Table Of ContentVector definitions, identities, and theorems
Definitions
Rectangular coordinates (a:, y^ z)
w -f-f.^-%y-%^
dAx dAy dA,
(2) V A = +
dx dy dz
dA^ dAy ^^^^^^A^_^A^.^ dAy dA^
(3) V x A =
dy dz dz dx dx dy
(4) ^^f^^+'f ^^'f+^ ^'f
dx^ dy^ dz^
(5) V^A = (V^^x) X + (VMj,) y + (V^yl,) z = V(V • A) - V x (V x A)
Cylindrical coordinates (p, 0, z)
(7) V.A = 1A(,^,) + 1 ^ + ^
pdp '^ p d(j) dz
l a ^ dA^\^ (dA 9 , . , dAp
(8) VxA: P +
p d<j) dz J dz dp J p dp d(t)
Id ( df 1 av , 9V
(9) V2/ = -)^?
pdp\ dp
(10) V^A = V(V • A) - V X (V x A)
Spherical coordinates (r, 6, 0)
(„) v/=§^.+ l | « + ^ | *
ar r w r sm 0 a<^
1 a^^
(12) V«A=-||-(r'A,) + -i^^(A,sin0) +
r^ or ^ ^ rsmOdO rsin0 d(f>
. 1 1 aA^ d{rA^)
(13) V x A -(A,sm0)-—J 0
rsin^ r + r- [sin0 Si^ dr J
1 a(rAe) dAr
+-
r dr de
1 a 1 a
(14) VV ^2 Qr [^ dr ) ^ r^ sin 0 90
(15) V^A - V(V • A) - V X (V X A)
Identities
*Starred equations are valid only in Cartesian coordinates.
(1) (A X B) • C = A • (B X C)
(2) Ax{BxC) = B{A • C) - C{A • B)
(3) '^{f9) = fV9 + g'^f
(4) V(a/6) = (1/6) Va - {a/b^) Vb
(5) * V(A •B) = {B- V)A + {A • V)B + B x (V x A) + A x (V x B)
(6) V.(/A) = (V/)-A + /(V-A)
(7) V • {Ax B) = B • {V X A) - A- {V X B)
(8) (V.V)/ = VV
(9) Vx(V/) = 0
(10) V • (V X A) = 0
(11) V X (/A) = (V/) X A + /(V X A)
(12) * Vx{AxB) = {B- V)A - (A • V)B + (V • B)A - (V • A)B
(13) V X (V X A) = V(V • A) - V^A
The above equation defines the Laplacian of a vector in other than Cartesian
coordinates.
(14) (A.V)B=(^^^ ,^ ++ A A ,^^ + + AA , ^—) x
dx ^ dy '^ dz
+ ( A .^ + A ,^ + A . ^ )y
dy ^ dz
(, ddBB^, ,. ddBB,^ .. dB,
dz
Theorems
The divergence theorem:
f AdA= f'
AdA= V-Adv
J A Jv
where A is the area of the closed surface that bounds the volume v.
Stokes's theorem:
i Adl= f {V xA)-dA,
where A is the open surface bounded by the closed cvirve C.
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Continued after Index
Paul Lorrain
Frangois Lorrain
Stephane Houle
Magneto-Fluid
Dynamics
Fundamentals and Case Studies of
Natural Phenomena
^ Sprin ger
Paul Lorrain died peacefully on June 29, 2006. After his retirement from the
Departement de physique of the Universite de Montreal, he worked many years
in the Department of Earth and Planetary Sciences at McGill University (also in
Montreal), where he did most of the research that led to this book.
Francois Lorrain Stephane Houle
Departement de mathematiques et Departement de physique
d'informatique College Jean-de-Brebeuf
College Jean-de-Brebeuf 3200, chemin de la Cote-Sainte-Catherine
3200, chemin de la Cote-Sainte-Catherine Montreal (Quebec)
Montreal (Quebec), Canada H3T ICl Canada H3T1C1
florrain @ brebeuf.qc.ca Stephane.Houle® brebeuf .qc.ca
Cover illustration: Solar spicules near the limb of the Sun - the luminous jets of plasma can be as much
as 10 000 kilometers long. See Chapter 14.
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Contents
Preface XV
Looking Ahead XXI
List of Symbols XXIX
Part I The Early History
1 The Early History 3
1.1 Magnetism in China 4
1.2 Magnetism in Europe 5
1.2.1 Navigation by Compass in Europe 5
1.2.2 Pierre de Maricourt (Petrus Peregrinus) 6
1.2.3 Prom Gilbert on 7
1.3 Faraday 8
1.4 The Story of the "Maxwell" Equations 9
1.5 Maxwell 10
1.6 Heaviside 11
1.7 Zeeman, Hale, Gouy, and Larmor 13
Part II Fundamentals
2 The Maxwell Equations 19
2.1 Introduction 20
2.2 The Equations in Differential Form 20
2.2.1 Example: The Differential Form of Gauss's Law for
Electric Fields 23
2.2.2 Example: The Differential Form of Gauss's Law for
Magnetic Fields 23
2.2.3 Example: The Differential Form of the Law of Faraday 23
2.2.4 Example: The Differential Form of Ampere's
Circuital Law 23
2.3 The Equations in Integral Form 24
VI Contents
2.3.1 Example: The Integral Form of Gauss's Law for
Electric Fields 25
2.3.2 Example: The Integral Form of Gauss's Law for
Magnetic Fields 26
2.3.3 Example: The Integral Form of the Law of Faraday . . 26
2.3.4 Example: The Integral Form of Ampere's Circuital
Law 27
2.4 The Displacement Current Density 27
2.5 The Convection Current Density 28
2.6 Summary 29
3 Electric Fields 31
3.1 Electric Fields and Forces 31
3.2 Electric Currents 33
3.2.1 Example: The Drift Speeds of Charge Carriers 34
3.3 The Conservation of Electric Charge 34
3.3.1 Example: The Relaxation Time of a Conductor 35
3.4 The Electric Potential V 36
3.5 The Equations of Poisson and of Laplace 36
3.6 Joule Losses 37
3.7 Electric Energy 37
3.8 Example: A Proton Beam I 37
3.9 Case Study: V at the Surface of the Sun 41
3.10 Summary 44
4 Constant Magnetic Fields 45
4.1 Magnetic Field Lines 46
4.2 The Magnetic Flux Density ^ at a Point 46
4.2.1 Example: Calculating the Field of a Coil 47
4.3 Magnetic Flux # 48
4.4 Ampere's Circuital Law 48
4.4.1 Example: The Thin Solenoid 48
4.4.2 Example: The Thick Solenoid 49
4.5 The Vector Potential 50
4.5.1 Example: The Vector Potential near a Circuit 51
4.5.2 Example: The Thin Solenoid 51
4.6 Magnetic Multipoles 51
4.7 Case Study: The Earth's Magnetic Field 52
4.8 The Magnetic Force 56
4.8.1 Example: The Electromagnetic Pump 57
4.8.2 Magnetic Field Lines Again 57
4.9 Magnetic Pressure 59
4.9.1 Example 59
4.10 Magnetic Energy 59
Contents VII
4.11 Example: A Proton Beam II 60
4.12 Summary 61
Time-dependent Magnetic Fields: The Law of Faraday .... 63
5.1 The Electric Field Strength 63
5.2 The Law of Faraday 64
5.2.1 Lenz's Law 64
5.2.2 Example: The Thin Solenoid 65
5.2.3 Example: A Secondary on a Long Solenoid 65
5.3 Self-inductance 65
5.3.1 Magnetic Energy in a Self-inductance 66
5.4 Mutual Inductance 66
5.5 Electromagnetic Waves 67
5.5.1 Case Study: The Earth's Magnetic Field 67
5.6 Summary 68
Part III Moving Conductors
6 Ohm's Law for Moving Conductors 73
6.1 Reference Frames 74
6.2 Special Relativity 74
6.2.1 The Kinematic Transformation Equations 75
6.2.2 The Transformation Equations of Electromagnetic
Quantities 75
6.2.3 Example: Straight Wire Carrying a Current 79
6.3 The Lorentz Force 79
6.4 Ohm's Law for Moving Conductors 79
6.5 The Induction Equation 82
6.6 The Magnetic Reynolds Number 83
6.6.1 Examples 84
6.7 Magnetic Forces on Moving Conductors 85
6.8 Summary 86
7 Charges Inside Moving Conductors 87
7.1 Introduction 88
7.2 Electric Charges in Moving Conductors 88
7.2.1 Example: Rotating Solid Conductor 90
7.2.2 Time-dependent Situations 91
7.2.3 The Magnetic Field of the Convection Current 92
7.3 Example: The Faraday Disk 92
7.3.1 The Faraday Disk with the Switch SW Open 93
7.3.2 The Faraday Disk as a Generator 94
7.3.3 The Faraday Disk as a Motor 95
7.3.4 The Currents in the Axle and in the Disk 96
