The multipole expansion circumvents this difficulty by expanding not E or B, but r ⋅ E or r ⋅ B into spherical harmonics. These expansions still solve the original Helmholtz equations for E and B because for a divergence-free field F, ∇ (r ⋅ F) = r ⋅ (∇ F). The resulting expressions for a generic electromagnetic field are: