"Electromagnetic theory is a peculiar subject. The peculiarity resides not so much in the stratification - superposed layers of electrostatics. magnetostatics. steady currents and time-varying fields - as in the failure that has attended all attempts to weld these layers into a logical whole. The lowest layer. electrostatics. defines certain concepts. such as E. D, ~, in a way that is generally satisfactory only for the static case. Yet the attempt is made to force these specialised definitions into the higher strata, with ad hoc modifications when necessary. The student, in looking through his text books on electromagnetics, can find general definitions only with difficulty. if at all; and even the most advanced treatises fail to present a rigorously logical development of the subject". 1 So wrote Moon and Spencer some 30 years ago; and their criticism continues to be pertinent today. 2 More recently. a senior physicist of the National Bureau of Standards has expressed his concern in similar terms: "A logically consistent set of definitions of the electromagnetic field quantities is extremely difficult to find in the literature. Most text books either evade the problem or present definitions that are applicable only to special cases".
1 The Differential and Integral Calculus of Vectors.- 1.1 Scalar and Vector Fields.- 1.2 Directional Derivative of a Scalar Field Gradient.- 1.3 Directional Derivative of a Vector Field.- 1.4 Differentiability of a Scalar Point Function Gradient and Directional Derivative of Combined Scalar Fields.- 1.5 Scalar and Vector Line Integrals.- 1.6 Scalar and Vector Surface Integrals.- 1.7 Scalar and Vector Volume Integrals.- 1.8 Stokes's Theorem Curl of a Vector Field.- 1.9 Alternative Approach to Stokes's Theorem.- 1.10 Application of Stokes's Theorem.- 1.11 The Irrotational Vector Field.- 1.12 Flux Through a Closed Surface The Divergence Theorem.- 1.13 Alternative Approach to the Divergence Theorem.- 1.14 Application of the Divergence Theorem.- 1.15 The Solenoidal Vector Field.- 1.16 Expansion Formulae for Gradient, Curl and Divergence.- 1.17 Deductions from Stokes's Theorem and the Divergence Theorem.- 1.18 The Laplacian Operator ?2.- 1.19 Invariance of Grad, Div, Curl and ?2 With Respect to Choice of Rectangular Axes.- 1.20 Moving Systems and Time-Dependent Fields.- 1.21 Time Rates of Change of a Vector Quantity Referred to Coordinate Systems in Relative Motion.- 1.22 Complex Scalar and Vector Fields.- 2 Curvilinear Coordinate Systems.- 2.1 Curvilinear Coordinates.- 2.2 Cylindrical Coordinates.- 2.3 Spherical Coordinates.- 2.4 Line, Surface and Volume Integration in Cylindrical and Spherical Coordinates.- 2.5 Grad V, Curl $$overline {text{F}}$$, Div F and ?2V in Orthogonal Curvilinear Coordinates.- 2.6 Grad V, Curl $$overline {text{F}}$$, Div $$overline {text{F}}$$ and ?2V in Cylindrical and Spherical Coordinates.- 2.7 Derivation of Grad V, Div $$overline {text{F}}$$ and Curl $$[overline {text{F}}$$ in Cylindrical Coordinates by Transformation of Axes.- 2.8 Derivation of Grad V, Div $$overline {text{F}}$$ and Curl $$overline {text{F}}$$ in Spherical Coordinates by Transformation of Axes.- 2.9 Derivation of Curl $$overline {text{F}}$$ and Div $$overline {text{F}}$$ in Orthogonal Curvilinear Coordinates via Line and Surface Integration.- 2.10 ?2$$overline {text{F}}$$ in General Orthogonal, Cylindrical and Spherical Coordinates.- 2.11 Change of Volume Resulting from Transformation of Coordinate Values.- 2.12 Surface Relationships.- 3 Green's Theorem and Allied Topics.- 3.1 Green's Theorem.- 3.2 The Harmonic Function.- 3.3 Green's Formula.- 3.4 Gauss's Integral Solid Angle.- 3.5 Treatment of Surface and Point Discontinuities in Scalar Fields.- 3.6 Uniqueness Theorem for Scalar Fields.- 3.7 Theorems Relating to Vector Fields Vector Analogue of Green's Theorem.- 3.8 Green's Function The Dirichlet and Neumann Problems.- 3.9 Scalar Fields in Plane Regions.- 3.10 Minimal Theorems.- 4 Unretarded Potential Theory.- 4.1 The Scalar Potential of Point Sources.- 4.2 The Scalar Potential of Line Sources.- 4.3 The Scalar Potential of Surface Sources.- 4.4 The Scalar Potential of a Volume Source.- 4.5 The Representation of a Scalar Point Function as the Combined Potentials of Surface and Volume Sources.- 4.6 The Gradient and Laplacian of the Scalar Potential of Point Sources Gauss's Law.- 4.7 The Gradient and Laplacian of the Scalar Potential of Line and Surface Sources.- 4.8 The Gradient of the Scalar Potential of a Volume Source.- 4.9 The Laplacian of the Scalar Potential of a Volume Source Poisson's Equation Extension of Gauss's Law.- 4.10 Equivalent Layer Theorems in Scalar Potential Theory.- 4.11 The Method of Images in Scalar Potential Theory.- 4.12 The Vector Potential of Line, Surface and Volume Sources.- 4.13 Reciprocal Relationships in Scalar and Vector Potential Theory.- 4.14 The Divergence, Curl and Laplacian of the Vector Potential of Simple Line and Surface Sources.- 4.15 The Divergence, Curl and Laplacian of the Vector Potential of a Volume Source.- 4.16 Equivalent Layers and Image Systems in Vector Potential Theory.- 4.17 The Grad-Curl Theorem.- 4.18 The Gradient and Laplacian of the Scalar Point Function $$int {bar P} {text{.grad }}frac{{text{1}}}{{text{r}}}{text{d}}tau $$.- 4.19 The Divergence, Curl and Laplacian of the Vector Point Function $$int {bar M} times gradfrac{{text{1}}}{{text{r}}}{text{d}}tau$$.- 4.20 Introduction to the Macroscopic Potentials.- 4.21 Inverse-Square Vector Fields and their Relationship to the Potential Functions.- 5 Retarded Potential Theory.- 5.1 Retarded Scalar and Vector Fields.- 5.2 Expansion of Grad [V], Div [$$overline {text{F}} $$] and Curl [$$overline {text{F}}$$].- 5.3 Dynamical Extension of Green's Formula.- 5.4 Uniqueness Theorems for Time-Dependent Fields.- 5.5 The Retarded Potentials of Scalar and Vector Sources.- 5.6 The Gradient, Divergence and Curl of Retarded Potentials.- 5.7 The d'Alembertian of the Retarded Potentials.- 5.8 The Gradient and d'Alembertian of the Scalar Point Function $$int {left{ {left[ {bar P} right]} right}} .grad{text{ }}frac{1}{r}{text{ }} - {text{ }}left[ {frac{{partial bar P}}{{partial t}}} right].frac{{bar r}}{{c{r^2}}}dtau $$.- 5.9 The Divergence, Curl and d'Alembertian of the Vector Point Function $$int {left{ {left[ {bar M} right] times gradfrac{1}{r}{text{ }} - {text{ }}left[ {frac{{partial bar M}}{{partial t}}} right]{text{ }} times {text{ }}frac{{bar r}}{{c{r^2}}}} right}} dtau $$.- 5.10 The Liénard-Wiechert Potentials.- 5.11 Space and Time Derivatives of the Liénard-Wiechert Potentials.- 5.12 Approximations for the Lignard-Wiechert Potentials and Their Derivatives in Terms of Unretarded Quantities.- 5.13 The Retarded Potentials of an Oscillating Point Doublet with Time-Dependent Orientation.- 5.14 The Retarded Potentials of a Point Whirl of Constant Moment.- 5.15 The Retarded Vector Potential of a Point Whirl of Time-Dependent Orientation.- 5.16 The ? and $$ overline {text{B}} $$ Fields of Time-Dependent Doublets and Whirls.- 5.17 The Retarded Densities and Potentials of a Statistically-Regular Configuration of Point Singlets in Motion The Equation of Continuity.- 5.18 Construction of the Macroscopic Density and Potential Functions for Singlet, Doublet and Whirl Distributions.- 5.19 The Macroscopic Potentials of a Composite Source System The Polarisation Potentials The Lorentz Gauge.- 5.20 Microscopic/Macroscopic Relationships for ? and $$ overline {text{B}} $$ Fields within Volume Distributions of Doublets and Whirls.- 5.21 Maxwell's Equations.- 5.22 The Macroscopic Vector Fields ?, $$ overline {text{D}} $$, $$ overline {text{B}} $$, $$bar H $$.- 6 Helmholtz's Formula and Allied Topics.- 6.1 Helmholtz's Equation Helmholtz's Formula Conditions for Uniqueness.- 6.2 Scalar Green's Functions for Helmholtz's Equation.- 6.3 Vector Green's Functions for the Equation: Curl curl $$ tilde bar F - {k^2}tilde bar F = bar 0$$.- 6.4 Surface/Volume Integral Formulations for Complex Vector Fields.- 6.5 Time-Harmonic Fields and their Representation by Complex Quantities.- 6.6 Time-Averaged Products of Time-Harmonic Quantities.- 6.7 Uniqueness Criteria for Time-Harmonic Fields.- 7 Exponential Potential Theory.- 7.1 Introduction.- 7.2 The Scalar Exponential Potential and its Derivatives.- 7.3 The Vector Exponential Potential and its Derivatives.- 7.4 The Representation of a Complex Field as the Exponential Potential of Surface and Volume Sources.- 7.5 Equivalent Layers in Scalar Exponential Potential Theory.- 7.6 The Complex Form of Maxwell's Equations.- 7.7 The Macroscopic Fields $$ tilde bar E,tilde bar B,tilde bar D,tilde bar H $$.- 7.8 The Diffraction Integrals.- 7.9 Introduction to the Auxiliary Potentials.- Appendices.- A.1 The Activity Equation for Point Sources.- A.2 The Linear Momentum Equation for Point Sources.- A.3 The Angular Momentum Equation for Point Sources.- Useful Transformations.- Addenda to Tables.
Show more"Electromagnetic theory is a peculiar subject. The peculiarity resides not so much in the stratification - superposed layers of electrostatics. magnetostatics. steady currents and time-varying fields - as in the failure that has attended all attempts to weld these layers into a logical whole. The lowest layer. electrostatics. defines certain concepts. such as E. D, ~, in a way that is generally satisfactory only for the static case. Yet the attempt is made to force these specialised definitions into the higher strata, with ad hoc modifications when necessary. The student, in looking through his text books on electromagnetics, can find general definitions only with difficulty. if at all; and even the most advanced treatises fail to present a rigorously logical development of the subject". 1 So wrote Moon and Spencer some 30 years ago; and their criticism continues to be pertinent today. 2 More recently. a senior physicist of the National Bureau of Standards has expressed his concern in similar terms: "A logically consistent set of definitions of the electromagnetic field quantities is extremely difficult to find in the literature. Most text books either evade the problem or present definitions that are applicable only to special cases".
1 The Differential and Integral Calculus of Vectors.- 1.1 Scalar and Vector Fields.- 1.2 Directional Derivative of a Scalar Field Gradient.- 1.3 Directional Derivative of a Vector Field.- 1.4 Differentiability of a Scalar Point Function Gradient and Directional Derivative of Combined Scalar Fields.- 1.5 Scalar and Vector Line Integrals.- 1.6 Scalar and Vector Surface Integrals.- 1.7 Scalar and Vector Volume Integrals.- 1.8 Stokes's Theorem Curl of a Vector Field.- 1.9 Alternative Approach to Stokes's Theorem.- 1.10 Application of Stokes's Theorem.- 1.11 The Irrotational Vector Field.- 1.12 Flux Through a Closed Surface The Divergence Theorem.- 1.13 Alternative Approach to the Divergence Theorem.- 1.14 Application of the Divergence Theorem.- 1.15 The Solenoidal Vector Field.- 1.16 Expansion Formulae for Gradient, Curl and Divergence.- 1.17 Deductions from Stokes's Theorem and the Divergence Theorem.- 1.18 The Laplacian Operator ?2.- 1.19 Invariance of Grad, Div, Curl and ?2 With Respect to Choice of Rectangular Axes.- 1.20 Moving Systems and Time-Dependent Fields.- 1.21 Time Rates of Change of a Vector Quantity Referred to Coordinate Systems in Relative Motion.- 1.22 Complex Scalar and Vector Fields.- 2 Curvilinear Coordinate Systems.- 2.1 Curvilinear Coordinates.- 2.2 Cylindrical Coordinates.- 2.3 Spherical Coordinates.- 2.4 Line, Surface and Volume Integration in Cylindrical and Spherical Coordinates.- 2.5 Grad V, Curl $$overline {text{F}}$$, Div F and ?2V in Orthogonal Curvilinear Coordinates.- 2.6 Grad V, Curl $$overline {text{F}}$$, Div $$overline {text{F}}$$ and ?2V in Cylindrical and Spherical Coordinates.- 2.7 Derivation of Grad V, Div $$overline {text{F}}$$ and Curl $$[overline {text{F}}$$ in Cylindrical Coordinates by Transformation of Axes.- 2.8 Derivation of Grad V, Div $$overline {text{F}}$$ and Curl $$overline {text{F}}$$ in Spherical Coordinates by Transformation of Axes.- 2.9 Derivation of Curl $$overline {text{F}}$$ and Div $$overline {text{F}}$$ in Orthogonal Curvilinear Coordinates via Line and Surface Integration.- 2.10 ?2$$overline {text{F}}$$ in General Orthogonal, Cylindrical and Spherical Coordinates.- 2.11 Change of Volume Resulting from Transformation of Coordinate Values.- 2.12 Surface Relationships.- 3 Green's Theorem and Allied Topics.- 3.1 Green's Theorem.- 3.2 The Harmonic Function.- 3.3 Green's Formula.- 3.4 Gauss's Integral Solid Angle.- 3.5 Treatment of Surface and Point Discontinuities in Scalar Fields.- 3.6 Uniqueness Theorem for Scalar Fields.- 3.7 Theorems Relating to Vector Fields Vector Analogue of Green's Theorem.- 3.8 Green's Function The Dirichlet and Neumann Problems.- 3.9 Scalar Fields in Plane Regions.- 3.10 Minimal Theorems.- 4 Unretarded Potential Theory.- 4.1 The Scalar Potential of Point Sources.- 4.2 The Scalar Potential of Line Sources.- 4.3 The Scalar Potential of Surface Sources.- 4.4 The Scalar Potential of a Volume Source.- 4.5 The Representation of a Scalar Point Function as the Combined Potentials of Surface and Volume Sources.- 4.6 The Gradient and Laplacian of the Scalar Potential of Point Sources Gauss's Law.- 4.7 The Gradient and Laplacian of the Scalar Potential of Line and Surface Sources.- 4.8 The Gradient of the Scalar Potential of a Volume Source.- 4.9 The Laplacian of the Scalar Potential of a Volume Source Poisson's Equation Extension of Gauss's Law.- 4.10 Equivalent Layer Theorems in Scalar Potential Theory.- 4.11 The Method of Images in Scalar Potential Theory.- 4.12 The Vector Potential of Line, Surface and Volume Sources.- 4.13 Reciprocal Relationships in Scalar and Vector Potential Theory.- 4.14 The Divergence, Curl and Laplacian of the Vector Potential of Simple Line and Surface Sources.- 4.15 The Divergence, Curl and Laplacian of the Vector Potential of a Volume Source.- 4.16 Equivalent Layers and Image Systems in Vector Potential Theory.- 4.17 The Grad-Curl Theorem.- 4.18 The Gradient and Laplacian of the Scalar Point Function $$int {bar P} {text{.grad }}frac{{text{1}}}{{text{r}}}{text{d}}tau $$.- 4.19 The Divergence, Curl and Laplacian of the Vector Point Function $$int {bar M} times gradfrac{{text{1}}}{{text{r}}}{text{d}}tau$$.- 4.20 Introduction to the Macroscopic Potentials.- 4.21 Inverse-Square Vector Fields and their Relationship to the Potential Functions.- 5 Retarded Potential Theory.- 5.1 Retarded Scalar and Vector Fields.- 5.2 Expansion of Grad [V], Div [$$overline {text{F}} $$] and Curl [$$overline {text{F}}$$].- 5.3 Dynamical Extension of Green's Formula.- 5.4 Uniqueness Theorems for Time-Dependent Fields.- 5.5 The Retarded Potentials of Scalar and Vector Sources.- 5.6 The Gradient, Divergence and Curl of Retarded Potentials.- 5.7 The d'Alembertian of the Retarded Potentials.- 5.8 The Gradient and d'Alembertian of the Scalar Point Function $$int {left{ {left[ {bar P} right]} right}} .grad{text{ }}frac{1}{r}{text{ }} - {text{ }}left[ {frac{{partial bar P}}{{partial t}}} right].frac{{bar r}}{{c{r^2}}}dtau $$.- 5.9 The Divergence, Curl and d'Alembertian of the Vector Point Function $$int {left{ {left[ {bar M} right] times gradfrac{1}{r}{text{ }} - {text{ }}left[ {frac{{partial bar M}}{{partial t}}} right]{text{ }} times {text{ }}frac{{bar r}}{{c{r^2}}}} right}} dtau $$.- 5.10 The Liénard-Wiechert Potentials.- 5.11 Space and Time Derivatives of the Liénard-Wiechert Potentials.- 5.12 Approximations for the Lignard-Wiechert Potentials and Their Derivatives in Terms of Unretarded Quantities.- 5.13 The Retarded Potentials of an Oscillating Point Doublet with Time-Dependent Orientation.- 5.14 The Retarded Potentials of a Point Whirl of Constant Moment.- 5.15 The Retarded Vector Potential of a Point Whirl of Time-Dependent Orientation.- 5.16 The ? and $$ overline {text{B}} $$ Fields of Time-Dependent Doublets and Whirls.- 5.17 The Retarded Densities and Potentials of a Statistically-Regular Configuration of Point Singlets in Motion The Equation of Continuity.- 5.18 Construction of the Macroscopic Density and Potential Functions for Singlet, Doublet and Whirl Distributions.- 5.19 The Macroscopic Potentials of a Composite Source System The Polarisation Potentials The Lorentz Gauge.- 5.20 Microscopic/Macroscopic Relationships for ? and $$ overline {text{B}} $$ Fields within Volume Distributions of Doublets and Whirls.- 5.21 Maxwell's Equations.- 5.22 The Macroscopic Vector Fields ?, $$ overline {text{D}} $$, $$ overline {text{B}} $$, $$bar H $$.- 6 Helmholtz's Formula and Allied Topics.- 6.1 Helmholtz's Equation Helmholtz's Formula Conditions for Uniqueness.- 6.2 Scalar Green's Functions for Helmholtz's Equation.- 6.3 Vector Green's Functions for the Equation: Curl curl $$ tilde bar F - {k^2}tilde bar F = bar 0$$.- 6.4 Surface/Volume Integral Formulations for Complex Vector Fields.- 6.5 Time-Harmonic Fields and their Representation by Complex Quantities.- 6.6 Time-Averaged Products of Time-Harmonic Quantities.- 6.7 Uniqueness Criteria for Time-Harmonic Fields.- 7 Exponential Potential Theory.- 7.1 Introduction.- 7.2 The Scalar Exponential Potential and its Derivatives.- 7.3 The Vector Exponential Potential and its Derivatives.- 7.4 The Representation of a Complex Field as the Exponential Potential of Surface and Volume Sources.- 7.5 Equivalent Layers in Scalar Exponential Potential Theory.- 7.6 The Complex Form of Maxwell's Equations.- 7.7 The Macroscopic Fields $$ tilde bar E,tilde bar B,tilde bar D,tilde bar H $$.- 7.8 The Diffraction Integrals.- 7.9 Introduction to the Auxiliary Potentials.- Appendices.- A.1 The Activity Equation for Point Sources.- A.2 The Linear Momentum Equation for Point Sources.- A.3 The Angular Momentum Equation for Point Sources.- Useful Transformations.- Addenda to Tables.
Show more1 The Differential and Integral Calculus of Vectors.- 2 Curvilinear Coordinate Systems.- 3 Green’s Theorem and Allied Topics.- 4 Unretarded Potential Theory.- 5 Retarded Potential Theory.- 6 Helmholtz’s Formula and Allied Topics.- 7 Exponential Potential Theory.- Appendices.- A.1 The Activity Equation for Point Sources.- A.2 The Linear Momentum Equation for Point Sources.- A.3 The Angular Momentum Equation for Point Sources.- Useful Transformations.- Addenda to Tables.
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