2.1.5 The Distance Between Points Within and on the Toroid2.2 Useful Expressions of Maxwell's Equations for Toroidal Rings; 2.2.1 Materials Linear in Maxwell's Equations; 2.2.2 Linear Wave Equations; 2.3 Propagation Parameters; 2.3.1 Material Impedance and Absorption; 2.3.2 Frequency Dependence of Material Parameters; 2.3.3 Characteristics of Metals and Dielectrics From RF to Optical Wavelengths; 2.4 Vector and Scalar Potentials; 2.5 The Governing Equation of the Loop Antenna and Nano-Scaled Ring; 2.5.1 The General Equation

2.5.2 The Governing Equation for Perfectly Conducting (PEC) Metals2.5.3 The Governing Equation for Thin-Wire, PEC Loops and Rings; 2.5.4 The Governing Equation for Thick-Wire, PEC Loops and Rings; 2.6 Coupling a Driving Source to the Loop or Ring; 2.6.1 Direct Coupling; 2.6.2 Inductive Coupling; 2.6.3 Illuminated Coupling; References; Part II Standalone Loop Antennas and Rings; 3 Thin-Wire Perfectly Conducting Loops and Rings; 3.1 The Early History; 3.2 The Governing Equation and Solutions; 3.3 Determining the Coefficients, an and Kn; 3.3.1 Storer's Recursive Solution

3.3.2 Storer's Non-recursive Solution3.3.3 Wu's Solution; 3.4 An Elliptical Solution; 3.5 Determining Non-solvability and Convergence; 3.6 Summary of Solutions to the Thin-Wire PEC Loop; 3.6.1 The Symmetry of the Closed Loop Coefficients, In, Around Mode n=0; References; 4 The Driving Point Impedance and Admittance of Thin, PEC Loops and Rings; 4.1 Formation of the Input Impedance; 4.2 The Circuit Element Representation of the Loop; 4.2.1 The Closed Loop as an R, L and C Circuit at Any kb; 4.2.2 Difficulties with the Series Resonant Model of the Loop; 4.3 The Subwavelength Anti-resonance