filings
Figure c compares the field lines to that of a bar magnet

78. Magnetic Force Between Two Parallel Conductors
Two parallel wires each carry a steady current
The field due to the current in wire 2 exerts a force on wire 1 of F1 = I1l B2

79. Magnetic Force Between Two Parallel Conductors, cont
Substituting the equation for B2 gives
Parallel conductors carrying currents in the same direction attract each other
Parallel conductors carrying current in opposite directions repel each other

80. Magnetic Force Between Two Parallel Conductors, final
The result is often expressed as the magnetic force between the two wires, FB
This can also be given as the force per unit length, FB/l

81. Definition of the Ampere
The force between two parallel wires can be used to define the ampere
When the magnitude of the force per unit length between two long parallel wires that carry identical currents and are separated by 1 m is 2 x 10-7 N/m, the current in each wire is defined to be 1 A

82. Definition of the Coulomb
The SI unit of charge, the coulomb, is defined in terms of the ampere
When a conductor carries a steady current of 1 A, the quantity of charge that flows through a cross section of the conductor in 1 s is 1 C

83. Magnetic Field of a Wire
A compass can be used to detect the magnetic field
When there is no current in the wire, there is no field due to the current
The compass needles all point toward the earth’s north pole
Due to the earth’s magnetic field

84. Magnetic Field of a Wire, 2
The wire carries a strong current
The compass needles deflect in a direction tangent to the circle
This shows the direction of the magnetic field produced by the wire

85. Magnetic Field of a Wire, 3
The circular magnetic field around the wire is shown by the iron filings

86. André-Marie Ampère
1775 –1836
Credited with the discovery of electromagnetism
The relationship between electric currents and magnetic fields
Died of pneumonia

87. Ampere’s Law
The product of can be evaluated for small length elements on the circular path defined by the compass needles for the long straight wire
Ampere’s Law states that the line integral of
around any closed path equals moI where I is the total steady current passing through any surface bounded by the closed path

88. Ampere’s Law, cont
Ampere’s Law describes the creation of magnetic fields by all continuous current configurations
Most useful for this course if the current configuration has a high degree of symmetry
Put the thumb of your right hand in the direction of the current through the amperian loop and your figures curl in the direction you should integrate around the loop

89. Amperian Loops
Each portion of the path satisfies one or more of the following conditions:
The value of the magnetic field can be argued by symmetry to be constant over the portion of the path
The dot product can be expressed as a simple algebraic product B ds
The vectors are parallel

90. Amperian Loops, cont Conditions: The dot product is zero
The vectors are perpendicular
The magnetic field can be argued to be zero at all points on the portion of the path

91. Field Due to a Long Straight Wire – From Ampere’s Law
Want to calculate the magnetic field at a distance r from the center of a wire carrying a steady current I
The current is uniformly distributed through the cross section of the wire

92. Field Due to a Long Straight Wire – Results From Ampere’s Law
Outside of the wire, r > R
Inside the wire, we need I’, the current inside the amperian circle

93. Field Due to a Long Straight Wire – Results Summary
The field is proportional to r inside the wire
The field varies as 1/r outside the wire
Both equations are equal at r = R

94. Magnetic Field of a Toroid
Find the field at a point at distance r from the center of the toroid
The toroid has N turns of wire

95. Magnetic Field of a Solenoid
A solenoid is a long wire wound in the form of a helix
A reasonably uniform magnetic field can be produced in the space surrounded by the turns of the wire
Each of the turns can be modeled as a circular loop
The net magnetic field is the vector sum of all the fields due to all the turns

96. Magnetic Field of a Solenoid, Description
The field lines in the interior are
Approximately parallel to each other
Uniformly distributed
Close together
This indicates the field is strong and almost uniform

97. Magnetic Field of a Tightly Wound Solenoid
The field distribution is similar to that of a bar magnet
As the length of the solenoid increases
The interior field becomes more uniform
The exterior field becomes weaker

98. Ideal Solenoid – Characteristics
An ideal solenoid is approached when
The turns are closely spaced
The length is much greater than the radius of the turns
For an ideal solenoid, the field outside of solenoid is negligible
The field inside is uniform

99. Ampere’s Law Applied to a Solenoid
Ampere’s Law can be used to find the interior magnetic field of the solenoid
Consider a rectangle with side l parallel to the interior field and side w perpendicular to the field
The side of length l inside the solenoid contributes to the field
This is path 1 in the diagram

100. Ampere’s Law Applied to a Solenoid, cont
Applying Ampere’s Law gives
The total current through the rectangular path equals the current through each turn multiplied by the number of turns

101. Magnetic Field of a Solenoid, final
Solving Ampere’s Law for the magnetic field is
n = N / l is the number of turns per unit length
This is valid only at points near the center of a very long solenoid

102. Magnetic Moment – Bohr Atom
The electrons move in circular orbits
The orbiting electron constitutes a tiny current loop
The magnetic moment of the electron is associated with this orbital motion
The angular momentum and magnetic moment are in opposite directions due to the electron’s negative charge

103. Magnetic Moments of Multiple Electrons
In most substances, the magnetic moment of one electron is canceled by that of another electron orbiting in the opposite direction
The net result is that the magnetic effect produced by the orbital motion of the electrons is either zero or very small

104. Electron Spin
Electrons (and other particles) have an intrinsic property called spin that also contributes to its magnetic moment
The electron is not physically spinning
It has an intrinsic angular momentum as if it were spinning
Spin angular momentum is actually a relativistic effect

105. Electron Magnetic Moment, final
In atoms with multiple electrons, many electrons are paired up with their spins in opposite directions
The spin magnetic moments cancel
Those with an “odd” electron will have a net moment
Some moments are given in the table

106. Ferromagnetic Materials
Some examples of ferromagnetic materials are
Iron
Cobalt
Nickel
Gadolinium
Dysprosium
They contain permanent atomic magnetic moments that tend to align parallel to each other even in a weak external magnetic field

107. Domains
All ferromagnetic materials are made up of microscopic regions called domains
The domain is an area within which all magnetic moments are aligned
The boundaries between various domains having different orientations are called domain walls

108. Domains, Unmagnetized Material
The magnetic moments in the domains are randomly aligned
The net magnetic moment is zero

109. Domains, External Field Applied
A sample is placed in an external magnetic field
The size of the domains with magnetic moments aligned with the field grows
The sample is magnetized

110. Domains, External Field Applied, cont
The material is placed in a stronger field
The domains not aligned with the field become very small
When the external field is removed, the material may retain most of its magnetism

111. Magnetic Levitation
The Electromagnetic System (EMS) is one design model for magnetic levitation
The magnets supporting the vehicle are located below the track because the attractive force between these magnets and those in the track lift the vehicle

112. EMS, cont
The proximity detector uses magnetic induction to measure the magnet-rail separation
The power supply is adjusted to maintain proper separation

113. EMS, final Disadvantages Advantage
Instability caused by the variation of magnetic force with distance
Compensated for by the proximity detector
Relatively small separation between the magnets and the tracks
Usually about 10 mm
Track needs high maintenance
Advantage
Independent of speed, so wheels are not needed
Wheels are in place for “emergency landing” system

114. German Transrapid – EMS Example