The Hertzian potentials. With a little manipulation and the introduction of a new notation, we can maintain the wave nature of the potential functions and still provide a decomposition into purely lamellar and solenoidal components. In this analysis we shall assume lossless media only. When we chose the lorentz gauge to remove the arbitrariness of the divergence of the vector potential, we established a relationship between Ae and e. Thus we should be able to write both the electric and magnetic fields in terms of a single potential function. From the lorentz gauge we can write e 中(r,t) V·A2(r,t)dt By(5.17)and (.18)we can thus write the EM fields as E V·Ad B=V (5.34) The integro-differential representation of E in(5.33) is somewhat clumsy in appear ance. We can make it easier to manipulate by defining the Hertzian potential In differential form 4=i (5.35) With this, (5. 33)and(5.34)become E=V(V.le)-u (5.36) die (5.37) An equation for Ile in terms of the source current can be found by substituting (5. 35) nto(5.31) ∈(v2Ⅱ-k∈%I Let us define For general impressed current sources(5. 38)is just a convenient notation. However,we can conceive of an impressed polarization current that is independent of e and defined through the relation D= EoE +P+P. Then(.38)has a physical interpretation as described in(2. 119). We now have which is a wave equation for Ile. Thus the Hertzian potential has the same wave behavior as the vector potential under the lorentz gauge ②2001The Hertzian potentials. With a little manipulation and the introduction of a new notation, we can maintain the wave nature of the potential functions and still provide a decomposition into purely lamellar and solenoidal components. In this analysis we shall assume lossless media only. When we chose the Lorentz gauge to remove the arbitrariness of the divergence of the vector potential, we established a relationship between Ae and φe. Thus we should be able to write both the electric and magnetic fields in terms of a single potential function. From the Lorentz gauge we can write φe as φe(r, t) = − 1 µ  t −∞ ∇ · Ae(r, t) dt. By (5.17) and (5.18) we can thus write the EM fields as E = 1 µ ∇  t −∞ ∇ · Aedt − ∂Ae ∂t , (5.33) B =∇× Ae. (5.34) The integro-differential representation of E in (5.33) is somewhat clumsy in appear￾ance. We can make it easier to manipulate by defining the Hertzian potential Πe = 1 µ  t −∞ Ae dt. In differential form Ae = µ ∂Πe dt . (5.35) With this, (5.33) and (5.34) become E = ∇(∇ · Πe) − µ ∂2 ∂t 2 Πe, (5.36) B = µ∇ × ∂Πe ∂t . (5.37) An equation for Πe in terms of the source current can be found by substituting (5.35) into (5.31): µ ∂ ∂t ∇2 Πe − µ ∂2 ∂t 2 Πe  = −µJi . Let us define Ji = ∂Pi ∂t . (5.38) For general impressed current sources (5.38) is just a convenient notation. However, we can conceive of an impressed polarization current that is independent of E and defined through the relation D = 0E + P + Pi . Then (5.38) has a physical interpretation as described in (2.119). We nowhave ∇2 Πe − µ ∂2 ∂t 2 Πe = −1 Pi , (5.39) which is a wave equation for Πe. Thus the Hertzian potential has the same wave behavior as the vector potential under the Lorentz gauge.