A Student's Guide to Maxwell's Equations

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