THE NATURE OF AN INDUCTOR
INTRODUCTION
The primary object of our attention in this essay is the inductor. This circuit theory element is ordinarily
introduced to all of us in a study of AC circuit theory. We are told of Michael Faraday, Joseph Henry and
Heinrich Lenz. We learn that Lenz’ law explains the signature of the induced current.
An induced electromotive force (emf) gives rise to a current whose magnetic field
opposes the original change in magnetic flux.
As we shall soon demonstrate, there is no obvious way to employ this early-nineteenth century explanation
for the behavior of the simplest inductor in a circuit. We shall show that in a series RCL circuit, holding an
ideal inductor 𝐿, that there is no electric field inside the inductor of any kind. Our illustrations in this paper
shall depict a DC voltage source (battery), but the methodology and the details of all explanations are
identical for an AC voltage source (generator).
Definitions
Voltage 𝑣𝑎𝑏 , from a to b: line integral from b to a of an electric field.
EMF : open-line integral in direction of an energy converter’s electric field
Induced EMF: closed-line integral in direction of an energy converter’s electric field
Notes
Voltage is a well-defined term, for either a conservative or non-conservative electric field.
The phrase “from a to b” signifies that point 𝑎 is selected as the reference point.
Potential difference is only defined (outside this paper!) for a conservative field.
An energy converter, such as a battery or electric generator, develops an electric field of nonelectrostatic origin as well as an electric field of electrostatic origin.
Starting with Field Theory, we interpret how an inductor works in an electrical circuit. We also show that
Kirchhoff’s Voltage Law is valid for an RCL circuit. Doubt had been cast on this latter point by a Professor at
M.I.T.
As is customary in discussions of “lumped” circuit elements, we assume the dimensions of our circuit are
small compared to a wavelength of radiation.
To avoid a commonly made error in such discussions, we show that care must be taken with the (real)
meaning of a voltmeter as opposed to merely discussing “voltage” as a Field Theory concept.
DISCUSSION
A sketch is now provided to show the objects of this discussion: an ideal battery with no internal resistance, a
resistor of conductivity 𝜎, a capacitor of permittivity 𝜖, and a large single-turn inductor. A SPST switch is also
1
THE NATURE OF AN INDUCTOR
provided, so that transient response can be followed. The battery could just as well be a voltage generator in
all the following discussion.
Figure 1. Sketch of a Single Loop Inductor Circuit


J
Our circuit has four elements: a battery, a resistor, a capacitor and a large, single-turn loop. We imagine this
one equivalent inductor accounts for an actual inductor and for a second inductor that is a parasitic inductor
due to the rectangular circuit loop. Mutual inductance between these two inductors is ignored. In
consideration of this abstraction of adding two inductors, we now regard the rectangular loop lying to the left
⃗ threading it. The depiction shown here is
of the much larger circular loop as having no induction field 𝐵
“physical,” as in a snapshot. One should studiously avoid mixing circuit theory symbols with field theory
symbols. Mixing of such pictures causes an error made by many writers.
The Maxwell-Faraday Law, like any law, provides no “why.” A physical law, by definition, is but a concise
mathematical equation relating physical quantities. In words, the Maxwell-Faraday vector equation states
⃗ is accompanied by a non-conservative electric field 𝐸⃗ and
that a time-changing magnetic induction field 𝐵
vice versa. By “non-conservative,” we mean that whenever the right-hand side of the following equation is
nonzero, then this electric field 𝐸⃗ cannot be described by the gradient of a scalar potential:
∇𝑥𝐸⃗ = −
⃗
𝜕𝐵
𝜕𝑡
(1)
Note well that there is no causality implied by this equation. No general statement can be made about which
2
THE NATURE OF AN INDUCTOR
side of this equation creates the other side! The nature of the electric field is not addressed; that is, parts of
the electric field appearing on the left-hand side of this equation can be electrostatic or non-electrostatic in
origin. Yes, this James Clerk Maxwell 1861 expression certainly had its origin in Michael Faraday’s (Joseph
Henry’s) 1831 work, but this Equation (1) is far more general in scope.
By use of the Kevin-Stokes theorem, this vector differential Equation (1) can also be written in integral form.
Our focus is stationary circuits, so the time derivative simplifies. One obtains:
∮ 𝐸⃗ ∙ 𝑑𝑙 = −
⃗ ∙ 𝑑𝑎
𝑑∯𝐵
𝑑𝑡
(2)
The next pair of sketches shows the sign conventions used with this Equation (2):
⃗ of the Maxwell-Faraday Law in Integral Form
Figure 2. Sign Conventions for Magnetic Induction 𝐵
dl


da
B
J
Notes
The line element 𝑑𝑙 and area element 𝑑𝑎 are related by the right-hand rule according to the KevinStokes theorem. (The clockwise direction of 𝑑𝑙 is arbitrarily chosen.) Placing one’s thumb in the
direction of 𝑑𝑙 and grasping the wire loop shows that area element 𝑑𝑎 points into the page. The
⃗ also point into the page.
direction of the current density 𝐽 makes the magnetic induction field 𝐵
3
THE NATURE OF AN INDUCTOR
Figure 3. Sign Conventions for Electric Field 𝐸⃗ of the Maxwell-Faraday Law in Integral Form
E
b
b
c
E
Area = A, Length = l
Ee
E
En
Ee
d
c
E
a
Area = S, Gap = d
d
b
a

c

d
E
e
e
Notes
1) Circuit element nodes 𝑎, 𝑏, 𝑐, 𝑑 and 𝑒 are defined in the lower, left inset.
2) Every battery has three(3) associated electric fields, as illustrated. We choose the internal
⃗⃗⃗⃗𝑒 ; otherwise, 𝑞𝐸
⃗⃗⃗⃗𝑛 = −(𝑞𝐸
⃗⃗⃗⃗𝑛 + 𝑓), where
resistance 𝑟 of this battery to be zero so that ⃗⃗⃗⃗
𝐸𝑛 = −𝐸
𝑓 is the viscous force associated with internal resistance 𝑟, so that | ⃗⃗⃗⃗
𝐸𝑛 | > | ⃗⃗⃗⃗
𝐸𝑒 | .
3) The direction of the electric field from point 𝑑 to 𝑒 is explained in the body text.
Two Loop Circuit: one non-conservative and the other conservative
Here we further interpret the Faraday-Maxwell Equation (2), for this circuit.
4
THE NATURE OF AN INDUCTOR
According to Circuit Theory, we have a one loop circuit. According to Field Theory, we have (at least) two
closed loops. Recall that a loop, according to conventional interpretation of Equation (2), can consist entirely
of wire or it can consist of wire and air or it can consist entirely of air.
Consider the large closed circular loop, lying to the right. This is our inductor. By closed, for this loop, we
mean a line integral can be taken entirely around the wire and the air gap combination. We shall find
(below) that an induced voltage exists across this air gap. A voltmeter placed in this air gap will indicate the
magnitude and direction of this voltage. This closed circular loop has a non-conservative electric field,
because the right-hand side of Equation (2) does not vanish.
Consider the smaller closed rectangular loop, lying to the left. As earlier assumed, this loop has no induction
⃗ threading it. We suppose the height of this rectangle is as small as we please; further, we suppose the
field 𝐵
three elements (battery, resistor and capacitor) are as small as we please. We seem “to hit a wall” that was
mentioned in Carson’s 1927 Bell Labs paper on the foundation of Circuit Theory. Maxwell’s (macroscopic)
Equations cannot deal with chemical EMFs (batteries)! Although no magnetic induction threads this
rectangular loop, the line integral of electric field around this rectangular loop cannot vanish, even though
the right-hand side of Equation (2) does vanish for this rectangular loop! So it becomes clear that the
Maxwell Equations cannot deal with the EMF of a battery.
To make this point yet clearer, consider the case where the induced voltage across the air gap is zero. That is,
consider the case where the circuit has no changing current. The switch has been closed for a long time and
the transient behavior has decayed. The current in this circuit is zero, the right-hand side of Equation (2) for
this rectangular loop is zero, but the left-hand side of Equation (2) is the line integral for this rectangular loop.
This closed loop line integral has the value of the EMF of the battery.
Are we stumped? Not in the least! Consider that we have no interest whatever in the value of a line integral
though the left-hand, rectangular loop. In building a map from Field Theory to Circuit Theory, we must forgo
interest in “theoretical” line integrals and consider line integrals taken through the paths occupied by
voltmeters. See Figure 4.
We need only bear in mind that even in the absence of a real (extended) voltmeter, all the electric fields still
exist. None of these electric fields that traverse any one of these voltmeters pass though the interior of our
battery! Thusly, the theoretical problem with Maxwell’s Equations not being able to deal with the EMF of a
battery is obviated!
This key point has apparently been overlooked by too many other authors! Figure 4 or, optionally, Figure 5,
shows the circuit as it must be considered. Real voltmeters across circuit elements are employed in Circuit
Theory, not ill-considered line integrals.
What do we know of voltmeters? We know they display the values of line integrals taken through devices
ideally having infinite impedance. Our rectangular loop contains no magnetic flux, so this rectangular loop
has zero for the right-hand side of Equation (2). Our loop of four voltmeters in this rectangular loop contains
three line integrals of electric fields of electrostatic origin (voltage 𝑉𝐵 , voltage 𝑣𝑅 , voltage 𝑣𝐶 ) and one line
integral of an electric field of non-electrostatic origin (voltage 𝑣𝐿 ).
5
THE NATURE OF AN INDUCTOR
Voltage is but work per unit charge, and we show below that Kirchhoff’s Second Law (KVL) works perfectly
fine when the algebra is done correctly for these four voltages. (Three of these voltages are indeed
expressible as potential differences, but the induced voltage cannot be expressed as a potential difference.)
Maxwell-Faraday Equation for the Non-Conservative Right-Lying Loop
In this section, we shall find an expression for the value of voltmeter voltage 𝑣𝐿 .
Inductor Voltage 𝒗𝑳
Consider the right-lying circular loop as a closed (Maxwell-Faraday) loop consists of the coil and its highimpedance voltmeter 𝑣𝐿 alone.
The left-hand side of Equation (2) has no contribution from the large circular loop of the inductor, because
the electric field inside a perfect inductor vanishes. This vanishing of the electric field inside the inductor
wire is the result of Ohm’s Law in field vector format
𝐽 = 𝜎𝐸⃗
(9)
so that the electric field 𝐸⃗ inside the inductor becomes
𝐸⃗ =
𝐽
,
𝜎
(10)
which goes to zero as the conductivity 𝜎 goes to infinity.
Another argument can be provided to show that the field inside a perfect conductor vanishes. A perfect
conductor cannot contain an electric field because the skin depth is zero.
Note Skin depth is 𝛿 = √2/(𝜔𝜇𝜎) , which vanishes at all angular frequencies except
for 𝜔 = 0 or DC.
Re-express Equation (2) as
∮ 𝐸⃗ ∙ 𝑑𝑙 = −
⃗ ∙ 𝑑𝑎
𝑑∯𝐵
𝑑𝜑
= −
,
𝑑𝑡
𝑑𝑡
(11)
where the flux 𝜑 is related to the loop current
𝑖 = 𝐽∙𝐴 ,
and where the directed cross-sectional area of the wire is designated by vector 𝐴 .
6
(12)
THE NATURE OF AN INDUCTOR
Under the assumption that the dimensions of our circuit are small compared to a wavelength of radiation,
the ratio of magnetic flux 𝜑 to current 𝑖 is a constant called inductance 𝐿. Thus, for this right-lying circular
loop, the right-hand side of Equation (2) becomes simply
−
⃗ ∙ 𝑑𝑎
𝑑∯𝐵
𝑑𝜑
𝑑(𝐿𝑖)
𝑑𝑖
= −
= −
= −𝐿
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
(13)
In this case, the left-hand side of Equation (2) becomes two clockwise integrals.
𝑑
𝑒
𝑑
∮ 𝐸⃗ ∙ 𝑑𝑙 = ∫ 𝐸⃗ ∙ 𝑑𝑙⃗ + ∫ 𝐸⃗ ∙ 𝑑𝑙⃗ = ∫ 𝐸⃗ ∙ 𝑑𝑙⃗ ,
𝑒
𝑑
(14)
𝑒
because the electric field within the wire of our perfect conductor loop is zero. With the assumption that the
current is increasing so that 𝑑𝑖 ⁄𝑑𝑡 > 0, Equation (13) informs us that the electric field through the voltmeter
points downward. Thus this clockwise line integral is taken against the induced electric field 𝐸⃗ . According to
our voltage definition we find that
𝑒
𝑣𝐿 ≡ 𝑣𝑒𝑑 = ∫ 𝐸⃗ ∙ 𝑑𝑙
(15)
𝑑
is therefore positively valued, as illustrated in Figure 4.
Note In Equation (15) the integration element 𝑑𝑙 is a dummy variable having no necessary
relationship to the element 𝑑𝑙 appearing in Equation (2). In fact the element 𝑑𝑙 of Equation (15)
happens to be the negative of the element 𝑑𝑙 appearing in Equation (2). Equation (2) was freely
selected as clockwise, and here the direction of the line integral is freely chosen to be
counterclockwise from point 𝑑 to point 𝑒.
Substitution of Equations (13) and (15) into (11) yields the sought inductor voltage 𝑣𝐿 indicated on a
voltmeter in Figure 4 as
𝑣𝐿 = 𝐿
𝑑𝑖
.
𝑑𝑡
(16)
Maxwell-Faraday Equation for the Conservative Left-Lying Loop
Here, Equation (2) becomes simply
∮ 𝐸⃗ ∙ 𝑑𝑙 = 0.
(17)
.
Next we determine the three remaining voltmeter readings 𝑉𝐵 , 𝑣𝑅 and 𝑣𝐶 , by examining three clockwise line
integrals.
7
THE NATURE OF AN INDUCTOR
Battery Voltage 𝑽𝑩
By definition, the voltmeter reading or voltage 𝑉𝐵 across the battery in Figure 4 is given by
𝑎
𝑉𝐵 ≡ 𝑉𝑎𝑏 = ∫ ⃗⃗⃗⃗
𝐸𝑒 ∙ 𝑑𝑙 .
(18)
𝑏
Note that a line integral through a battery vanishes (unless it has internal resistance); this is the reason a
voltmeter is placed across a battery. The “currency” of Circuit Theory consists of voltmeters and ammeters.
We obviate the need for an ammeter in this discussion, because a resistor is present in the left-hand loop.
The voltmeter 𝑣𝑅 across this resistor can also inform us if the circuit current is increasing (decreasing).
Resistor Voltage 𝑽𝑹
By definition, the voltmeter reading or voltage 𝑣𝑅 across the resistor in Figure 4 is given by
𝑐
𝑣𝑅 ≡ 𝑣𝑐𝑏 = ∫ 𝐸⃗ ∙ 𝑑𝑙 .
(19)
𝑏
Note that 𝐽 (and therefore 𝐸⃗ , as illustrated) are pointing clockwise.
This line integral can be simplified to
𝑐
𝑐
⃗⃗⃗ ∙ 𝑑𝑙 = ∫
𝑣𝑅 = ∫ 𝐸
𝑏
𝑏
𝐽
𝐽
𝐼
𝑑𝑙 = (∆𝑙) =
∆𝑙 = 𝐼𝑅 .
𝜎
𝜎
𝐴𝜎
(20)
Capacitor Voltage 𝑽𝑪
By definition, the voltmeter reading or voltage 𝑣𝑅 across the resistor in Figure 4 is given by
𝑑
𝑣𝐶 = 𝑣𝑑𝑐 = ∫ 𝐸⃗ ∙ 𝑑𝑙 .
(21)
𝑐
Consistently with the assumption that a positive charge 𝑄 sits on the left-hand plate of this capacitor, a real
voltmeter placed externally across capacitor 𝐶 “reads” the value given by
𝑑
𝑑
𝑣𝐶 = ∫ 𝐸⃗ ∙ 𝑑𝑙 = ∫
𝑐
𝑐
𝑄
∆𝑙
𝑄
1 𝑡
𝑄(0)
𝑑𝑙 = 𝑄
=
=
∫ 𝑖𝑑𝑡 +
.
𝜀𝐴
𝜀𝐴
𝐶
𝐶 0
𝐶
(22)
Circuit Fields are Very Narrow and Very Harmonious
The non-conservative inductor loop is one-dimensional (parameter) contour through includes an open loop
through an ideal conductor that is closed by voltmeter 𝑣𝐿 path.
8
THE NATURE OF AN INDUCTOR
By contrast, the conservative loop is a one-dimensional contour that includes a serial voltmeters paths
(𝑉𝐵 , 𝑣𝑅 and 𝑣𝐶 ) that are closed by voltmeter 𝑣𝐿 path.
At any instant, a test charge 𝑞 that is imagined to traverse the right-hand loop consisting of a perfect
conductor in series with a voltmeter 𝑣𝐿 acquires energy 𝑞𝜀𝐿 , because the electric field on this contour is
non-conservative.
At any instant, a test charge 𝑞 that is imagined to traverse the left-hand loop consisting of four voltmeters
acquires no energy, because this contour is conservative.
No manner of conflict exists with the intersection of these two contours, due to the fact that the voltmeter
𝑣𝐿 path is common to both loops.
Figure 4. Resulting Circuit Theory Picture
LEFT-LYING LOOP
VB
RIGHT-LYING LOOP
vR
vC
R
C
vL
i
Collecting the voltmeter indications for the left-lying “conservative” loop yields
𝑉𝐵 = 𝑣𝑅 + 𝑣𝐶 + 𝑣𝐿 ,
9
(23)
THE NATURE OF AN INDUCTOR
for time 𝑡 ≥ 0. Substitution of Equations 16, 19 and 22 into this expression yields
𝑉𝐵 = 𝑖𝑅 +
1 𝑡
𝑄(0)
𝑑𝑖
∫ 𝑖𝑑𝑡 +
+ 𝐿 .
𝐶 0
𝐶
𝑑𝑡
(24)
Differentiation of Equation (24) and division by 𝐿 yields the familiar expression
0 =
𝑑2 𝑖
𝑅 𝑑𝑖
1
+
+
𝑖 .
2
𝑑𝑡
𝐿 𝑑𝑡
𝐿𝐶
(25)
Note that each of the four circuit voltages has been determined by eliminating any unnecessary (induction)
area between the voltmeter and a particular circuit element. By eliminating unnecessary voltmeter lead loop
areas, all four voltages are identically the same as given by the “Laws” of Circuit Theory. Equation (2) teaches
us, for three of the four circuit elements (battery, resistor, and capacitor), that the electric field of
electrostatic origin is indeed conservative for the respectively indicated zero-area voltmeter “loops.” Thus, in
each of these three cases, the electric field of electrostatic origin just outside any of these three circuit
elements is identical to the electric field of electrostatic origin within that same element.
One should not connect a voltmeter to the inductor loop except as illustrated: from terminal 𝑒 to terminal 𝑑
of the inductor. The length of the leads of a voltmeter must always be minimized when connecting them to a
circuit element; otherwise, one will not get what is expected from their measurement.
It is noted that the constant battery voltage 𝑉 can alternatively be described in terms of the battery’s EMF
𝜀𝐵 = 𝑉𝐵 , because
𝑏
𝑏
⃗⃗⃗⃗𝑒 ) ∙ 𝑑𝑙 = 𝑉𝐵 .
𝜀𝐵 = ∫ ⃗⃗⃗⃗
𝐸𝑛 ∙ 𝑑𝑙 = ∫ (−𝐸
𝑎
𝑎
for an ideal battery with no internal resistance. Thus we can construct the next figure as an optional
description of this circuit:
10
(26)
THE NATURE OF AN INDUCTOR
Figure 5. Alternative Depiction of the Resulting Circuit Theory Picture

B
vR
vC
R
C
vL
i
The historical form of Kirchhoff’s Second circuit law (KVL) might have been:
The sum of the EMF’s in any closed loop is equivalent to the sum of the voltages in that loop.
This 1845 law was developed long before the vector calculus was developed by Josiah Willard Gibbs at the
close of the nineteenth century. Thus, it might be unfair to say that the (development of the) KVL is originally
based upon the fact that the line integral of the electrostatic electric field vanishes.
Proceeding clockwise, one applies Kirchhoff’s Second Law, in historical voltage format, to find
𝑛
𝑚
∑ 𝜀𝑖 = ∑ 𝑣𝑚 .
𝑖=1
(27)
𝑗=1
so that
𝜀𝐵 = 𝑣𝑅 + 𝑣𝐶 + 𝑣𝐿 .
(28)
It follows that the historical form of the KVL is obeyed perfectly!
Now what did become of the (“Back”) EMF opposing the change in current? Did we learn all that Lenz’s Law
stuff for nothing?! In the limit of the inductor being a perfect conductor, it is not possible to find this EMF
inside the large wire loop. Nonetheless, a phantom or vestige can be conjured in the following way. Assume
11
THE NATURE OF AN INDUCTOR
the conductor is slightly imperfect; i.e., assume that the conductor has a large but finite conductivity.
Consistently with this provisional assumption, an electric field can now exist within the inductor loop. This
assumed field has the identical direction (counterclockwise) as the illustrated field extending from point 𝑑 to
point 𝑒. What was intended (1835) by Heinrich Lenz has been explicitly found. This induced EMF, in turn,
induces a current that opposes the change in current 𝑑𝑖 ⁄𝑑𝑡 > 0 we earlier assumed occurred in the
clockwise direction.
Analogously to Newton’s third Law, but now expressed in terms of continuous electric fields,
a force cannot be applied to (work done on) an inductor unless the inductor reacts by
counter-posing that applied force.
Interpretation of the ideal inductor based upon Superposition
In our idealized version of this perfectly conducting inductor, we earlier stated that Ohm’s law (vector form)
informed us that the electric field inside the inductor wire is zero. As an alternative argument for the electric
field vanishing inside the wire loop, we stated that the skin depth of an ideal inductor is zero.
In earlier discussion, we also found an induced electric field pointing from node 𝑑 to 𝑒. Again see Figure 3
and Equation 15.
Although the electric field inside the inductor wire is zero, this electric field can be regarded as the net
electric field based upon two counter-posed components: a counterclockwise induced field ⃗⃗⃗⃗
𝐸𝑛 of non⃗⃗⃗⃗
⃗⃗⃗⃗
electrostatic origin and a clockwise field 𝐸𝑒 = −𝐸𝑛 of electrostatic origin. Thus, superposition yields a zero
field inside this wire loop.
⃗⃗⃗⃗𝑒 arises from surface charges induced at the pair of terminals (nodes 𝑑 and 𝑒) of the inductor.
The field 𝐸
Positive charge 𝑄𝐿 sits on node 𝑑 and an opposite charge −𝑄𝐿 sits on node 𝑒. This pair of charges have
been separated by the induced field ⃗⃗⃗⃗
𝐸𝑛 .
Notes
Again see the second Note beneath Figure 3.
An inductor, then, is like a battery having a variable magnitude. (If the circuit voltage source is a generator
with alternating polarity, then the polarity of this inductor can alternate, too.) Particularly, three electric
fields are associated with this energy converter: a pair of counter-posed fields ⃗⃗⃗⃗
𝐸𝑛 and ⃗⃗⃗⃗
𝐸𝑒 within the wire
and a third field ⃗⃗⃗⃗
𝐸𝑒 that is external to the inductor. A component of this external field is illustrated in Figure
3, pointing from node 𝑑 to node 𝑒. [If this inductor is non ideal, then the magnitude of ⃗⃗⃗⃗
𝐸𝑛 exceeds the
⃗⃗⃗⃗
magnitude of 𝐸𝑒 .]
We have just created a linguistic issue by stating the field established externally to this inductor is
electrostatic in origin. A re-read of recent sentences shows that this external (third) electric field ⃗⃗⃗⃗
𝐸𝑒
has its ultimate origin in the non-electrostatic field ⃗⃗⃗⃗
𝐸𝑛 within the inductor wire.
By means of this third electric field ⃗⃗⃗⃗
𝐸𝑒 , which lies externally to the wire of the inductor loop, the inductor
exerts its reaction forces on the rest of the electric circuit.
12
THE NATURE OF AN INDUCTOR
Just as the left-lying loop has a conservative electric field, the inductor has its own conservative electric field
⃗⃗⃗⃗
𝐸𝑒 whose closed-line integral around the inductor loop vanishes! In fact a simple, closed-line integral of all
electrostatic fields contained in both loop contours vanishes! That is, clockwise (or counter-clockwise)
integration from node 𝑎 to 𝑏 to 𝑐 to 𝑑 to 𝑒 (around the big loop) followed by return through the closed
switch to node 𝑎 vanishes.
13