Mutual Impedance of Grounded Wires Lying on the Surface of the Earth

HE mutual impedance of grounded circuits may be derived from certain results obtained by A. Sommerfeld, 1 who has developed formula? for the electric and magnetic fields in the earth and in the air due to horizontal and vertical electric and magnetic antennae situated at the surface of the earth. For our present problem we use his formulae for the electric field in the earth due to a horizontal electric doublet, since this doublet may be regarded as a short element dS of a wire of negligible diameter carrying a finite current. At the end of this present paper we shall show how the same formula for the mutual impedance may be obtained directly from first principles. Sommerfeld uses rectangular coordinates (x, y, z) and the corresponding cylindrical coordinates (r, £, z), the surface of the earth, assumed flat, being the xy plane, and the z axis extending upward into the air. The doublet is at the origin, and its axis along the x axis. Then the components of the Hertzian vector 2 in the earth (s 0) from which the electric field is determined are 3 (l) (2)