Vanishing electric field divergence

Conventional Case of a Vanishing Electric Field Divergence... 

When D Er and D E differ from zero, Eqs. (77)-(80) can be satisfied only when div E = 0. This, in turn, implies that the right-hand members of Eqs. (74)-(76) all disappear. Consequently, this branch represents a classical electromagnetic (EM) mode with vanishing electric field divergence. 

For branch 1 of a vanishing electric field divergence, the corresponding axisymmetric EM mode is obtained from Eqs. (45) and (74)-(76). Since no dispersion relation for such a mode is available at this point of the deductions, we first introduce the notation... 

A direct consequence of Maxwell s equations is that the electric and magnetic fields can be expressed in terms of the scalar and vector potentials , A, which now include both spatial and temporal dependence. From the second Maxwell equation, Eq. (A.22), we conclude that we can express the magnetic field as the curl of the vector potential A(r, t), since the divergence of the curl of a vector potential vanishes identically (see Appendix G) ... 

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