This chapter discusses simple wire antennas based on electric and magnetic dipoles. Starting from Maxwell's equations, the electromagnetic fields due to an ideal current element are calculated. The ideal current element then serves as a building block for more complicated antennas. A variety of practical wire antennas will be analyzed and evaluated. None of these will turn out by themselves to have very high gain, however. Methods of achieving the high gain required by radars will be taken up in Chapters 3 and 4. While much of the analysis concerns using antennas for transmission, the reciprocity theorem will provide a recipe for evaluating performance for reception.

Hertzian Dipole

All time-varying currents radiate as antennas, and the simplest radiating system is an ideal current element or a Hertzian electric dipole. A Hertzian dipole is a uniform current density Jflowing in an infinitesimal volume with cross-section A and differential length dl. The current is assumed to have a sinusoidal time variation given by the implicit exp(jωt) factor, and so applying phasor notation is natural. The differential length of the element is regarded as being very small compared to a wavelength.We also regard the current density as being distributed uniformly in space in the infinitesimal volume. This is an idealization; one does not set out to construct an ideal dipole, although larger antennas can be regarded as being assembled from them. It is expedient to use a combination of spherical and rectangular coordinates and to alternate between them as needed. Be aware that the unit vectors associated with spherical coordinates vary with position.

Calculating the fields arising from a Hertzian dipole is a fundamental physics problem addressed by most textbooks on antennas and electricity and magnetism. An abbreviated, intuitive derivation is provided below. Readers interested in more details should consult the Appendix B or one of the titles under Notes and Additional Reading at the end of this chapter.

Vector Potential: Phasor Form

Maxwell's equations for electromagnetic fields in their native form are not amenable to solution by familiar mathematical methods, and it is expedient to reformulate them in terms of vector and scalar potentials along with a gauge or condition that completes their definitions.