1. Chapter 28 Magnetic Field and Magnetic Forces

2. Physicist: Hans Christian Orsted
Born: 14 August 1777
Died: 9 March Dane
What Orsted saw...

3. Magnetical Field Intensity of Magnetic field Magnetic Force
Interaction
with
Magnetic material
Biot-Savart law
To Moving
Particle
To Coil
Guass’s law
Circular law
Hall
Effect

4. Key words permanent magnet(永磁体) magnetic monopole(磁单极)
magnetic field(磁场)
tesla(特斯拉)
magnetic field line(磁场线)
gauss(高斯)
magnetic flux(磁通量)
weber(韦伯)
magnetic flux density(磁通密度)

5. Key words cyclotron frequency(回旋频率) mass spectronmeter(质谱仪)
isotope(同位素)
magnetic dipole(磁偶极子)
Hall effect(霍耳效应)
magnetic moment(磁矩)
solenoid(螺线管)
magnetic dipole moment(磁偶极矩)

6. 28.1 Magnetism
Permanent magnets.
Magnetic poles: a north pole(N-pole) and a south pole(S-pole).
Compass.
Opposite poles attract each, and like poles repel each other.
There are not magnetic monopoles.

7. 28.2 Magnetic Field
A moving charge or a current creates a magnetic field in the surrounding space(in addition to its electric field).
The magnetic field exerts a force F on any other moving charge or current that is present in the field.

8. Magnetic field (cont) Magnetic field (B) is a vector field.
At any position the direction of B is defined as that in which the north pole of a compass needle tends to point.
For any magnet, B points out of its north pole and into its south pole.

9. Units are those of 28.3 Magnetic Force on a Moving Particle
Consider a particle of charge q and velocity v at a point in a magnetic field with induction B.
The magnetic force exerted on the charge is given by the definition of B,
Units are those of

10. Magnetic force on a moving particle (cont)

11. Orders of Magnitude Magnetic field at surface of Earth 10-4T.
Bar magnet 10-2T.
Write head for hard disk 10-1T.
Lab electromagnet 1T.
Superconducting magnet 101T.
Interior of atoms, up to 101T.
Surface of a pulsar(脉冲星) 108T.

12. 28.4 Magnetic Field Lines Magnetic flux
Basic concept is as electric field lines but:
they have the direction a compass needle would point to at each location;
tangent to the field lines gives the direction of B at a point;
they never cross and
they form closed loops - a continuous field.

14. Magnetic flux Magnetic flux is a scalar quantity.
This unit is the Weber (1Wb)
1Wb=1T· m2=1N· m/A

15. Magnetic field B is sometimes called magnetic flux density.
28.5 Gauss’s Law for Magnetism
The total magnetic flux through a closed surface is always zero.
(magnetic flux through any closed surface).
Magnetic field B is sometimes called magnetic flux density.

16. 28.6 Moving Charges in B Fields
Uniform field of B.
Non-uniform field of B.
Combined E and B fields.
Force on a current carrying conductor.

17. Basic Motion in Magnetic Field
The velocity of a charged particle can be divided into two components one parallel and one perpendicular to the field.
The perpendicular component gives circular motion in a plane perpendicular to B.
The parallel component gives constant linear motion along B (no force).
Combination is a helical path.

18. Uniform Field of B +

19. Uniform Field of B (cont)
If v is perpendicular to B
F=qvB the force is perpendicular to the velocity giving circular motion.
Modulus of force is qvB=mv2/r  r=(mv)/(qB).
Define cyclotron frequency
Independent of speed and radius of orbit, particles of same q/m have same fc.

20. Uniform Field of B (cont)

21. Uniform Field of B (cont)
If v is not perpendicular to B.
So we have circular motion from the perpendicular component.
The parallel component gives constant linear motion along B (no force).
Combination is the helix.

22. Non-Uniform field of B Effect can be to confine particles
Examples magnetic bottles and van Allen belts.

23. Charged Particles from Sun Enter Earth’s Magnetic Field

24. Example: Helical particle motion
At t=0, the proton has velocity components vx=1.50x105m/s, vy=0, and vz=2.00x105m/s
At t=0, find the force on the proton ant its acceleration.
Find the radius of helical path, the angular speed of the proton, and the pitch of the helix (the distance traveled along the helix axis per revolution).

25. Example: Helical particle motion (cont)
The magnetic force on the proton is
The acceleration is

26. Example: Helical particle motion (cont)
The force is perpendicular to the velocity, so the speed of the proton does not change.
The radius of the helical trajectory is given by
The angular speed is

27. Example: Helical particle motion (cont)
The time required for one revolution (the period) is
The pitch is the distance traveled along the x-axis during this time, so

28. 28.7 Combined E and B Fields
The velocity selector.
Uniform, perpendicular B and E; v perpendicular to both.
Balance when q[E+(vB)]=0  v=E/B
Undeflected particles have been velocity selected.

29. Combined E and B fields (cont)
e.g. Thomson's Experiment.
First accelerate particles so ½mv2=eV.
Then force straight motion so E/B=v=(2eV/m)½.
Thus gives e/m, which is constant!

30. Combined E and B fields (cont)
e.g. The Bainbridge Mass Spectrometer.
First a velocity selector.
Then uniform magnetic field as r=(mv)/(qB) m/q.

31. 28.8 Magnetic Force on Current Carrying Conductor

32. Magnetic force on current carrying conductor (cont)
Where vd is the drift velocity. If n conduction electrons per unit volume, force summed over all the electrons is
Since I=nAvde,
For variable magnetic fields, or curved wires

34. Example
The conductor has a straight segment with length L perpendicular to the plane of the figure on the right, with the current opposite to B.
Find the total magnetic field force on these three segments of wire.

35. Example (cont)
There is no force on the segment on the right perpendicular to the plane of the figure on the right.
For the straight segment on the left, points to the left, perpendicular to . The force has magnitude F=ILB, and its direction is up.
Choose a segment with length on the semicircle, so magnitude dF of the force on the segment is

36. Example (cont) The components of the force on segment are
To Find the component of the total force, we integral these expression.
Adding the forces on the straight and semicircular segments, we find the total force:

37. 28.9 Torque on a Current Loop

38. Torque on a current loop (cont)
Consider a square current loop. Each side
parallel to axis produces a force IaB,
perpendicular to axis produces a force IbBcos().
Forces balance but parallel sides yield a torque  =2(½bIaBsin())=IABsin()=|IAB|.
So define  = B.
 is called the magnetic dipole moment. Units?

39. Torque on a current loop (cont)

40. 28.9 Hall Effect

41. Hall Effect
Charges in a wire within a B field will be pushed to one side of wire.
Charge will build up until E and B forces match i.e. FB=qvdB=FE=qE=qVH/h. (h is the height of the bar.)
Recall that I=nqvdht, so VH=(IB)/(nqt). (t is the thickness of the bar, so ht is the cross sectional area; vd is the drift velocity, n the carrier density.)
Measures field or charge carrier density.
Sign of Hall voltage gives sign of charge carrier.

42. Electrostatic vs Magnetic Fields (I)
Magnetic monopoles do not exist
Created by moving charge (or current) through B field.
The field exerts a force (F=qvB) on a charge moving at velocity v.
Electrostatic Field
Electric monopoles exist (charges, +q, -q)
Created by charges at rest through E field
The field exerts a force (F=qE) on a static charge.

43. Electrostatic vs Magnetic Fields (II)
B field is a vector (F is perpendicular to v and B).
The magnetic force does no work on a charged particle.
The energy of the particle does not change.
E field is a vector (F is parallel to E).
The electric force does work on a charged particle.
The energy of a charged particle changes.

44. Electrostatic vs Magnetic Fields (III)
E field lines start at +ve and end on -ve charges.
Gauss's law tells us
B field lines are continuous (form loops).
Since B lines have no end points
Since
the E field is conservative, hence definition of V.
Wait and see…

45. Chapter 29 Sources of Magnetic

46. Andre Marie Ampere
Born: 20 January 1775
Died: 10 June 1836
French

47. Key words principle of superposition of magnetic fields(磁场叠加原理)
law of Biot and Savart (毕奥－萨伐定律)
source point(源点)
field point (场点)
ampere (安培)
Ampere’s law (安培定律)
toroidal solenoid(螺线管)
Bohr magneton(波尔磁子)

48. Key words paramagnetic(顺磁性的) relative permeability(相对渗透性)
permeability(渗透性) magnetic susceptibility(磁化率)
diamagnetic(抗磁性的) ferromagnetic(铁磁体)
magnetic domain(磁畴) hysteresis(磁滞)
magnetization(磁化) displacement current(位移电流)

49. 29.1 Magnetic Field of a Moving Charge
Proportional to q.
Proportional to 1/r2.
AND to v.
AND to sin().
ELECTRIC FIELD
Proportional to q.
Proportional to 1/r2.
 is the angle between and .

50. Magnetic field of a moving charge (cont)
ELECTRIC FIELD
For a point charge q.
Field is radial, i.e. the direction of E is along r.
MAGNETIC FIELD
q moves with velocity v.
The field is circular, i.e in the direction of vr.

51. Magnetic field of a moving charge (cont)
The field line directions are given by the right-hand rule.

52. In SI unit the numerical value of is exactly

53. Example: Forces between two moving protons
Find the electric and magnetic forces on the upper proton, and determine the ratio of their magnitudes. + +

54. Example: Forces between two moving protons (cont)
Solution:
By Coulomb’s law, the magnitude of electric force is
The force on the upper proton is vertically upward.
The magnitude of B caused by the lower proton is
From the right-hand rule, the direction of B is in the +z direction.

55. Example: Forces between two moving protons (cont)
The magnitude of the magnetic force on the upper proton is
The ratio of the magnitude of the two forces is
So in non-relativity analysis, FB<<FE, FB can be ignored.

56. 29.2 Magnetic Field of a Current Element
Law of Biot and Savart

57. 29.3 Magnetic Field of a Straight Current Carrying Conductor
When a>>x,

58. Magnetic field of a straight current carrying conductor (cont)
For a long straight current-carrying conductor, at all points on a circle of radius r around the conductor, the magnitude B is

59. Example
Wire 1
Wire 2
Find the magnitude and direction of B at points P1, P2, and P3.
Find the magnitude and direction of B at any point on the x-axis to the right of wire 2 in terms of the x-coordinate of the point.

60. Example (cont)
Solution:
a) At pint 1:
At pint 2:
At pint 3:

61. Example (cont)

62. Example (cont) b) The magnitudes of the fields due to each wire are
The total field is in the negative y-direction, and has magnitude
At points very far from wires, so that x>>d, and
( a long, straight, current-carrying conductor)

63. 29.3 Force between Parallel Conductors

64. Force between parallel conductors (cont)
The magnitude of B field produced by the lower conductor, at the position of the upper conductor, is
The magnitude of the force exerted on a length L of the upper conductor is

65. Force between parallel conductors (cont)
The force per unit length F/L is
(two long, parallel, current-carrying conductors)
Similar result for force on the lower conductor.
Like currents attract, unlike currents repel.

66. Definition of the Ampere
One ampere is that unvarying current that, if present in each of two parallel conductors of infinite length and one meter apart in empty space, cause each conductor to experience a force of exactly 2x10-7 Newtons per meter of length.

67. 29.4 Magnetic field of a circular current loop
Due to rotational symmetry about x-axis, the y-component cancel.
(on the axis of a circular loop)

68. Magnetic field of a circular current loop (cont)
For N loop,
(on the axis of N circular loop)
The maximum value of the field is at x=0, the center of the loop.
(at the center of N circular loops)

69. Magnetic field of a circular current loop (cont)

70. 29.5 Ampere’s Law
Recall Gauss's law vs. Coulomb's Law.
Ampere's law is the alternative to the B-S law.
Use to find B fields from highly symmetric current distributions.
Find current distributions for particular B fields.

71. Ampere's Law (cont)
The line integral is independent of the radius of the circle and is equal to multiplied by the current passing through the area bounded by the circle.

72. and are antiparallel, so
Ampere's Law (cont)
and are antiparallel, so
The sign depends on the direction of the current relative to the direction of integration.
Like directions is a positive sign, unlike direction a negative sign.

73. Ampere's Law (cont)

74. Ampere's Law (cont)

75. Ampere's Law (cont)

76. Ampere's Law (cont)
It relates the tangential component of B summed around a closed curve (C) to the current (I) which passes through the curve.
(Ampere’s Law)
Holds for any curve provided the current is continuous, i.e. it does not begin or end at any point.

77. Ampere's Law (cont)
If , it does not necessarily mean that everywhere along the path, only that the total current through an area bounded by the path is zero.

78. Ampere's Law - The recipe
Let geometric closed path pass through the required field point and lie in a plane.
And encompass the current
Ensure B is constant along the path, and that B is tangential to the line or
it is perpendicular to the field lines (zero contrib.)
or it is in a region where B=0.
To achieve this use the symmetry of the field lines.

79. Example: inside a long cylindrical conductor
The current is uniformly distributed over the cross-section area of the conductor.
Find the magnetic field as a function of the distance r from the conductor axis for points inside (r<R) and outside (r>r) the conductor.

80. Example: inside a long cylindrical conductor (cont)
Solution:
i) r<R Choose integral path a circle with radius r<R.
By Ampere’s law,

81. Example: inside a long cylindrical conductor (cont)
ii) r>R
Choose integral path a circle with radius r>R.
By Ampere’s law,

82. Example: inside a long cylindrical conductor (cont)

83. Example: Field of a solenoid
The solenoid has n turns of wire per unit length and carries a current I.
Use Ampere’s law to find the field at or near the center of the solenoid.

84. Example: Field of a solenoid (cont)
Solution:
Choose the integration path the rectangle abcd.
Side ab, with length L, is parallel to the axis of the solenoid.
Side bc and da are taken to be very long so that side cd is far from the solenoid; the the field at side cd is negligibly small.

85. Example: Field of a solenoid (cont)

86. Example: field of a toroidal(环行) solenoid
A toroid is wound with N turns of wire carrying a current I.
Find the magnetic field at all points.

87. Example: field of a toroidal solenoid (cont)
Solution:
To find the field using Ampere’s law, we’ll use the integration paths shown as black lines in the right-hand picture.
i) Path 1

88. Example: field of a toroidal solenoid (cont)
ii) Path 2
iii) Path 3

89. This is a clear contradiction.
29.5 Displacement Current
This is a clear contradiction.
As the capacitor charges, the electric field and electric flux through the bulged-out surface are increasing.

90. Displacement current (cont)
The capacitor charge q is
As the capacitor charges, the rate of q is the conduction current.
We invent a fictitious current iD in the region between the plates, defined as
(displacement current)

91. Displacement current (cont)
We include displacement current iD, along with conduction current iC, in Ampere’s law:
(generalized Ampere’s law)

92. Displacement current (cont)
Apply Ampere’s law to a circle of radius r, or
For r>R, B is the same as through the wire were continues and plates not present at all.

93. Summary The magnetic force exerted on the charge is given by B,
The force on a straight segment of a conductor current I in a uniformly B is
For a infinitesimal segment dl the force is
Gauss’s law for magnetism,

94. Summary Magnetic field of a moving charge
Magnetic field of a current element
Magnetic field of a long straight current
Ampere’s law

95. 29.6 Magnetic Materials
A extern magnetic field B0
Put a magnetic material into the magnetic field.
Interaction between the magnetic field and the magnetic material.
The accessional magnetic field by the magnetic material is B¹.
The total magnetic field:

96. Magnetic materials (cont)
Define the relative permeability:
Paramagnetic(顺磁性) material: μr>1
Diamagnetic(反磁性) material: μr<1

97. The total magnetic field B in the material is
Molecule current
The molecule current can produce a magnetic moment.
Define the magnetization of the material:
The total magnetic field B in the material is

98. Molecule current (cont)
Molecule current line density: js

99. Ampere’s law in material
Define:
So:
(Ampere’s law in material)

100. Boundary conditions of magnetic field

101. Boundary conditions of magnetic field (cont)