A Student's Guide to Maxwell's Equations

As individual needs have arisen in the fields of physics, electrical engineering and computational science, each has created its own theories of information to serve as conceptual instruments for advancing developments. This book provides a coherent consolidation of information theories from these different fields. The author gives a survey of current theories and then introduces the underlying notion of symmetry, showing how information is related to the capacity of a system to distinguish itself. A formal methodology using group theory is employed and leads to the application of Burnside's Lemma to count distinguishable states. This provides a versatile tool for quantifying complexity and information capacity in any physical system. Written in an informal style, the book is accessible to all researchers in the fields of physics, chemistry, biology, computational science as well as many others. 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 This book gathers concepts of information across diverse fields physics, electrical engineering and computational science surveying current theories, discussing underlying notions of symmetry, and showing how the capacity of a system to distinguish itself relates to information. The author develops a formal methodology using group theory, leading to the application of Burnside's Lemma to count distinguishable states. This provides a tool to quantify complexity and information capacity in any physical system."