Modeling of electromagnetic radiation using a dual four-potential representation: from dipole blade radiators to ribbon loop-like antennas

Abstract In this paper, we explore classical electromagnetic radiation using a dual four-dimensional potential Θ μ approach. Our focus is on the Planar Dipole Blade Antenna (PDBA), a system consisting of two flat perfect conductive regions on the xy -plane, separated by a gap <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic">G</mml:mi> </mml:math> , with alternating potentials applied to the conductors. This method emphasizes the use of the scalar magnetic potential Ψ( r , t ) and the electric vector potential Θ , which generates the electric field E ( r , t ) = ∇ × Θ ( r , t ) in free space. These potentials replace the standard magnetic vector potential A and the scalar electric potential Φ in our analysis. For harmonic radiation, the electromagnetic field can be expressed in terms of the electric vector potential Θ ( r , t ). We derive a corresponding retarded vector potential for Θ in terms of a two-dimensional vector field W ( r , t ), which flows through the gap region <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic">G</mml:mi> </mml:math> . This dual analytical approach yields mathematically equivalent expressions for modeling Planar Blade Antennas, analogous to those used for ribbons in the region <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic">G</mml:mi> </mml:math> , simplifying the mathematical problem. In the gapless limit, this approach reduces the two-dimensional radiator (PDBA) to a one-dimensional wire-loop-like antenna, significantly simplifying the problem's dimensionality. This leads to a dual version of Jefimenko's equations for the electric field, where W behaves like a surface current in the gap region and satisfies a continuity condition. To demonstrate the utility of this approach, we provide an analytical solution for a PDBA with a thin annular gap at low frequency.