Three-vector and scalar field identities and uniqueness theorems in ...

where the unprimed time derivative of F(x ′ ) is zero and where a minus sign has been introduced in the substitutionsE(ct, x, y, z) = −∇φ − ∇ × K − ∂A∂t , (29) be adequate to generate a new uniqueness theorem of thetype of theorem U1. To demonstrate this deficiency weEq. (22b) has been used in Eq. (25a). The time integrationfor the second and third terms of Eq. (29) to empha-of the first term of Eq. (25b) can be assumed tovanish as t ′ → ±∞ for the field F which is assumed tobe bounded in time. Therefore, only the second term ofEq. (25b) contributes to Eq. (24), reducing it tosize its similarity to a well-known definition of the electricfield in terms of the electric and magnetic potentials. 20A compatible definition of the magnetic field 20 with appropriatevariable and sign changes would be∫F(x) = −∇ ∇ ′ · F(x ′ )G(x, x ′ ) dV ′ c dt ′B(ct, x, y, z) = −∇ξ + ∇ × A − ∂K∂t . (30)V 4∫′+ ∇ × ∇ ′ × F(x ′ )G(x, x ′ ) dV ′ c dt ′ In a different calculation on a related subject, Eq. (13) inRef. 4 [similar to Eq. (28) here] was used to derive an extensionof Jefimenko’s time-dependent generalizations ofV 4′+ 1 ∫∂ ∂F(x ′ )c 2 ∂t V 4′ ∂t ′ G(x, x ′ ) dV ′ c dt ′ , (26) the Coulomb and Biot-Savart laws 21 to include magneticcharge and current sources. The time-dependent Helmholtzidentity (26) therefore has interesting applications.which can be regarded as a time-dependent generalizationof the (static) Helmholtz identity (13) for any tion of Eq. (2) via the time-dependent Helmholtz iden-Although Eq. (29) is an example of the generaliza-suitable Green’s function G(x, x ′ ) that is a solution of tity (26), it is the possibility of using Eq. (26) to obtainEq. (17b). [A result like Eq. (26), but in terms of a retardedGreen’s function, was given in Ref. 2 with a com-the claim by Heras 2 that the terms of Eq. (26) could beuniqueness theorems that is of interest here. Considerment that the proof followed from Eq. (23b); we have labeled (preserving the order of the terms) as follows:verified this proof in detail here.] If we substitute theretarded Green’s function (18) into Eq. (26) we obtainE(ct, x, y, z) = E ‖ + E ⊥ + E T . (31)∫F(x) = −∇ ∇ ′ · F(x ′ ) δ(r − c(t − t′ ))dV ′ c dt ′ The longitudinal component E ‖ is irrotational and satisfiesV 4′4πr∫+ ∇ × ∇ ′ × F(x ′ ) δ(r − c(t − t′ ))dV ′ c dt ′∇ × E ‖ = 0, (32)V 4′4πr+ 1 ∫∂ ∂F(x ′ ) δ(r − c(t − t ′ and the transverse component E))⊥ is solenoidal and satisfiesc 2 ∂t V 4′ ∂t ′ dV ′ c dt ′ .4πr(27)∇ · E ⊥ = 0. (33)When the time integration in Eq. (27) is performed,the argument of the delta function implies that all t ′dependent variables in the integrand are evaluated att − r/c. If we use a retarded potential bracket notationThe time component E T may have both transverse andlongitudinal components. The first term of Eq. (26) is alongitudinal component via Eq. (3) and the second termof Eq. (26) is a transverse component via Eq. (4). However,Heras 4 states in this context that “the specificationwhere [P(r ′ , ct ′ )] ret denotes evaluation at the retardedtime t ′ = t − r/c, then Eq. (27) can be rewritten as of ∂F/∂t is not available in general and there is not asimple approach for obtaining it.”∫[∇ ′ · F(r ′ , ct ′ )] retThe salient issue here is that the divergence and curlF(x) = −c∇dV ′V 4πroperations are sufficient to produce all the possible projectionsof vector fields in R 3 . That is, the application of∫′[∇ ′ × F(r ′ , ct ′ )] ret+ c∇ ×dV ′a time partial derivative cannot project out a time componentlongitudinal to the time dimension from a three-V 4πr′+ 1 ∫ [∂ ∂F(r ′ , ct ′ ) 1c ∂t V ∂t]ret′ 4πr dV ′ vector field that has only spatial dimensions. Four-vector, (28)fields are required for that. All that has been done here′is that an independent (time) variable has been included,which is similar to Eq. (13) in Ref. 4. The result (28) isobtained here in a more easily verified manner becausewithout forming a vector space of one more dimension.The absence of a unique specification of a time projectiontricky retarded potential calculations are avoided by retainingfor the hypothetical time component E T belonging tothe general Green’s function G(x, x ′ ) through tothe time-dependent Helmholtz identity (26).the three-vector space R 3 makes it problematic for thereto be a time-dependent version of theorem H1 based onWith suitable substitutions the identity (26) can be Eq. (31) [or Eq. (26)].rewritten in a form resembling Eq. (2) asNeither does Eq. (29), as a representative example ofa choice of variables following from Eq. (26), appear to5