This paper is concerned with the electric and magnetic field close to conductors which carry an alternating current. Hertz's famous solution of Maxwell's equations gave the field at a great distance from a magnetic or electric dipole whose moment was alternating harmonically: but from its very character it could not describe the field at points which were insufficiently remote from the origin for the source to appear as a dipole. The great development of radio communication has made important certain problems about the field in the vicinity of wires which are carrying an alternating current. Thus imagine that an aerial of some thousand feet in height and very many acres in extent has been erected. Before the designer of this vast and costly structure can calculate the rate of energy output, he must suppose himself so far removed that the aerial has dwindled to a mere speck in the aether. Such a process must seem both unsatisfactory and unsatisfying to those who take the responsibility of obtaining radio communication. It is already known that the field at any point can be expressed formally when the distribution of current and charge in the source is postulated: it is shown in this paper that the field at all points very near to the source can be expressed by simple formulae. Of course it is not an exact solution for the field due to a conductor of specified shape, because the postulated distribution of current and charge is incorrect.