What do voltmeters measure?
Alain Bossavit
LGEP, 11 Rue Joliot-Curie, 91192 Gif-sur-Yvette CEDEX, France
Bossavit@lgep.supelec.fr
Abstract–We show that a voltmeter's reading is the
electromotive force, as it existed before branching it, along
the path γ traced out by the connectors between the contact
points. So, remarkably, a device which largely perturbs the
electric field is still able to give accurate information on
this very field as it were before the device was installed. We
give first a rough analysis, based on Faraday's law, which
is enough to make the result plausible, then a more detailed
asymptotic analysis, with the radius of the leads (assumed
perfectly conductive) as small parameter
Should the question in the title sound preposterous, let us
stress its legitimacy by discussing the following idealization.
In otherwise empty 3D Euclidean space E, we have a
massive conductor C (Fig. 1) and a coil Cs, that is the
support of a known source current j s (AC at some angular
frequency ω , and complex-valued). With σ > 0 inside
C and σ = 0 outside, the eddy current equations,
rot H = J = σE + Js, iω B + rot
(1)
E
= 0,
B
= µ0 H,
uniquely determine H (subject to the finite energy condition
∫E µ0 |H|2 < ∞) and determine E up to some curl-free field
supported outside C. For definiteness, we assume that
electric charge q = div(ε 0 E) is specified outside C (yet
not on the air-conductor interface ∂C, the boundary of C),
which lifts the ambiguity, again under the reasonable energy
condition ∫E ε 0 |E|2 < ∞.
V
B
J
s
C
A
s
C
Fig. 1. The setup. An induction coil C s generates eddy currents
in conductor C. What's the meaning of V, as displayed by a
voltmeter connected between points A and B of the surface?
Suppose now a voltmeter and its two connecting leads
are branched between two points A and B of ∂C. Like
all real-life voltmeters, that one has an adjustable inner
resistance R, which can be made high enough to reduce
the current I through the voltmeter to a negligible value.
The reading is then V, equal to RI. How does V relate
to the physical situation without the measuring device? Is
it correct to say that "V is the voltage drop between A
and B"? (The short answer is: No.) Is "voltage drop",
for that matter, a well defined notion? (Not always.) And
even if so, does that make the reading dependent on the
position of points A and B only (No) or is the position
of the leads relevant? (Yes—definitely.) These conclusions
can be found in the literature [1–6], but the underlying
rationales are not always convincing, and the very fact that
the problem is raised again and again (cf. [7], and the 2004
Web debate, http://www.physicsforums.com/archive/index.
php/t-24022.html) betrays some persistent uneasiness.
The paper is in two parts. First, we assume a "negligible"
current through the voltmeter, which means that the magnetic
field in presence of the connected voltmeter and its threads
is about the same as it was before their introduction. (Note,
immediately, that no such assumption can be made about
the electric field, which is widely changed by the fact of
connecting the voltmeter, thin as the leads may be.) Under
this hypothesis, we establish the rule quoted in the Abstract.
A second part addresses the "negligible" proviso:
Assuming the leads are perfect conductors, but with a
nonzero radius r, problem (1) is embedded in a family of
problems, indexed by the small non-dimensional parameter
α = r/<size of the device>. It is then proved that the
corresponding magnetic induction bα converges, in a
mathematically definite sense, towards a limit b 0. We'll
see in conclusion how this results validates the rough
analysis, thus making the intended point.
A word on notation: Although we use vector fields E,
etc., according to tradition, we put emphasis on the
physical entities they are meant to represent, such as
electromotive forces, fluxes, etc., i.e. to the differential
forms e, b, etc., they "stand proxy" for. Hence notations
such as " ∫γ e", " ∫S b", etc., meaning "the circulation of E
along γ ", "the flux of B embraced by surface S", etc.,
leaving unmentioned the tangent and/or normal vector fields,
dot product, etc., that would be necessary for mathematical
definiteness (better achieved, anyway, if e, b, etc., are
conceived as differential forms).
B,
I. ROUGH ANALYSIS
Call γ the path, oriented, from A to B, along the leads
and through the voltmeter, and ζ some path, from B to
A, entirely contained in the surface ∂C of the conductor.
Let S be any (oriented) surface the boundary of which be
γ + ζ. Consider the situation before (Fig. 2, left) and
after (Fig. 2, right) connecting the voltmeter, all other
things unchanged. The induction flux Φ = ∫S b is the
same in both cases, since currents do not change. By
Faraday's law,
(2)
iω Φ + ∫γ e + ∫ζ e = 0
always holds. Thanks to Ohm's law, the electric field e
in the conductor, and hence its tangential component on
the conductor's surface, doesn't change either. Therefore,
∫γ e is the same before and after, though e did change.
Since, as we assumed, the leads are perfect conductors,
where e vanishes, this circulation is the one that exists
inside the voltmeter, between its two ports, whose value
it's what the voltmeter is engineered to show on its dial, as
generally agreed (cf. [4]). We conclude that the voltmeter's
reading is the electromotive force, as it existed before
branching it, along the circuit γ traced out between
contact points A and B by the leads and voltmeter. It
is fairly remarkable that a property of the electric field can
thus be measured in spite of the considerable modification
of this very field brought in by the measuring apparatus.
γ
B
ζ
S
V
B
plane, containing both points A and B, with respect to
which (vectorial) values of B are mirrored; then, placing
γ in this plane will do the trick [5], since Φ = 0 in this
case, and W = – V [4]. Or else, the flux Φ can be
negligible for the same reasons the flux through ∂C is,
just because only a weak, negligible magnetic field exists
in the region where A and B lie, and where one branches
the voltmeter. Note that, if so, E is approximately
curl-free in this region, hence E = – grad v there, so that
V = ∫γ e = v(A) – v(B) is effectively the voltage drop one
wished to measure.
There is another way to make Φ vanish, thus radically
lifting the uncertainty about its value, as suggested by
Fig. 3. If a part of γ , lying in the air but near the surface
of the conductor, closely follows ζ, and if the other part,
joining the voltmeter's terminals, is properly intertwined,
Φ will be zero. Any preassigned path ζ can thus be
followed, which takes care of case (b). This applies, a
fortiori, to case (a), too: If the conductor's surface lets no
induction flux leak through,3 V does equal the surface
voltage drop between A and B, whatever the path ζ
followed on the surface. Beware of loops in such reasoning,
though (Fig. 3, right).
A
A
V
B
ζ
Fig. 2. V is equal to the emf along γ as it existed before
branching the voltmeter—the same as after branching it. The
effect of the measuring device is, so to speak, to lump this emf
between the voltmeter's terminals.
So the naive preconception that V would be "the
voltage 'between' A and B" is refuted: V does depend
on the path γ , and hence, on how the voltmeter is connected.
Anyway, does this "voltage between" notion make any
sense? Only in two cases can this be asserted: (a) When
the electric field, or at least, its tangential part on ∂C,
derives from a potential, so we are talking of a potential
difference, (b) When one has a specific path ζ in mind, so
that by "voltage" one really means the electromotive force,
i.e., the integral of e (or circulation1 of its proxy field
E), along ζ. Let's review both possibilities.
Case (a): It may happen, in special circumstances, that
no induction flux crosses ∂C, or only a negligible amount
of it, in which case the curl of the tangential part of E
vanishes. Then, an electric potential v can be defined, up
to some additive constant, on ∂C — locally,2 at least,
and the voltage drop W, relative to the surface, between A
and B (that is, – ∫ζ e) is well defined. Since W = V –
iω Φ, one can get W if information about Φ is available.
This may happen: For instance, when there is some symmetry
1
Which seems to be the right way to understand "voltage" [8][2].
"Voltage drop", on the other hand [5], evokes a difference in potential
values (also measured in volts), but such a potential may fail to exist.
2
If C presents "loops", v may be multi-valued. Then W depends on
the so-called "homology class" of ζ on the surface, according to around
which loops (embracing nonzero flux, as a rule) ζ goes (cf. Fig. 3,
right).
ζ ''
B
V
ζ'
ζ
A
A
Fig. 3. Left: How to effectively measure W = – ∫ζ e. Right:
How non-trivial topology can affect the issue, even when no
flux crosses the conductor's surface; ∫ζ' e = ∫ζ e, but ∫ζ'' e ≠ ∫ζ e,
owing to the loop (cf. Note 1), because ζ", contrary to ζ', is not
homologous to ζ.
So what to do when a specific voltage is one of the
quantities a numerical simulation is supposed to deliver?
Modelling the whole situation, including voltmeter and
wires with their precise geometry, would most often be
overkilling. One will rather ignore them, and just compute
the electric field where required, that is, along path γ
(some chain of edges of the mesh, in practice). Yet, this
means E must be computed in the air, which some
popular eddy-current methods (based on H, or on J ) avoid
to do. With such methods, a complementary computation
in the air will be needed (a problem of the form rot E = –
iω B , D = ε 0E, div D = q, outside C, with B and the
tangential part of E on ∂C coming from the first
computation). One may therefore prefer an "e-oriented"
method (over a large enough computational domain—this
also incurs some cost, and is part of the trade-off), in terms
of either e directly or of some a–v combination. In the
latter case, remind that V = – ∫γ (iω a + grad v), the
circulation of the total electric field, not only the so-called
"Coulomb voltage" v(A) – v(B). (The latter, gauge3
See [9] for an interesting application where this assumption is not
warranted.
dependent, is not physically observable anyway, so its
determination may fail to serve any useful purpose.)
Let's now turn a critical eye to the main part of our
argument: that "Φ = ∫S b is 'about the same' in both cases
[with or without wires and voltmeter]". To turn this into a
dignified mathematical assertion, we must parameterize the
situation by a real α > 0, meant to tend to 0, and prove
that the corresponding induction b α tends to b 0, the one
that exists without wires and voltmeter, which entails that
∫S bα tend to ∫S b0, and if so we are done. As such a
parameterization involves lots of details (size of the voltmeter,
its inner resistance, shape of the wires, contact resistances
at A and B, etc.), not all relevant to the same degree, we
shall rather introduce a clearcut auxiliary problem, of interest
in its own right, discuss its relevance, and prove the needed
convergence result.
II. DETAILED ANALYSIS
Our model problem in this part is magnetostatics in all
space,
(3)
rot H = J,
H
= ν 0 B , div
B
= 0,
with J as data and ν 0 = 1/µ0, under the constraint that B
= 0 in M α, owing to the presence of a perfectly diamagnetic
material in M α. Let's call B the functional space {B ∈
L2(E) : div B = 0} and set Bα = {B ∈ B : B = 0 in
Mα}. The weak form of (3) consists in finding B Α in B α
such that
(4)
∫E ν 0
Bα
· B ' = ∫E Hj · B ' for all
B'
in Bα,
where HJ is some "source field", so chosen that rot HJ = J.
(Alternatively, this is (3) outside the perfect diamagnet,
with n · B = 0 on its boundary ∂Bα.) There is a unique
solution B α, and we investigate its convergence, in the
sense of the L2-norm (or rather, || B || = (∫E ν 0 | B |2)1/2)
towards some limit B 0 when α → 0. In that case, fluxes
∫S bα tend to ∫S b0, by well known trace theorems, for any
well-behaved surface S.
x
uα (x)
Mα
y
uα (y)
α
z
uα (z)
M0
Fig. 4. The mapping u α, shrinking the diamagnetic region Mα
onto the subset M0 (meant to represent the path γ, or part of it,
of the previous section).
The set M α is a regular domain intended to model
one of the leads,4 and B = 0 stems from the assumed
perfect conductivity of this wire (zero skin depth, layer of
4
So we need two of them, one on each side of the voltmeter. The
latter is also in need of a similar asymptotic study, under hypotheses that
factor in its "smallness" (in size) and its high internal resistance. We
omit this, which would not bring in new ideas, from the present paper.
surface currents which screens out the magnetic field). To
account for the "small radius" of the wire, we introduce a
closed set M 0 ⊂ Mα with empty interior (for definiteness,
imagine M0 as a smooth curve segment—it will be the
above path γ , or rather, a half of it) and a family u α of
smooth mappings from E to itself, depending smoothly
on α, with the following properties: (H1) uα(Mα) = M 0,
(H2) u0 is the identity. (Figure 4 illustrates the idea.)
Moreover: (H3) the restriction of u α to E – Mα is a
diffeomorphism (so the derivative Duα(x) is an invertible
linear map, from 3D vectors to 3D vectors, at each point),
(H4) limα = 0 Duα(x) is the identity, (H5) As a map on
L2(E – M0), (Duα)–1 is uniformly bounded.
Some clarifications are needed: Imagine M0 (meant
to model a part of the above path γ ) cut out from space,
and the rest of space "retracted" to E – M α, so that the
"buttonhole" M0 "opens up" to a three-dimensional elongated
domain Mα: this describes the (restriction to E – M0 of
the) inverse mapping u α–1. On the other hand, the direct
map uα "stitches up the cut", and retracts Mα to M0.
The actual thread, with its effective dimensions, is Mα,
filled by perfectly diamagnetic stuff, for some definite value
of α. The asymptotic analysis is meant to justify the
approximation consisting in substituting M 0 to Mα.
Hypotheses (H3) to (H5) are technical requirements.
Let's recall that, if b is a 2-form (with proxy vector
in L2(E)), its "pullback" u α* b is the 2-form defined by
⟨(uα* b)(x) ; v, w⟩ = ⟨b(x) ; (Duα(x))v, (Duα(x))w⟩, where v
and w are two vectors at x. Abusing the notation, we
shall denote the proxy vector of uα* b by uα* B . Pullback
and exterior derivative d commute, so if div B = 0, i.e.,
db = 0, the proxy uα* B is also divergence-free. We note
that uα* B (x) is well-defined for x in M α, but null in the
case we consider: For if v and w are two vectors
anchored at x ∈ Mα, their images under u α are collinear.5
We conclude that if B 0 is the solution, in B, of
B
(5)
∫E ν 0
B
· B ' = ∫E Hj · B ' for all
B'
in B
(i.e., the magnetic field without any dimagnetic inclusion
present), then u α* B 0 belongs to the above-defined space
B α. Now compare (4) and (5), setting B ' = B – B α in (5)
and B ' = B α – uα* B in (4), which results in
∫E ν 0 |Bα – B |2 = ∫E (Hj + ν 0
B α )· (B α
– uα* B ).
Thanks to hypotheses (H4) and (H5)—made for that
purpose—the right-hand side tends to 0, therefore B α tends
to B 0, as announced, in L2(E). To sum up:
Proposition 1. Under hypotheses (H1–5), and assuming
uα* bα vanishes in Mα, fluxes ∫S b α tend to ∫S b0 for
regular surfaces S.
It remains to show that hypotheses (H1–5) are reasonable.
Let's begin with the (not yet realistic) case where M 0 is a
single point in space E, which we can take as origin.
Define then Mα = {x ∈ E : |x| ≤ α}, and uα(x) = x –
α x/|x|, i.e., "draw all points outside Mα towards the
5
Because M 0 is a curve, one-dimensional, here. In different
contexts, if M0 was a flat disk, for instance, a weaker conclusion
would follow: that uα* B is orthogonal to the surface of Mα.
origin by the same amount", and shrink M0 to the origin.
Call this "spherical stitching": it clearly satisfies our
hypotheses. Second case, an infinite straight line l, and
Mα is the tube of radius α around this line: use the
previous transformation in planes orthogonal to l, call
that "cylindrical stitching". Next, in the more realistic case
of a finite segment from point A to point B: If x
projects to line AB between points A and B, bring it
closer to its projection by cylindrical stitching, otherwise
push it towards A or B, whichever is closer, by spherical
stitching. (Then, Mα is a cylinder of radius α with
rounded ends.) Last, in the case of a smooth curve from A
to B, first build an isotopy (a "good" map from E to
itself) to rectify the curve, then compose with the previous
transformation.
And so on: Whatever the shape of the wire (non-uniform
radius, varying cross-section, etc.), a suitable family uα
that shrinks it to a curve γ can be found, such that Prop.
1 hold—in other words, when the perfectly conducting
wire reduces to a curve, the magnetic field tends to what it
would have been, had this curve not been there.
Remark. Therefore, "line defects", and a fortiori, "point
defects", with perfect diamagnetism, can be ignored in
magnetostatics. This conclusion cannot be drawn for surface
defects, however, because the property "uα* B 0 ∈ Bα" does
not hold. This is intuitively obvious: A perfectly
diamagnetic surface, however thin, will make a barrier to
magnetic flux, and "force flux lines to turn around it". ◊
Remark. It may be instructive to interpret our technique
of proof in this spirit: The homotopy uα is constructed
in order to "warp the flux tubes" of the induction field B 0
to push them out of the diamagnetic region. When the
latter shrinks to a curve, the "no defect" situation is recovered,
but not when the residual defect is a surface. ◊
III. DISCUSSION AND CONCLUSION
What precedes should be enough to support the main thesis:
What a voltmeter measures is the voltage, as it existed
before branching the device, along the path traced out by
the two wires and the voltmeter itself between the two
measurement points. We have proved the main asymptotic
theorem on which to base this conclusion. A lot of details
should be dealt with, however, using similar techniques.
First, treating the problem as one in magnetostatics, instead
of eddy currents, as we did, is justified by the fact that
placing the voltmeter does not modify the currents in the
main conductor C. Of course, this is not rigorously true,
and a limit analysis of this effect (a second order perturbation)
should be done. Also, to assume perfect conductivity, and
hence perfect diamagnetism, is abusive, and again, a limit
analysis (with a second small parameter, resistivity of the
wires, subordinate to the above α) should take care of
that. The voltmeter itself ("small and highly resistive")
should receive similar attention. What happens at the
contact points A and B is also matter for investigation.
And beyond convergence results, one might very well push
further the Taylor expansions of the fields involved, in
terms of the small parameter(s), in order to reach estimates
of the errors incurred. Let us just mention these directions
of research, and express the hope that this subject, modelling
asymptotics, will be developed as a more rigorous substitute
to the intuition-based procedures we use nowadays in
modelling.
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