To obtain a formula for VI we take the divergence of(5. 9 )and use(.11)to get v=vvt=V.Vφ=Vφ The result may be regarded as Poissons equation for the unknown This equation is solved in Chapter 3. By (3.61)we have V.V(r) 4丌R where R= r-r, and we have V. V(r v1(r)=-V Similarly, a formula for Vs can be obtained by taking the curl of (5.9)to get VxV=VXV Substituting(. 10)we have xv=V×(V×A)=V(·A)-V2A We may choose any value we wish for V. A, since this does not alter Vs=VxA (We discuss such "gauge transformations"in greater detail later in this chapter. )With V.A=O we obtain VxV=V-A This is Poissons equation for each rectangular component of A; therefore V'xV(r) 4丌R and we have V' x V(r) V,(r=VX Summing the results we obtain the Helmholtz decomposition V=V+V dV′+V 4T R (5.13) Identification of the electromagnetic potentials. Let us write the electromagnetic fields as a general superposition of solenoidal and lamellar components E=V×AE+VφE (5.14) B=V×AB+Vφ (5.15) One possible form of the potentials AE, AB, E, and B appears in(5.13).However, because e and B are related by Maxwells equations, the potentials should be related to the sources. We can determine the explicit relationship by substituting(5. 14) and (5.15) ②2001To obtain a formula for Vl we take the divergence of (5.9) and use (5.11) to get ∇ · V =∇· Vl =∇·∇φ = ∇2 φ. The result, ∇2 φ =∇· V, may be regarded as Poisson’s equation for the unknown φ. This equation is solved in Chapter 3. By (3.61) we have φ(r) = −  V ∇ · V(r ) 4π R dV , where R = |r − r |, and we have Vl(r) = −∇  V ∇ · V(r ) 4π R dV . (5.12) Similarly, a formula for Vs can be obtained by taking the curl of (5.9) to get ∇ × V =∇× Vs. Substituting (5.10) we have ∇ × V =∇× (∇ × A) = ∇(∇ · A) − ∇2 A. We may choose any value we wish for ∇ · A, since this does not alter Vs =∇× A. (We discuss such “gauge transformations” in greater detail later in this chapter.) With ∇ · A = 0 we obtain −∇ × V = ∇2 A. This is Poisson’s equation for each rectangular component of A; therefore A(r) =  V ∇ × V(r ) 4π R dV , and we have Vs(r) =∇×  V ∇ × V(r ) 4π R dV . Summing the results we obtain the Helmholtz decomposition V = Vl + Vs = −∇  V ∇ · V(r ) 4π R dV +∇×  V ∇ × V(r ) 4π R dV . (5.13) Identification of the electromagnetic potentials. Let us write the electromagnetic fields as a general superposition of solenoidal and lamellar components: E =∇× AE + ∇φE , (5.14) B =∇× AB + ∇φB. (5.15) One possible form of the potentials AE , AB, φE , and φB appears in (5.13). However, because E and B are related by Maxwell’s equations, the potentials should be related to the sources. We can determine the explicit relationship by substituting (5.14) and (5.15)