A Student's Guide to Maxwell's Equations

A Student's Guide to Maxwell's Equations - Daniel Fleisch......Page 1
Contents......Page 5
Preface......Page 7
Acknowledgments......Page 9
1.1 The integral form of Gauss's law......Page 10
The electric field......Page 12
The dot product......Page 15
The unit normal vector......Page 16
The component of E normal to a surface......Page 17
The surface integral......Page 18
The flux of a vector field......Page 19
The electric flux through a closed surface......Page 22
The enclosed charge......Page 25
The permittivity of free space......Page 27
Example 1.1: Given a charge distribution, find the flux through a closed surface surrounding that charge.......Page 29
Example 1.3: Find the flux through a section of a closed surface.......Page 30
Example 1.4: Given E over a surface, find the flux through the surface and the charge enclosed by the surface.......Page 32
Example 1.5: Given a symmetric charge distribution, find E.......Page 34
1.2 The differential form of Gauss's law......Page 38
Nabla - the del operator......Page 40
Del dot - the divergence......Page 41
The divergence of the electric field......Page 45
Example 1.6: Given an expression for the vector electric field, find the divergence of the field at a specified location.......Page 47
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 48
2.1 The integral form of Gauss's law......Page 52
The magnetic field......Page 54
The magnetic flux through a closed surface......Page 57
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 59
Example 2.2: Given the current in a long wire, find the magnetic flux through nearby surfaces......Page 60
2.2 The differential form of Gauss's law......Page 62
The divergence of the magnetic field......Page 63
Example 2.4: Given an expression for a vector field, determine whether that field could be a magnetic field.......Page 64
3.1 The integral form of Faraday's law......Page 67
The induced electric field......Page 71
The line integral......Page 73
The path integral of a vector field......Page 74
The electric field circulation......Page 77
The rate of change of flux......Page 78
Lenz's law......Page 80
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 81
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 82
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 83
3.2 The differential form of Faraday's law......Page 84
Del cross - the curl......Page 85
The curl of the electric field......Page 88
Example 3.4: Given an expression for the magnetic field as a function of time, find the curl of the electric field.......Page 89
Example 3.5: Given an expression for the induced electric field, find the time rate of change of the magnetic field.......Page 90
4.1 The integral form of the Ampere-Maxwell law......Page 92
The magnetic field circulation......Page 94
The permeability of free space......Page 96
The enclosed electric current......Page 98
The rate of change of flux......Page 100
Applying the Ampere-Maxwell law (integral form)......Page 104
Example 4.1: Given the current in a wire, find the magnetic field within and outside the wire.......Page 106
Example 4.2: Given the time-dependent charge on a capacitor, find the rate of change of the electric flux between…......Page 108
4.2 The differential form of the Ampere-Maxwell law......Page 110
The curl of the magnetic field......Page 111
The electric current density......Page 114
The displacement current density......Page 116
Example 4.3: Given the magnetic field, find the current density at a specified location.......Page 117
Example 4.4: Given the magnetic field, find the displacement current density.......Page 118
5 From Maxwell's Equations to the wave equation......Page 121
The divergence theorem......Page 123
Stokes' theorem......Page 125
The gradient......Page 128
Some useful identities......Page 129
The wave equation......Page 131
Appendix: Maxwell’s Equations in matter......Page 134
Further reading......Page 140
Index......Page 141