On Equivalent Expressions for the Faraday's Law of Induction

In this paper we give a rigorous proof of the equivalence of some different forms of Faraday's law of induction clarifying some misconceptions on the subject and emphasizing that many derivations of this law appearing in textbooks and papers are only valid under very special circumstances and not satisfactory under a mathematical point of view.


Introduction
Let Γ t be a smooth closed curve in R 3 with parametrization x(t,ℓ) which is here supposed to represent a filamentary closed circuit which is moving in an a convex and simply-connected (open) region U ⊂ R 3 where at time t as measured in an inertial frame 1 , there are an electric and a magnetic fields (described after a space orientation is fixed by) E : R × R 3 → R 3 , (t, x) → E(t, x) ∈ R 3 and B : R × R 3 → R 3 , (t, x) → B(t, x) ∈ R 3 . We suppose that when in motion the closed circuit may be eventually deforming. Let Γ be a smooth closed curve in R 3 with parametrization x(ℓ) representing the filamentary circuit at t = 0. Then, the smooth curve Γ t is given by Γ t = σ t (Γ) where σ t (see details below) is the flow of a velocity vector field v : R × R 3 → R 3 , which describes the motion (and deformation) of the closed circuit. It is an empirical fact known as Faraday's law of induction that on the closed loop Γ t acts an induced electromotive force, E, such that where S t is a smooth surface on R 3 such that Γ t is its boundary and n is the normal vector field on S t . We write Γ t = ∂S t with Γ = ∂S. Now, on each element of Γ t the force acting on a unit charge which is moving with velocity v(t, x(t, ℓ)) is given by the Lorentz force law. Thus 2 the E is by definition: where dl := ∂x(t,ℓ) ∂ℓ dℓ and Faraday's law reads: We want to prove that Eq. (3) is equivalent to from where it trivially follows the differential form of Faraday's law, i.e., Those statements will be proved in Section 3, but first we shall need to recall a few mathematical results concerning differentiable vector fields, in Section 2.

Some Identities Involving the Integration of Differentiable Vector Fields
Let U ⊂ R 3 be a convex and simply-connected (open) region, X : R×U → R 3 ,(t, x) → X(t, x) be a generic differentiable vector field and let v : R×U → R 3 be a differentiable velocity vector field of a fluid flow. A trajetory (of a "fluid particle") 3 associated to v passing through a given x ∈ R 3 is a smooth curve σ x : R →R 3 , t → σ x (t) = σ(t, x) which at t = 0 is at x (i.e., σ x (0)= x ) and such that its tangent vector at σ(t, x) is Let moreover σ t : U → R 3 , σ t (x) = σ(t, x). We call σ t the fluid flow map. Let J = (0, 1) ∈ R and let Γ be a closed loop parametrized by Γ : J → R 3 , ℓ → Γ(ℓ) := x(ℓ) and denote by Γ t = σ t (Γ) the loop transported by the flow. Then is clearly a parametrization of Γ t . We have the proposition: where is the so-called material derivative 4 and Now, taking into account that for each hence, the first term in the right side of Eq.(11) can be written as 4 Mind that the material derivative is a derivative taken along a path σt with tangent vector v| σx . It is frequently used in fluid mechanics, where it describes the total time rate of change of a given quantity as viewed by a fluid particle moving on σx. 5 Take notice that dl is not an explicit function of the cartesian coordinates (x, y, z).
Also writing σ(t, x(ℓ)) = (x 1 (t, ℓ), x 2 (t, ℓ), x 3 (t, ℓ)) we see that the last term in Eq.(11) can be written as: We now recall that for arbitrary differentiable vector fields a, b : Setting a = dl and b = v and noting that We need also to recall the well known identity and also the not so well known identity 6 to write that Finally, using Eq. (13) and Eq.(20) completes the proof of Eq.(8a) and Eq.(8b). Also, from Eq.(8b) if we use Eq.(15), setting a = X and b = v and noting that from where the proof of Eq.(8c) follows immediately. 6 See the Appendix for a proof of this identity.
Remark 1 Before proceeding, we recall that if X = v we have a result that is known in fluid mechanics as Kelvin's circulation theorem (see, e.g., [2,16]).
where, if S is a smooth surface such that ∂S = Γ, then S t = σ t (S). Also n is the normal vector field to S t . Then using Eq.(8c) we can write: Also, denoting Y : = ∇ × X we can write Despite Eq.(24), for a general differentiable vector field Z : R×U → R 3 such that ∇ · Z = 0 we have the so-called Helmholtz identity [4]. Note that the identity is also mentioned in [5]. A proof of Helmholtz identity can be obtained using arguments similar to the ones used in the proof of Eq.(8a). Some textbooks quoting Helmholtz identity are [1,6,10,17,18]. However, we emphasize that the proof of Faraday's law of induction presented in all the textbooks just quoted are always for very particular situations and definitively not satisfactory from a mathematical point of view. We now want to use the above results to prove Eq.(3) and Eq.(4).

Proofs of Eq.(3) and Eq.(4)
We start remembering that in Maxwell theory we have that the E and B fields are derived from potentials, i.e., where φ : R × R 3 → R is the scalar potential and A : R × R 3 → R is the (magnetic) vector potential. If Eq.(26) is taken into account we can immediately derive Eq.(3). All we need is to use the results just derived in Section 2 taking X = A. Indeed, the first line of Eq.(23) then becomes To obtain Eq.(4) we recall that from the second line of Eq.(23) we can write (using Stokes theorem) Comparing the second member of Eq.(27) and Eq.(28) we get Eq.(4), i.e., from where the differential form of Faraday's law follows.
Remark 2 We end this section by recalling that in the physical world the real circuits are not filamentary and worse, are not described by smooth closed curves. However, if the closed curve representing a 'filamentary circuit' is made of finite number of sections that are smooth, we can yet apply the above formulas with the integrals meaning Lebesgue integrals.

Conclusions
Recently a paper [12] titled 'Faraday's Law via the Magnetic Vector Potential', has been commented in [7] and replied in [13]. Thus, the author of [12], claims to have presented an "alternative" derivation for Faraday's law for a filamentary circuit which is moving with an arbitrary velocity and which is changing its shape, using directly the vector potential A instead of the magnetic field B and the electric field E (which is the one presented in almost all textbooks). Now, [7] correctly identified that the derivation in [12] is wrong, and that author agreed with that in [13]. Here we want to recall that a presentation of Faraday's law in terms of the magnetic vector potential A already appeared in Maxwell treatise [8], using big formulas involving the components of the vector fields involved. We recall also that a formulation of Faraday's law in terms of A using modern vector calculus has been given by Gamo more than 30 years ago [3]. In Gamo's paper (not quoted in [7,12,13]) Eqs.(8c) appear for the special case in which X = A (the vector potential) and B =∇×A (the magnetic field), i.e., Thus, Eq.(30) also appears in [12] (it is there Eq. (9)). However, in footnote 3 of [12] it is said that Eq.(30) is equivalent to " d dt Γt A · dl = Γt D Dt A · dl ", where the term Γ [(A · ∇)v)] · dl is missing. This is the error that has been observed by authors [7], which also presented a proof of Eq.(8b), which however is not very satisfactory from a mathematical point of view, that being one of the reasons why we decided to write this note presenting a correct derivation of Faraday's law in terms of A and its relation with Helmholtz formula. Another reason is that there are still people (e.g., [11]) that do not understand that Eq.(3) and Eq.(4) are equivalent and think that Eq.(3) implies the form of Maxwell equations as given by Hertz, something that we know since a long time that is wrong [9].
We also want to observe that Jackson's proof of Faraday's law using 'Galilean invariance' is valid only for a filamentary circuit moving without deformation with a constant velocity. The proof we presented is general and valid in Special Relativity, since it is based on trustful mathematical identities and in the Lorentz force law applied in the laboratory frame with the motion and deformation of the filamentary circuit mathematically well described.
On the other hand, considering dl = (dl 1 , dl 2 , dl 3 ) = dl i e i , we have Hence, substituting Eq.(34) and Eq.(35) in Eq.(31), we can rewrite it as From this last result, it is easy to see that where X = (X 1 , X 2 , X 3 ) = X i e i , e i · e j = δ j i .