FUNDAMENTALS OF ELECTROMAGNETIC RADIATION

 2.5 RADIATION FROM A SMALL CURRENT LOOP

 Figure 2.8 shows a small current loop of radius , area , and with a current I. The axis of the loop is oriented in the z direction. For �the loop may be treated as a point source. A small loop of current is called a magnetic dipole, and its magnetic dipole moment equals the product of the area with the current; thus

Figure 2.8 A small current loop in the xy plane.

The field radiated by a small magnetic dipole is the dual of that radiated by a small electric dipole, i.e., the short current filament. The roles of electric and magnetic fields are interchanged so the radiation field has an �and an �component only.

With reference to Fig. 2.8, consider the current filament at , which has the vector orientation . The contribution of the current filament of strength �to the total vector potential may be found by using the fundamental solution [Eq. (2.24)] derived earlier. Thus from �we obtain a contribution given by

where . The total vector potential is obtained by integrating over the current loop; thus

This integral is difficult to evaluate unless we make certain approximations in the expression for R. We are primarily interested in the far‑zone radiation field, so w e ran assume that . We have already assumed that , so . By imposing the above conditions we can replace R by r in the amplitude factor 1/R. With reference to a spherical coordinate system , , and , so the expression for R can be rewritten in the form

We now drop �relative to �and use the binomial expansion �for �to obtain

 In the exponential function we will have a term involving , when we substitute our approximate expression for R. But , so we can use the approximation �for �to obtain the following simplified form:

 By using these approximations the integral [Eq. (2.41)] for �becomes

 The only terms that do not integrate to zero are the �and �terms, both of which give a factor of . Hence the final expression for the vector potential becomes

where we have also put .

We can find the magnetic intensity H by using Eq. (2.13), which gives

where �and is the dipole moment of the small current loop. In the radiation zone the electric field is related to �by the simple expression (2.30a), which gives

 These expressions show that the role of the electric and magnetic fields for magnetic dipole radiation have been interchanged from their role in electric dipole radiation. However, the radiation pattern and directivity have not changed.

 The total radiated power is given by

 The radiation resistance of the loop may be found by equating �to . After simplification we find that

     As an example, consider a loop with �at 1MHz. For this loop, .

     It is obvious that a small loop antenna is a very poor radiator. If �turns of wire are used, the radiation resistance is increased by a factor of . Small loop antennas are often used as receiving antennas for portablc radios. Although they are very inefficient, they do give an acceptable performance because of the large available signal level. In a later chapter we will find that at low �frequencies atmospheric noise is often the limiting factor, so a more efficient antenna does not necessarily give better reception. Of course, a small loop antenna would not be used for transmitting purposes unless very short distances were involved and the poor gain could be tolerated. The gain of a small loop antenna is very low because the ohmic resistance of the wire is generally much greater than the radiation resistance.