Electromagnetismo

ELECTROMAGNETISM

Gerald L. Pollack

Michigan State University
Daniel R. Stump

Michigan State University

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ISBN 0-8053-8567-3
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Contents

1 • History and Perspective

1

2 • Vector Calculus

9

1.1
1 .2

2 .1

2.2

2 .3

2.4
2 .5

Brief History of the Science of Electromagnetism 1
Electromagnetism in the Standard Model 5
Vector Algebra 10
2 .1 . 1 Definitions . . . . . . . . . . . . . . .
2. 1.2 Addition and Multiplication of Vectors
2. 1 .3 VectorProduct Identities
2.1 .4 Geometric Meanings . . . . .
Vector Differential Operators 1 8
2.2.1 Gradient of a Scalar Function
2.2.2 Divergence of a Vector Function .
2.2.3 Curl of a Vector Function .
2.2.4 Del Identities . . . . . . . . . . .
Integral Theorems 25
2.3. 1 Gauss's Theorem . . . . . . . . . .
2.3.2 Stokes's Theorem . . . . . . . . . .
2.3.3 Vector Calculus in Fluid MechanicsCurvilinear Coordinates 30
2.4.1 General Derivations . . . . . . . . . . . . . . . .
2.4.2 Cartesian, Cylindrical, and Spherical Coordinates .
The Helmholtz Theorem 37

3 • Basic Principles of Electrostatics

3 .1

3.2

Coulomb's Law 44
3 . 1 . 1 The Superposition Principle
The Electric Field 46
3.2.1 Definition . . . . . . . . .
3 .2.2 Charge as the Source of E .
3 .2.3 Field of aCharge Continuum .
.

10
13
14
16
18
19
20
23
26
27
29
30
33
44

. . . . . . 46
46
47
49
v

vi

Contents

Curl and Divergence of E 5 4
3.3. 1 Field Theory Versus Action at a Distance
3.3.2 Boundary Conditions of the Electrostatic Field
3 .4 The Integral Form of Gauss's Law 57
3.4. 1 Flux and Charge . . . . . . . . . .
3.4.2 Proof of Gauss's Law . . . . . . . .3.4.3 Calculations Based on Gauss's Law
3.5 Green's Function and the Dirac delta Function 62
3.5.1 The Dirac delta Function . . .
3.5.2 Another Proof of Gauss's Law
3.6 The Electric Potential 65
3.6. 1 Definition and Construction .
3.6.2 Poisson's Equation . . . . . .
3.6.3 Example Calculations of V (x)
3.7 Energy of the Electric Field 72
3.8 The Multipole Expansion 75
3.8. 1 '!\v o Charges. . . . . . . . . . . . . . . .
3.8.2 The Electric Dipole . . . . . . . . . . . . .
3.8.3 Moments of a General Charge Distribution
3.8.4 Equipotentials and Field Lines . . . . . . .
3.8.5 Torque and Potential Energy for a Dipole in an Electric
Field .
........................
3.9 Applications 82
3.10 Chapter Summary 83
3 .3

4 • Electrostatics and Conductors

4 .1
4.2

4 .3
4.4...