Two paths towards circulation time derivative (Maxwell's \mathfrak E revisited)
aa r X i v : . [ phy s i c s . h i s t - ph ] J un Two paths towards circulation time derivative(Maxwell’s E revisited) D V Redˇzi´c Faculty of Physics, University of Belgrade, PO Box 44, 11000 Beograd, SerbiaE-mail: [email protected]
Abstract.
The time derivative of the circulation of a vector field A over a movingand deforming closed curve, dd t H A · d r , is computed in two ways, with and withoutbringing the time derivative under the integral sign. As a by-product, the computationsreveal that the conceptualization of Faraday’s law of electromagnetic induction maydepend on which of the two methods is employed. The discussion presented providesan unexpected argument in favor of Maxwell’s mysterious choice for his electromotiveintensity E , made in Article 598 of his Treatise .
1. Introduction
Recently, we expounded how Maxwell had arrived, through an ingenious analysis ofFaraday’s law of electromagnetic induction given in Article 598 of his
Treatise [1], at ageneral expression for his electromotive intensity E in a moving medium: E = v × B − ∂ A ∂t − ∇ Ψ ; (1)here v is the velocity of an infinitesimal portion (‘particle’) of the medium, B = ∇ × A is the magnetic flux density, A is the vector potential, and a scalar field Ψ is Maxwell’s electric potential [2]. We recalled that various authors claimed that Maxwell should haveincluded a term − ∇ ( A · v ) in expression (1), as is strongly suggested by his derivationof Article 598. Namely, in Maxwell’s computation of the negative time derivativeof the circulation of A , − dd t H A · d r , two terms of his result are expressed through − H d ( A · v ) = − H ∇ ( A · v ) · d r . However, Maxwell mysteriously leaves out the gradientterm − ∇ ( A · v ) in his final version of the integrand, noting simply that it vanishes whenintegrated round a closed curve, and introduces a brand-new term − ∇ Ψ, ‘for the sakeof giving generality’ to the expression (1) for E . The situation is even more curious,taking into account that the alternative expression for the electromotive intensity, E HWT = v × B − ∂ A ∂t − ∇ Ψ − ∇ ( A · v ) , (2)as proposed by Helmholtz [3], Watson [4], and J J Thomson [1] (vol 2, p 260), see also[5, 6], complies perfectly with Maxwell’s general principle of relativity applied to theFaraday’s law, as is demonstrated in [7]. wo paths towards circulation time derivative ∇ ( A · v ) is anartefact of the specific path employed by Maxwell for computing dd t H A · d r . Namely, inan alternative computation path, the controversial term simply does not appear. Thisfact which seems to be little known, unfortunately, had escaped our attention duringthe writing of [2, 7], while it was implicit in [8].In the present note, we first outline Maxwell’s computation of dd t H A · d r over amoving and deforming closed curve, given in Article 598, which involves the non-obviousstep of bringing the time derivative under the integral sign. Then we give the alternative,simpler computation of dd t H A · d r , applying the Kelvin-Stokes theorem twice, whichavoids bringing the time derivative under the integral sign, and which is free from theterm ∇ ( A · v ). Both computations could be useful from didactic point of view. Also,it could be inspiring for the student to learn that the conceptualization of Faraday’slaw may depend on the specific path chosen for computing dd t H A · d r . Moreover, thesimpler computation appears to provide an unexpected vindication of Maxwell’s happyand controversial choice for E , one of the key concepts of his electromagnetic theory andthe progenitor of the Lorentz force expression.
2. Two paths for the computation of circulation time derivative
For the sake of completeness, we outline Maxwell’s computation of the total timederivative of the circulation of an arbitrary, continuous and differentiable vector field A ( r , t ) over a moving and deforming closed curve C ( t ) at the instant t . Contraryto Maxwell, who writes everything in the Cartesian form, we employ the modernvector notation, benefiting from Hamilton’s operator ∇ , keeping, however, the spiritof Maxwell’s argument. † Maxwell writes the circulation as, in modern notation, I C ( t ) A · d r = Z s max ( t )0 (cid:18) A x ∂x∂s + A y ∂y∂s + A z ∂z∂s (cid:19) d s , (3)where r = r ( s, t ) is the position vector of a point of the contour, parameter s is thearc length of the point considered at the instant t , and s max ( t ) is the total length ofthe contour at that instant. Since the circulation of A refers to the fixed t , d r is thepartial differential of r with respect to s , that is d r = ∂ r ∂s d s ≡ d s r . Note that Maxwelltakes tacitly that the parametrization which refers to the fixed instant t suffices fordescribing the moving and deforming contour also in subsequent instants so that s istime-independent. (This is of course correct; as can be seen, the fact that the total lengthof the moving contour is time-dependent is irrelevant, there is a bijection between thecorresponding two sets of points.) Thus, a point r ( s, t ) at the instant t + d t becomes † Maxwell’s original argument, free from ∇ , is presented in full detail in [2], and also, almost literally,and in a somewhat complemented form, in [9]. wo paths towards circulation time derivative r ( s, t + d t ) = r ( s, t ) + v ( s, t )d t where v ( s, t ) = ∂ r ( s,t ) ∂t is the instantaneous velocity ofthe point relative to the Cartesian coordinate system chosen.To compute the time derivative of the circulation in the case of a moving anddeforming contour C ( t ), Maxwell takes the time derivative inside the integral sign ‡ andthus dd t I C ( t ) A · d r = I C ( t ) dd t ( A · d r ) = I C ( t ) (cid:18) d A d t · d r + A · dd t d r (cid:19) , (4)The differentiations yieldd A d t = ∂ A ∂t + ( v · ∇ ) A , (5)and dd t d r = d v , (6)where d v = ∂ v ∂s d s ≡ d s v , since s is time-independent. Maxwell thus obtainsdd t I C ( t ) A · d r = I C ( t ) ∂ A ∂t · d r + I C ( t ) [( v · ∇ ) A ] · d r + I C ( t ) A · d v . (7)Equation (7) is, basically, Maxwell’s equation (2) of Art. 598, written in compact form,employing the modern vector notation. § Now express ( v · ∇ ) A via the well-known vector identity v × ( ∇ × A ) = ∇ ( v · A ∗ ) − ( v · ∇ ) A , (8)where the asterisk in the expression ∇ ( v · A ∗ ) indicates that ∇ operates only on A .Employing also equation ∇ ( v · A ∗ ) · d r = d( v · A ∗ ) = v · d A , (9)one obtains [( v · ∇ ) A ] · d r = [( ∇ × A ) × v ] · d r + v · d A , (10)Inserting (10) into (7) yieldsdd t I C ( t ) A · d r = I C ( t ) (cid:20) ∂ A ∂t + ( ∇ × A ) × v (cid:21) · d r + I C ( t ) d( A · v ) . (11) ‡ The validity of this step is not very obvious and a proof is given in the appendix of [2], arrivingat equation (7) directly from the definition of dd t H C ( t ) A · d r . An alternative proof, involving arenormalization of the variable s at each instant t is presented in [9]. While the renormalizationprocedure is mathematically expedient, it is not indispensable, the parametrization at one instantsuffices, cf the appendix of [2]. As can be seen, another way of vindicating this step would be toinvoke the Leibniz rule for differentiating an integral function, cf, e.g., [10], taking into account that s is time-independent. § Note that our expression dd t d r is nothing but Maxwell’s (cid:2) dd t (cid:0) ∂ r ∂s (cid:1)(cid:3) d s (clearly implicit in Art. 598),since s is time-independent. wo paths towards circulation time derivative t I C ( t ) A · d r = I C ( t ) (cid:20) ∂ A ∂t + ( ∇ × A ) × v + ∇ ( A · v ) (cid:21) · d r . (12)Finally, noting that the last integral in eq. (11) vanishes since it is taken round theclosed curve, Maxwell arrives atdd t I C ( t ) A · d r = I C ( t ) (cid:20) ∂ A ∂t + ( ∇ × A ) × v (cid:21) · d r . (13)Equation (13) is a purely mathematical and general result valid for an arbitrarymoving and deforming closed curve C ( t ) that remains continuous and closed during itsmotion, and for arbitrary, continuous and differentiable vector field A ( r , t ) and velocityfield v ( r , t ). Note that the appearance of the controversial term ∇ ( A · v ) in eq. (12) isa consequence of computing dd t ( A · d r ). Now we present a simpler computation of dd t H C ( t ) A · d r , which avoids bringing the timederivative under the integral sign, and avoids (explicit) parametrization of the curve C ( t ).The time derivative of the circulation of A is by definition:dd t I C ( t ) A ( r , t ) · d r = H C ( t +d t ) A ( r , t + d t ) · d r − H C ( t ) A ( r , t ) · d r d t . (14)A Taylor series expansion in the first integral on the right hand side of eq. (14)yields I C ( t +d t ) A ( r , t + d t ) · d r = I C ( t +d t ) A ( r , t ) · d r + I C ( t +d t ) ∂ A ( r , t ) ∂t d t · d r , (15)and applying the Kelvin-Stokes theorem to the second integral on the right hand sideof eq. (14) one has I C ( t ) A ( r , t ) · d r = Z S [ C ( t )] [ ∇ × A ( r , t )] · d S , (16)where S [ C ( t )] is any open surface bounded by the closed curve C ( t ). Choosing for S [ C ( t )] a surface which consists of a ribbon swept by the moving contour during thetime interval d t and a surface S [ C ( t + d t )] (any open surface bounded by the closedcurve C ( t + d t )), the surface integral becomes Z S [ C ( t )] [ ∇ × A ( r , t )] · d S = I C ( t ) [ ∇ × A ( r , t )] · (d r × v d t )+ Z S [ C ( t +d t )] [ ∇ × A ( r , t )] · d S , (17)where v is the instantaneous velocity of the circuit element d r at the instant t .Transforming the right hand side of eq. (17), rearranging terms in the first integral wo paths towards circulation time derivative I C ( t ) A ( r , t ) · d r = I C ( t ) d r · { v d t × [ ∇ × A ( r , t )] } + I C ( t +d t ) A ( r , t ) · d r . (18)Finally, inserting expressions (15) and (18) into the right-hand side of eq. (14),taking into account thatlim d t → I C ( t +d t ) ∂ A ( r , t ) ∂t · d r = I C ( t ) ∂ A ( r , t ) ∂t · d r , (19)the result (13) follows.
3. Concluding comments
The above discussion reveals that the conceptualization of Faraday’s induction lawmay depend on the specific path employed for computing dd t H A · d r . The simplercomputation path, applying the Kelvin-Stokes theorem twice, which avoids bringing thetime derivative under the integral sign, does not yield the controversial term ∇ ( A · v ).Thus the issue of its inclusion into Maxwell’s original expression for the electromotiveintensity E is basically a pseudo-problem. Namely, it seems reasonable to take thata quantity whose appearance depends on the specific path chosen for computing the physical quantity, dd t H A · d r , may have but a spurious physical meaning. Consequently,the present note provides an unexpected argument in favor of Maxwell’s mysteriouschoice for E . Acknowledgment
My work is supported by the Ministry of Science and Education of the Republic ofSerbia, project No. 171028.
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