A Student’s Guide to Maxwell’s Equations

Daniel Fleisch978-0-511-39308-2, 978-0-521-87761-9

Gauss’s law for electric fields, Gauss’s law for magnetic fields, Faraday’s law, and the Ampere-Maxwell law are four of the most influential equations in science. In this guide for students, each equation is the subject of an entire chapter, with detailed, plain-language explanations of the physical meaning of each symbol in the equation, for both the integral and differential forms. The final chapter shows how Maxwell’s equations may be combined to produce the wave equation, the basis for the electromagnetic theory of light. This book is a wonderful resource for undergraduate and graduate courses in electromagnetism and electromagnetics. A website hosted by the author at www.cambridge.org/9780521701471 contains interactive solutions to every problem in the text as well as audio podcasts to walk students through each chapter.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Contents……Page 7
Preface……Page 9
Acknowledgments……Page 11
1.1 The integral form of Gauss’s law……Page 13
The electric field……Page 15
The dot product……Page 18
The unit normal vector……Page 19
The component of E normal to a surface……Page 20
The surface integral……Page 21
The flux of a vector field……Page 22
The electric flux through a closed surface……Page 25
The enclosed charge……Page 28
The permittivity of free space……Page 30
Example 1.1: Given a charge distribution, find the flux through a closed surface surrounding that charge…….Page 32
Example 1.3: Find the flux through a section of a closed surface…….Page 33
Example 1.4: Given E over a surface, find the flux through the surface and the charge enclosed by the surface…….Page 35
Example 1.5: Given a symmetric charge distribution, find E…….Page 37
1.2 The differential form of Gauss’s law……Page 41
Nabla – the del operator……Page 43
Del dot – the divergence……Page 44
The divergence of the electric field……Page 48
Example 1.6: Given an expression for the vector electric field, find the divergence of the field at a specified location…….Page 50
Example 1.7: Given the vector electric field in a specified region, find the density of electric charge at a location within that region…….Page 51
2.1 The integral form of Gauss’s law……Page 55
The magnetic field……Page 57
The magnetic flux through a closed surface……Page 60
Example 2.1: Given an expression for the magnetic field and a surface geometry, find the flux through a specified portion of that surface…….Page 62
Example 2.2: Given the current in a long wire, find the magnetic flux through nearby surfaces……Page 63
2.2 The differential form of Gauss’s law……Page 65
The divergence of the magnetic field……Page 66
Example 2.4: Given an expression for a vector field, determine whether that field could be a magnetic field…….Page 67
3.1 The integral form of Faraday’s law……Page 70
The induced electric field……Page 74
The line integral……Page 76
The path integral of a vector field……Page 77
The electric field circulation……Page 80
The rate of change of flux……Page 81
Lenz’s law……Page 83
Example 3.1: Given an expression for the magnetic field as a function of time, determine the emf induced in a loop of specified size…….Page 84
Example 3.2: Given an expression for the change in orientation of a conducting loop in a fixed magnetic field, find the emf induced in the loop…….Page 85
Example 3.3: Given an expression for the change in size of a conducting loop in a fixed magnetic field, find the emf induced in the loop…….Page 86
3.2 The differential form of Faraday’s law……Page 87
Del cross – the curl……Page 88
The curl of the electric field……Page 91
Example 3.4: Given an expression for the magnetic field as a function of time, find the curl of the electric field…….Page 92
Example 3.5: Given an expression for the induced electric field, find the time rate of change of the magnetic field…….Page 93
4.1 The integral form of the Ampere-Maxwell law……Page 95
The magnetic field circulation……Page 97
The permeability of free space……Page 99
The enclosed electric current……Page 101
The rate of change of flux……Page 103
Applying the Ampere-Maxwell law (integral form)……Page 107
Example 4.1: Given the current in a wire, find the magnetic field within and outside the wire…….Page 109
Example 4.2: Given the time-dependent charge on a capacitor, find the rate of change of the electric flux between………Page 111
4.2 The differential form of the Ampere-Maxwell law……Page 113
The curl of the magnetic field……Page 114
The electric current density……Page 117
The displacement current density……Page 119
Example 4.3: Given the magnetic field, find the current density at a specified location…….Page 120
Example 4.4: Given the magnetic field, find the displacement current density…….Page 121
5 From Maxwell’s Equations to the wave equation……Page 124
The divergence theorem……Page 126
Stokes’ theorem……Page 128
The gradient……Page 131
Some useful identities……Page 132
The wave equation……Page 134
Appendix: Maxwell’s Equations in matter……Page 137
Further reading……Page 143
Index……Page 144