The Method of Moments in Electromagnetics

A Brief Review of Electromagnetics 11of practical interest, however, it may be difficult or impossible to directly solvethese equations for the fields. To remedy this, we derive a set of auxiliary vectorpotentials that can also be used to solve for the radiated fields. These potentials areobtained via integrals of the currents, and the radiated fields are obtained directlyfrom the potentials. Vector potential formulations are used extensively in the analysisof antenna radiation and scattering problems, and we will use them frequentlythroughout this book. These formulations are very similar to the integral equationsderived in Section 2.3, with slight differences that will be addressed.2.4.1 Magnetic Vector PotentialWe will first derive a magnetic vector potential for a homogeneous, source-freeregion. We begin by observing that since the magnetic field H is always solenoidal,it can be written as the curl of another vector A, which is arbitrary. Therefore, wewriteH ½ Ö¢A (2.53)Substituting the above into (2.1), we getwhich can be written asWe next use the identityÖ¢E Ö¢A (2.54)Ö¢´E · Aµ ¼ (2.55)Ö¢´ Ö¨µ ¼ (2.56)to writeE A Ö¨ (2.57)where ¨ is an arbitrary electric scalar potential. We next use the identityÖ¢Ö¢A Ö´Ö¡Aµ Ö ¾ A (2.58)and take the curl of both sides of (2.53), which allows us to writeand combining this with equation (2.2) leads toSubstituting (2.57) into the above leads toÖ¢H Ö´Ö¡Aµ Ö ¾ A (2.59)J · ¯E Ö´Ö¡Aµ Ö ¾ A (2.60)J · ¯´ A Ö¨µ Ö´Ö¡Aµ Ö ¾ A (2.61)which isÖ ¾ A · ¾ A J · Ö´Ö¡A · ¯¨µ (2.62)