arXiv:1002.2792v2 [physics.class-ph] 30 Mar 2010
On Equivalent Expressions for the Faraday’s Law
of Induction
Fabio G. Rodrigues
Institute of Mathematics, Statistics and Scientific Computation
Imecc-Unicamp
13083-859 Campinas SP, Brazil
e-mail: fabrod@ime.unicamp.br
March 31, 2010
Abstract
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 circunstances
and not satisfactory under a mathematical point of view.
1
Introduction
Let Γt be a smooth closed curve in R3 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 ⊂ R3 where at time t as measured
in an inertial frame1 , there are an electric and a magnetic fields (described after
a space orientation is fixed by) E : R × R3 → R3 , (t, x) 7→ E(t, x) ∈ R3
and B : R × R3 → R3 , (t, x) 7→ B(t, x) ∈ R3 . We suppose that when in
motion the closed circuit may be eventually deforming. Let Γ be a smooth closed
curve in R3 with parametrization x(ℓ) representing the filamentary circuit at
t = 0. Then, the smooth curve Γt is given by Γt = σt (Γ) where σt (see details
below) is the flow of a velocity vector field v : R × R3 7→ R3 , 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 Γt acts an induced
electromotive force, E, such that
Z
d
E =−
B · n da,
(1)
dt St
1 For a mathematical defintion of an inertial reference frame in Minkowski spacetime see,
e.g., [14, 15].
1
where St is a smooth surface on R3 such that Γt is its boundary and n is the
normal vector field on St . We write Γt = ∂St with Γ = ∂S. Now, on each
element of Γt the force acting on a unit charge which is moving with velocity
v(t, x(t, ℓ)) is given by the Lorentz force law. Thus2 the E is by definition:
Z
(2)
E = (E + v × B) · dl,
Γt
where dl :=
∂x(t,ℓ)
∂ℓ dℓ
and Faraday’s law reads:
Z
Z
d
B · n da.
(E + v × B) · dl = −
dt St
Γt
We want to prove that Eq.(3) is equivalent to
Z
Z
∂B
E · dl = −
· n da,
St ∂t
Γt
(3)
(4)
from where it trivially follows the differential form of Faraday’s law, i.e.,
∂B
= 0.
(5)
∂t
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.
∇×E+
2
Some Identities Involving the Integration of
Differentiable Vector Fields
Let U ⊂ R3 be a convex and simply-connected (open) region, X : R×U →
R3 ,(t, x) 7→ X(t, x) be a generic differentiable vector field and let v : R×U → R3
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 ∈ R3 is a smooth curve
σx : R →R3 , t → σx (t) = σ(t, x) which at t = 0 is at x (i.e., σx (0)= x ) and
such that its tangent vector at σ(t, x) is
∂
σ(t, x) = v(t, σ(t, x)).
∂t
(6)
Let moreover σt : U → R3 , σt (x) = σ(t, x). We call σt the fluid flow map. Let
J = (0, 1) ∈ R and let Γ be a closed loop parametrized by Γ : J → R3 , ℓ 7→
Γ(ℓ) := x(ℓ) and denote by Γt = σt (Γ) the loop transported by the flow. Then
σ(t, x(ℓ)) := x(t, ℓ)
(7)
is clearly a parametrization of Γt . We have the proposition:
2 In
this paper we use a system of units such that the numerical value of the speed of light
is c = 1.
3 An integral line (or a stream line) of v at fixed t is a mapping φ : R → R3 , s 7→ φ (s),
x
x
d
φ(0) = x, such that ds
φx (s) = v(t, φx (s)). Note that if v is stationary ( ∂v
= 0), then the
∂t
concepts of integral lines and trajectories coincide [2].
2
Proposition
d
dt
Z
X · dl =
Z
=
Z
=
Z
Γt
Γt
Γt
Γt
where
Z
D
X · [(dl · ∇)v],
(8a)
X · dl +
Dt
Γt
Z
Z
D
[(X · ∇)v)] · dl (8b)
[X × (∇ × v)] · dl +
X · dl +
Dt
Γt
Γt
Z
∂
[v × (∇ × X)] · dl,
(8c)
X · dl −
∂t
Γt
D
∂
X := X + (v · ∇)X
Dt
∂t
(9)
is the so-called material derivative4 and
dl =
∂
∂x(t, ℓ)
σ(t, x(ℓ))dℓ =
dℓ
∂ℓ
∂ℓ
(10)
is the tangent line element5 of Γt at σ(t, x(ℓ)).
Proof. We can write
d
dt
d
X · dl =
dt
Γt
Z
Z1
X(t, σ(t, x(ℓ))) ·
∂
σ(t, x(ℓ))dℓ
∂ℓ
0
=
Z1
d
∂
[X(t, σ(t, x(ℓ)))] · σ(t, x(ℓ))dℓ
dt
∂ℓ
+
Z1
X(t, σ(t, x(ℓ))) ·
0
∂ ∂
σ(t, x(ℓ))dℓ.
∂t ∂ℓ
(11)
0
Now, taking into account that for each x(ℓ),
∂
∂t σ(t, x)
= v(t, σ(t, x(ℓ))) we have
∂
D
[X(t, σ(t, x(ℓ)))] = X(t, σ(t, x(ℓ))) + (v · ∇)X(t, σ(t, x(ℓ))),
Dt
∂t
(12)
hence, the first term in the right side of Eq.(11) can be written as
Z1
d
∂
[X(t, σ(t, x(ℓ)))]· σ(t, x(ℓ))dℓ =
dt
∂ℓ
0
Z
Γt
[
∂
X + (v ·∇)X]·dl =
∂t
Z
Γt
D
X·dl.
Dt
(13)
4 Mind that the material derivative is a derivative taken along a path σ with tangent
t
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).
3
Also writing σ(t, x(ℓ)) = (x1 (t, ℓ), x2 (t, ℓ), x3 (t, ℓ)) we see that the last term
in Eq.(11) can be written as:
Z1
∂ ∂
X(t, σ(t, x(ℓ))) ·
σ(t, x(ℓ))dℓ =
∂t ∂ℓ
0
Z1
∂
X(t, σ(t, x(ℓ)))·
v(t, σ(t, x(ℓ)))dℓ
∂ℓ
0
=
Z
X · [(dl · ∇)v].
(14)
Γt
We now recall that for arbitrary differentiable vector fields a, b : U → R3 it
holds
∇(a · b) = (a · ∇)b + (b · ∇)a + a × (∇ × b) + b × (∇ × a).
(15)
Setting a = dl and b = v and noting that (v · ∇)dl = v × (∇×dl) = 0, it implies
that
(dl · ∇)v = ∇(dl · v)−dl × (∇ × v).
(16)
We need also to recall the well known identity
a · (b × c) = b · (c × a)
(17)
which implies setting a = X, b = dl and c = (∇ × v), that
−X · [dl × (∇ × v)] = −dl · [(∇ × v) × X],
= dl · [X × (∇ × v)],
(18)
and also the not so well known identity6
X · [∇(dl · v)] = [(X · ∇)v] · dl,
to write that
Z
Z
Z
X · [∇(dl · v)]
X · [dl × (∇ × v)]+
X · [(dl · ∇)v]= −
Γt
Γt
Γt
Z
Z
[(X · ∇)v)] · dl.
[X × (∇ × v)] · dl +
=
(19)
(20)
Γt
Γt
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
R
[∇(X
· v)] · dl = 0, it follows
Γt
Z
d
X · dl
dt Γt
Z
Z
Z
Z
∂
=
[X × (∇ × v)] · dl + [(X · ∇)v)] · dl
X · dl + [(v · ∇)X)] · dl +
Γ ∂t
Γ
Γt
Γ
Z t
Z
∂
=
[v×(∇ × X)] · dl.
X · dl −
Γt ∂t
Γt
from where the proof of Eq.(8c) follows immediately.
6 See
the Appendix for a proof of this identity.
4
Remark 1 Before proceeding, we recall that if X = v we have
Z
Z
d
D
v · dl,
v · dl =
dt Γt
Γt Dt
(21)
a result that is known in fluid mechanics as Kelvin’s circulation theorem (see,
e.g., [2, 16]).
Now,
d
dt
Z
X · dl =
Γt
d
dt
Z
(∇ × X) · n da,
(22)
St
where, if S is a smooth surface such that ∂S = Γ, then St = σt (S). Also n is
the normal vector field to St . Then using Eq.(8c) we can write:
d
dt
Z
(∇ × X) · n da =
Z
=
Z
Γt
St
St
Z
∂
[v×(∇ × X)] · dl
X · dl −
∂t
Γt
Z
∂
∇ × [v×(∇ × X)] · n da.
(∇ × X) · n da −
∂t
St
(23)
Also, denoting Y : = ∇ × X we can write
Z Z
∂
d
Y · n da =
Y − ∇ × (v × Y) · n da.
dt St
St ∂t
(24)
Despite Eq.(24), for a general differentiable vector field Z : R×U → R3 such
that ∇ · Z 6= 0 we have
Z Z
d
∂
Z · n da =
Z + v(∇ · Z) − ∇ × (v × Z) · n da,
(25)
dt St
St ∂t
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).
3
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.,
E = −∇φ −
B = ∇ × A,
5
∂A
,
∂t
(26)
where φ : R × R3 → R is the scalar potential and A : R × R3 7→ 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
Z
Z
Z
d
∂
[v×(∇ × A)] · dl,
A · dl −
(∇ × A) · n da =
dt St
Γt
Γt ∂t
or
d
dt
Z
Z
∂
(v × B] · dl
A · dl −
Γt
Γt ∂t
Z ∂
=
A + ∇φ − v × B · dl
∂t
Γt
Z
(E + v × B) · dl.
=−
B · n da =
St
Z
(27)
Γt
To obtain Eq.(4) we recall that from the second line of Eq.(23) we can write
(using Stokes theorem)
Z
Z
Z
∂
d
∇ × [v × B] · n da
B · n da −
B · n da =
dt St
St
St ∂t
Z
Z
∂
(v × B) · dl.
(28)
B · n da −
=
Γt
St ∂t
Comparing the second member of Eq.(27) and Eq.(28) we get Eq.(4), i.e.,
Z
Z
∂
E · dl = −
B · n da,
(29)
St ∂t
Γt
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.
4
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).
6
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.,
Z
Z
Z
d
∂
[v × (∇ × A)] · dl.
(30)
A · dl −
A · dl =
dt Γt
Γt
Γt ∂t
Thus, Eq.(30) also appears in [12] (it is there Eq.(9)).
However,
footnote
R in
R
D
d
A
·
dl
=
A
· dl ”,
3 of [12] it is said that Eq.(30) is equivalent to “ dt
Γt Dt
Γt
R
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.
A
Proof of the Identity in Eq.(19)
We know from Eq.(16) that
∇(dl · v) = (dl · ∇)v+dl × (∇ × v).
(31)
1 2 3
Let e , e , e be an orthonormal base of R3 . We can write, using Einstein
convention,
(∇ × v) = ei ∂i × v = ei × ∂i v,
(32)
∂
2 ∂
3 ∂
i
where ∇ = (∂1 , ∂2 , ∂3 ) = e1 ∂x
1 + e ∂x2 + e ∂x3 = e ∂i , with ∂i =
i = 1, 2, 3 are Cartesian coordinates. It follows then
dl × (∇ × v) = dl × (ei × ∂i v).
∂
∂xi
and {xi },
(33)
Using the known identity a × b × c = (a · c)b − (a · b)c in Eq.(33), we obtain
dl × (ei × ∂i v) = (dl · ∂ i v)ei − (dl · ei )∂i v.
7
(34)
On the other hand, considering dl = (dl1 , dl2 , dl3 ) = dli ei , we have
(dl · ∇)v = (dli ∂i )v = (dl · ei )∂i v.
(35)
Hence, substituting Eq.(34) and Eq.(35) in Eq.(31), we can rewrite it as
∇(dl · v)=(dl · ei )∂i v+(dl · ∂ i v)ei − (dl · ei )∂i v
= (dl · ∂ i v)ei .
(36)
From this last result, it is easy to see that
X · [∇ (dl · v)] = X · (dl · ∂ i v)ei = X i (dl · ∂ i )v =dl · (X i ∂i )v
= dl · [(X · ∇)v] = [(X · ∇)v] · dl,
where X = (X 1 , X 2 , X 3 ) = X i ei , ei · ej = δij .
References
[1] Abraham, M. and Becker, R., The Classical Theory of Electricity and Magnetism, Blackie, London, 1932.
[2] Chorin, A. J. and Mardsen, J. E., A Mathematical Introduction to Fluid
Mechanics (third edition), Springer-Verlag, New York, 1993.
[3] Gamo, S. A., General Formulation of Faraday’s Law of Induction, Proc.
IEEE 67, 676-677 (1979).
[4] Helmholtz, H., Gesammelt Scriften vol. I (3), pp. 597-603, OlmsWeidmann, Hildeshein 2003, (a reprint of Wissenschaftliche Abhandlungen
vol. 3, Johann Ambrosius Barth, Leipzig, 1895).
[5] Hertz, H. R., Electric Waves, MacMillan, London, pp. 243-247, 1893 (also,
Dover, New York, 1962).
[6] Jackson, J. D., Classical Electrodynamics (third edition) J., Wiley & Sons,
Inc., New York, 1999.
[7] Kholmetskii, A. K., Missevitch, O., and Yarman, T., Comment on the
Note: ‘Faraday’s Law via the Magnetic Vector Potential’ by Dragan V.
Redžić, Eur. J. Phys. 29, L1-L4 (2008).
[8] Maxwell, J. C., A Treatise of Electricity and Magnetism, pp. 238-243,
Dover, New York, 1954.
[9] Miller, A. I., Albert Einstein’s Special Theory of Relativity, pp. 16, AddisonWesley Publ. Co., Inc., Reading, Ma., 1981.
[10] Panofski, W. K. H. and Phillips, M., Classical Electricity and Magnetism,
pp. 160-162, Addison-Wesley Publ. Co., Reading MA, 1962.
8
[11] Phipps, T. E. Jr., Old Physics for New - A Worldview Alternative to Einstein’s Relativity Theory, Apeiron, Montreal, 2006.
[12] Redžić, D. V., Faraday’s Law via the Magnetic Vector Potential, Eur. J.
Phys., 28, N7-N10, (2007).
[13] Redžić, D. V., Reply to ‘Comment on ‘Faraday’s Law via the Magnetic
Vector Potential’ ’, Eur. J. Phys., 29, L5, (2008).
[14] Rodrigues, W. A. Jr. and Capelas de Oliveira, E., The Many Faces of
Maxwell, Dirac and Einstein Equations. A Clifford Bundle Approach, Lecture Notes in Physics 722, Springer, Heidelberg, 2007.
[15] Sachs, R. K., and Wu, H., General Relativity for Mathematicians, SpringerVerlag, New York, 1977.
[16] Saffman, P.G., Vortex Dynamics, Cambridge Monographs on Mechanics
and Applied Mathematics, Cambridge Univ. Press, Cambridge, 1992.
[17] Sommerfeld, A., Electrodynamics, pp. 285-287, Academic Press, New York,
1952.
[18] Paul, C. R., Whites, K. W. and Nasar, A. A., Introduction to Electromagnetic Fields (third edition), pp. 653-658 , McGraw-Hill, Boston, 1998.
9