FUNDAMENTALS OF ELECTROMAGNETIC RADIATION

2.6 RADIATION FROM ARBITRARY CURRENT DISTRIBUTIONS

 In this section we will present some useful formulas for calculating the far-zone radiation field from an arbitrary distribution of current. Consider a volume V with a current distribution , as shown in Fig. 2.9. The current element �will contribute an amount

 to the total vector potential where . In the far‑zone region for all r' in V. Thus all rays from the various current elements to the far-zone field point can be considered to be parallel to each other, as shown in Fig. 2.9. Thus a useful approximation for R is

 We can replace �by �in the amplitude term for the vector potential, since this has a negligible effect on the amplitude of each elementary contribution when . Hence in the far zone we obtain

 This equation superimposes the effects of each current element and takes into

�

 

Figure 2.9 An arbitrary distribution of current

 account the relative phase angle or path-length phase delay of each contribution. Since the current elements do not, in general, contribute in phase, interference effects are produced that may be exploited to control the shape of the radiation pattern. In the next chapter we will examine the use of such interference effects to produce high-gain directive radiation beams.

 We can find the fields �and �from Eq. (2.48) by using the relations (2.13) and (2.18). When only the terms varying as 1/r are retained, it is found that

 The form of the integrand in this expression shows that in a given direction, as specified by the unit vector a, it is only the current perpendicular to a, that contributes to the radiation field. The reason for this is that the radiation held along the axis of a current element is zero.

 When the current is a line current I along a contour C, then Ed. (2.49a ) can he expressed in the form

 where a is a unit vector along C in the direction of the current.

From Eqs. (2.49a) and (2.50) we see that the electric field has the form

 where , which is given by the integral, describes the radiation amplitude pattern or the angular dependence of the radiation distribution in space. The other factor is the outward-propagating spherical wave function.