The closed form expressions of the electromagnetic field in the earth by a low-frequency harmonic electric dipole

The investigation on electromagnetic fields by various dipole sources is of practical value in underground or submarine radio communication and geophysical exploration.The action of a dipole on a media results in the solution of Sommerfeld type integrals.In order to understand the laws of distribution and propagation of electromagnetic fields in the earth,we derive the closed form expressions of Sommerfeld type integrals subject to quasi static approximation for a low frequency transmitting antenna used in a losy medium for large prospecting depth and long range communication.These expressions are valid for any field zone and point except singularities and compared with the numerical integral they increase the speed of calculation.This paper also presents a new method of dividing field zones by using the closed form expressions.