The difference between voltage and potential difference

Abstract

In this paper some basic terms such as voltage and potential difference are presented. In many cases they are regarded as identical which leads to confusion with understanding of the fundamental concept of electromagnetic field. Related to this topic, some authors in their books and papers on electromagnetic theory have discussed what the voltmeter actually measures, which is resolved here in a simple way. In this paper it is shown that there is a difference between the terms voltage and potential difference depending on what is the observation point static fields or time-varying fields. Also in the transmission line model, the voltage between two points depends on the path of integration and, therefore, is ambiguous. What is commonly referred to as voltage, is transversal voltage that is a special case of voltage equal to the potential difference that is unique. Similarly, in electrical circuit analysis, branch voltages are unique and equal to difference of nodal voltages (nodal potentials).

Full text

The difference between voltage and potential
difference
Slavko Vujević, Tonći Modrić, Dino Lovrić
University of Split
Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture
Split, Croatia
vujevic@fesb.hr; tonci.modric@fesb.hr; dlovric@fesb.hr
Abstract—In this paper some basic terms such as voltage and
potential difference are presented. In many cases they are
regarded as identical which leads to confusion with
understanding of the fundamental concept of electromagnetic
field. Related to this topic, some authors in their books and
papers on electromagnetic theory have discussed what the
voltmeter actually measures, which is resolved here in a simple
way. In this paper it is shown that there is a difference between
the terms voltage and potential difference depending on what is
the observation point - static fields or time-varying fields. Also in
the transmission line model, the voltage between two points
depends on the path of integration and, therefore, is ambiguous.
What is commonly referred to as voltage, is transversal voltage
that is a special case of voltage equal to the potential difference
that is unique. Similarly, in electrical circuit analysis, branch
voltages are unique and equal to difference of nodal voltages
(nodal potentials).
Keywords—voltage; potential difference; induced electromotive
force; static fields; time-varying fields; electric field intensity
I. I
NTRODUCTION
There are different definitions of potential difference,
voltage and electromotive force in various textbooks [1] and
this leads to confusion with some basic notions. The
definitions, however, are often self-contradictory and contrary
to common usage, although they should be unambiguous and
easily understood. There must be a difference between these
terms in static fields and in time-varying fields [2-3]. In this
paper it will be shown that there is a difference between
voltage and potential difference, except in static fields, where
the two concepts are equivalent. Maxwell equations for static
electric fields describe the conservative nature of an
electrostatic field which implies that the electromotive force
for any closed curve is zero. However, in time-varying fields
there exists an induced electric field and therefore, the
electromotive force induced in the closed curve can be
expressed in terms of partial time derivative of the magnetic
flux. There are three different ways in which the partial time
derivative of the magnetic flux can be considered [4-6]. Time-
varying electric field is not a conservative field anymore and
the voltage between two points in a time-varying electric and
magnetic field depends on the choice of the integration path
between these two points.
Moreover, in numerous papers [7-9] that speak about
voltmeters readings, concepts such as voltage drop between
two points and other terms were used, other than the above
mentioned terms. The authors have discussed whether the
position of observed points or position of the voltmeter leads
affects the voltmeter readings. The conclusions obtained in
literature and used principles are not always convincing. It can
be shown that solution of this problem follows from Maxwell
equations and the most elementary properties of vector fields
and their line integrals.
In transmission line model voltage between conductors
and current in conductors depends on the position of the
observation point along the line. That voltage between two
points is ambiguous, and it is correct to call it transversal
voltage, as some authors do [10]. This will be explained on
simple examples and figures of the transmission line model.
II. S
TATIC FIELDS
Static fields are the simplest kind of fields, due to the fact
they do not change with time. Electrostatic fields are produced
by static electric charges; stationary currents are associated
with free charges moving along closed conductor circuits and
magnetostatic fields are due to motion of electric charges with
uniform velocity (direct current) or static magnetic charges
(magnetic poles). The electric field generated by a set of fixed
charges can be written as the gradient of a scalar field, known
as the electric scalar potential φ, which can be defined as the
amount of work per unit charge required to move a charge
from infinity to the given point:
ϕ−∇=E
r
(1)
where
E
r
is electric field intensity.
The negative sign in (1) shows that the direction of
E
r
is
opposite to the direction in which φ increases.
In Cartesian coordinates, (1) is equivalent to:
x
E
x
∂
ϕ
∂
−=
(2)

y
E
y
∂
ϕ
∂
−=
(3)
z
E
z
∂
ϕ
∂
−=
(4)
Equation (1) can be written inversely as:
iBA
B
A
AB
CdEu ∀ϕ−ϕ=⋅=
∫
;l
rr (5)
Equation (5) expresses the fact that a unique voltage u
AB
can be defined for any pair of points A and B independent of
the path of integration between them (Fig. 1).
Figure 1. Set of curves between two points.
The electric field generated by stationary charges is an
example of a conservative field. Equation (6) shows that the
line integral of
E
r
along any closed curve C
i
(Fig. 2) must be
zero. Physically, this implies that no work is done in moving a
charge along a closed curve in an electrostatic field.
i
C
CdE
i
∀=⋅
∫
;0l
r
r
(6)
Figure 2. Set of closed curves joined to point A.
The Stokes theorem states that the circulation of a electric
field intensity around a closed curve C is equal to the surface
integral of the curl of electric field intensity over the open
surface S bounded by closed curve C, provided that
E
r
and
curl of
E
r
are continuous on S.
Applying the Stokes theorem to (6), it can be obtained:
(
)
i
C S
CSdEdE
i i
∀=⋅×∇=⋅
∫ ∫∫
;0
r
r
l
r
r
(7)
Equation (5) can be written as:
0=×∇ E
r
(8)
It is well known that any vector field that satisfies (7) or
(8) is conservative, or irrotational. Equation (7) or (8) is
referred to as Maxwell equation for static electric fields.
Whether in integral form (7) or in differential form (8), they
depict the conservative nature of an electrostatic field.
The work done on the particle when it is taken around a
closed curve is zero, so the voltage around any contour C
i
can
be written as:
i
C
AAAA
CdEu
i
∀=ϕ−ϕ=⋅=
∫
;0l
r
r
(9)
III. T
IME
-
VARYING FIELDS
Time-varying fields can be generated by accelerated
charges or time-varying current. In the discussion of static
fields, voltage is defined to be the same as the potential
difference. Actually, the voltage between two points is defined
as the line integral of the total electric field intensity, from one
point to the other. In distinction from static fields described by
(8), in time-varying fields, the following equation is valid:
(
)
Bv
t
B
dt
Bd
Er
r
r
r
r××∇+
∂
∂
−=−=×∇ (10)
where
v
r
is relative velocity between magnetic field and
medium while
B
r
is the magnetic flux density and it can be
expressed as a curl of the other vector, ,A
r
which is known as
a magnetic vector potential:
AB
r
r
×∇= (11)
Equation (10) is one of the Maxwell equations for time-
varying fields, which states that the curl of the electric field
intensity is equal to the time rate of decrease of the magnetic
flux density. It also shows that the time varying electric field
is not conservative. The work done in taking a charge around a
closed curve in a time-varying electric field is a consequence

of the energy from the time-varying magnetic field. An
induced current in the contour would produce a magnetic field
in the opposite direction to the direction of increasing
magnetic field. Thus the induced current would reduce the rate
of change of the magnetic flux in the contour.
From (10) and (11) it is clear that the scalar electric
potential φ is now by itself insufficient to completely describe
the time-varying electric field because there is also direct
dependence on the magnetic field variations. By recalling the
definition of magnetic vector potential ,A
r
the electric field
intensity for time-varying fields is given by:
Bv
t
A
E
r
r
r
r×+
∂
∂
−ϕ∇−=
(12)
Total electric field intensity is defined by the sum of the
static part produced by charges and the induced part. In
electrostatics, the induced electric field does not exist, and
voltage does not depend on the integration path between these
points. This is not the case in a time-varying electric and
magnetic field [5, 6]:
indstat
EEE
r
r
r
+= (13)
where static part of the electric field intensity is defined by:
ϕ∇−=
stat
E
r
(14)
whereas the induced part of the electric field intensity can be
written as a sum of transformer and motional parts:
mtrind
EEE
r
r
r
+=
(15)
where transformer part is defined as:
t
A
E
tr
∂
∂
−=
r
r
(16)
and motional part is defined as:
BvE
m
r
r
r
×=
(17)
A. Closed curves
According to (13), the voltage around any contour C
i
(Fig. 2) can be written as:
∫∫∫
⋅+⋅=⋅=
iii
C
ind
C
stat
C
dEdEdEu l
r
r
l
r
r
l
r
r
(18)
In (18), integral of the static part of the electric field
intensity along any closed curve C
i
is zero:
iAA
C
stat
CdE
i
∀=ϕ−ϕ=⋅
∫
;0l
r
r
(19)
and voltage around any contour C
i
can be written as:
mtr
C
ind
C
eeedEdEu
ii
+==⋅=⋅=
∫∫
l
r
r
l
r
r
(20)
where e is the induced electromotive force, e
tr
is the
transformer electromotive force and e
m
is the motional
electromotive force. According to (15) - (17) and (20),
following equations can be written:
∫
⋅=
i
C
ind
dEe l
r
r
(21)
∫∫
⋅
∂
∂
−=⋅=
ii
CC
trtr
dA
t
dEe l
r
r
l
r
r
(22)
(
)
∫∫
⋅×=⋅=
ii
CC
mm
dBvdEe l
r
r
r
l
r
r
(23)
Therefore, for any contour C
i
,
voltage u is equal to induced
electromotive force e:
∫∫
⋅=⋅==
ii
ii
C
ind
C
C
AA
C
AA
dEdEeu l
r
r
l
r
r
(24)
where voltage and induced electromotive force depend on the
integration path.
Figure 3. Transformer electromotive force e
tr
induced in the contour C
i
.

Transformer electromotive force (22) can be expressed as
negative of partial time derivative of the magnetic flux
Φ
through the contour C
i
over the surface S
i
(Fig. 3):
t
SdB
t
dA
t
e
ii
SC
tr
∂
Φ
∂
−=⋅
∂
∂
−=⋅
∂
∂
−=
∫∫∫
r
r
l
r
r
(25)
B.
Open curves
In the case of time-varying electromagnetic field, voltage
u
AB
between any pair of points A and B (Fig. 1) can be defined
as:
∫∫∫
⋅+⋅=⋅=
B
A
ind
B
A
stat
B
A
AB
dEdEdEu l
rr
l
rr
l
rr
(26)
Integral of the static part of the electric field intensity
is
equal to the potential difference between points A and B:
iBA
B
A
stat
CdE ∀ϕ−ϕ=⋅
∫
;l
rr
(27)
which is independent of the integration path C
i
.
Integral of the induced part of the electric field intensity
between points A and B can is equal to induced electromotive
force between these points:
mABtrABAB
B
A
ind
eeedE +==⋅
∫
l
rr
(28)
where transformer electromotive force between points A and B
can be written as:
∫∫
⋅
∂
∂
−=⋅=
B
A
trtr
dA
t
dEe l
r
r
l
rr
B
A
(29)
whereas motional electromotive force between points A and B
can be written as:
( )
∫∫
⋅×=⋅=
B
A
B
A
l
rr
r
l
rr dBvdEe
mm
(30)
According to (26) - (28), the following equation is
obtained for voltage between points A and B:
ABBAAB
eu +ϕ−ϕ=
(31)
From (31) is evident that there exists a difference between
time-varying voltage and potential difference and these two
concepts are not equivalent:
BAAB
uϕ−ϕ≠
(32)
Moreover, potential difference between any two points A
and B is independent of the integration path C
i
. Otherwise,
voltage and induced electromotive force between any two
points A and B are not equal and depend on the integration
path C
i
:
BA
C
AB
C
AB
ii
eu ϕ−ϕ≠≠
(33)
IV.
A
C VOLTMETER READING
An alternating current (AC) voltmeter is device of high
impedance, which gives an indication, a deflection of a meter
needle, proportional to the current that passes through it. The
voltmeter reading was a topic of numerous papers, where
authors tried to explain what voltmeter measures. In some
examples [7-9], two identical voltmeters, both connected to
the same two points in the electrical network, didn't show
identical results. In conventional circuit analysis without time-
varying fields, one can apply Ohm law and Kirchhoff voltage
law. But, if time-harmonic electromagnetic field is present,
one must extend Ohm law and Kirchhoff voltage law with
Faraday law. The presence of a time-harmonic field produces
two different voltmeter results, even if the voltmeters are
equal and both are connected to the same nodes in the
electrical network. The authors in [7-9] concluded that the
position of the voltmeter leads affect the voltmeter readings,
which are path dependent. Therefore, the voltage between two
points in a time-harmonic electric and magnetic fields depends
on the choice of integration path between these two points.
The measured voltage depends on the rate of change of
magnetic flux through the surface defined by the voltmeter
leads and the electrical network. This effect is particularly
pronounced at high frequencies.
Figure 4. Voltmeter connected between two points of electrical network.
Time-harmonic electrical network currents and current
through the voltmeter, which is connected between points A
and B (Fig. 4), will induce a transformer electromotive force:

Φ⋅ω⋅−=ε j (34)
where
ε
is phasor of the induced electromotive force,
Φ
is
phasor of the magnetic flux through the contour formed by
voltmeter, voltmeter leads and electrical network (Fig. 4), j is
imaginary unit, whereas
ω
represents the angular frequency.
The simplest way to explain what voltmeter measures is to
use a Thevenin equivalent, which consists of Thevenin
electromotive force
T
E and Thevenin impedance
T
Z (Fig. 5).
Thevenin equivalent represents the electrical network between
points A and B.
Figure 5. Voltmeter connected to the Thevenin equivalent.
AC voltmeter reading depends on the Thevenin equivalent
parameters, induced electromotive force
ε
, voltmeter
impedance
V
Z
and internal impedance of voltmeters leads
L
Z
. Electromagnetic influence of the current through the
voltmeter on Thevenin equivalent parameters can be neglected
because current through the voltmeter is low in magnitude.
Thevenin electromotive force
,
T
E
induced electromotive
force ,
ε
magnetic flux
Φ
and current through the voltmeter
are phasors with magnitudes equal to effective values.
According to Fig. 5, current through the voltmeter can be
written as:
LVT
T
ZZZ
E
I++ ε+
= (35)
and voltage on voltmeter impedance can be written as:
V
LVT
T
VV
Z
ZZZ
E
ZIU ⋅
++ ε+
=⋅=
(36)
Voltmeter reading is equal to effective value of voltage on
voltmeter impedance:
V
LVT
T
VV
Z
ZZZ
E
UU ⋅
++ ε+
== (37)
Internal impedance of the voltmeters leads
L
Zhas
negligible value. If this impedance is ignored, voltmeter
reading can be written as:
V
VT
T
VV
Z
ZZ
E
UU ⋅
+ε+
== (38)
V.
T
RANSMISSION LINE MODEL
In the case of two-conductor transmission line model,
voltage u and current i along the line are
described by following equations [11, 12]:
t
i
LiR
x
u
∂
∂
⋅+⋅=
∂
∂
−
(39)
t
u
CuG
x
i
∂
∂
⋅+⋅=
∂
∂
−
(40)
Equations (39) and (40) follow from Maxwell equations
for the differential two-conductor transmission line segment
(Fig. 6). In time-varying electromagnetic field, voltage
between two points depends on integrating path. Voltage u
used in (39) and (40) can be called transversal voltage [10],
which is equal to potential difference between two joined
points (Fig. 6). Transversal voltage and current along the
transmission line depend on the x coordinate. Therefore, what
is referred to as transmission line voltage u is a special case of
voltage equal to the potential difference. Transversal voltage
is equal to potential difference because magnetic vector
potential ,A
r
which only has the x component, is perpendicular
to the used straight integration path.
Figure 6. Voltage and current distribution along the two-conductor
transmission line differential segment.
The first conductor with nodes 1 and 2 is the same as the
second with nodes 3 and 4. Fig. 6 shows transversal voltages
which can be written as:
∫
=ϕ−ϕ=⋅=
4
1
4114
udEu
l
rr (41)

dx
x
u
udEu
⋅
∂
∂
+=ϕ−ϕ=⋅=
∫
32
3
2
23
l
rr (42)
because:
∫ ∫
=⋅=⋅
4
1
3
2
0l
r
r
l
r
rdAdA
(43)
For closed rectangular contour C defined by nodes 1, 2, 3
and 4 (Fig. 7) the following equations can be written:
∫∫∫∫∫
⋅+⋅+⋅+⋅=⋅
1
4
4
3
3
2
2
1
l
rr
l
rr
l
rr
l
rr
l
rr
dEdEdEdEdE
C
(44)
t
i
dxLdx
t
dE
ext
ext
C
∂
∂
⋅⋅−=⋅
∂
Φ∂
−=⋅
∫
l
rr
(45)
where
ext
Φ
is external magnetic flux per-unit-length and
ext
L
is external inductance per-unit-length.
Figure 7. Longitudinal parameters of differential two-conductor transmission
line segment.
In Fig. 7,
E
x1
and
E
x2
are tangential components of the
electric field intensity along the external surface of the
conductors and they can be called longitudinal per-unit-length
voltages:
t
i
LiRE
x
∂
∂
⋅+⋅=
1int11
(46)
t
i
LiRE
x
∂
∂
⋅+⋅=
2int22
(47)
where
R
1
and
R
2
are resistances of the conductors, whereas
L
int1
and
L
int2
are internal inductances of the conductors per-
unit-length [13, 14].
The following equations for resistance per-unit-length
R
and inductance per-unit-length
L
are valid:
21
RRR +=
(48)
ext
LLLL ++=
2int1int
(49)
According to (41), (42) and Fig. 7, (44) can be written as:
( ) ( )
udxEdx
x
u
udxEdE
xx
C
−⋅+





⋅
∂
∂
++⋅=⋅
∫
21
l
rr
(50)
According to (45) – (47) and (50), following equation can
be obtained:
( ) ( )
t
i
LLLiRR
x
u
ext
∂
∂
⋅+++⋅+=
∂
∂
−
2int1int21
(51)
After substituting (48) and (49) into (51), transmission line
equation (39) can be obtained.
Applying Kirchhoff law or Maxwell continuity equation on
the differential line segment (Fig. 8), the following expression
can be obtained:
( ) ( )
t
u
dxCudxGdx
x
i
ii ∂
∂
⋅⋅+⋅⋅+





⋅
∂
∂
+=
(52)
where
G
is conductance per-unit-length, and
C
is capacitance
per-unit-length. From (52) follows the transmission line
equation (40).
Figure 8. Transversal parameters G⋅dx and C⋅dx of differential two-conductor
transmission line segment.
Single-conductor representation of the two-conductor
transmission line of length
ℓ
, with uniformly distributed per-
unit-length parameters
R
,
L
,
C
and
G
, is shown if Fig. 9. In
this case, transversal voltages
u
1
and
u
2
are equal to the
potentials
φ
1
and
φ
2
.

Figure 9. Single-conductor representation of two-conductor transmission line.
VI.
E
LECTRICAL CIRCUIT THEORY
Electrical circuit theory is not exact. It is an approximation
of electromagnetic field theory that can be obtained from
Maxwell equations [11]. Electric circuits consist of active
elements and passive elements. Active elements are current
and voltage sources. The passive circuit elements resistance
R
,
inductance
L
and capacitance
C
are defined by the manner in
which the voltage and current are related for the individual
element. In direct current, time-harmonic and transient
electrical circuit analysis, the voltage is unique [15]. One node
can be used as the reference node and all the other nodal
voltages are referenced to this common node. The voltage
across any network branch is equal to difference between
nodal voltages at the two branch ends. Nodal voltages are also
called nodal potentials [16]. Therefore, in electrical circuit
analysis, branch voltages are unique and equal to difference of
nodal potentials.
VII.
C
ONCLUSION
In this paper, some basic notions and equations about
potential difference, voltage and induced electromotive force
in electromagnetism are presented. Only in the case of static
fields, voltage is identical to the potential difference. Due to
conservative nature of static fields, voltage does not depend on
the integration path between any two points.
In the case of time-varying electromagnetic fields, voltage
and potential difference are not identical. Potential difference
between two points is unique, whereas voltage and induced
electromotive force between two points depend on the
integration path.
In the transmission line model, the time-varying voltage
between two points depends on the path of integration.
Therefore, voltage is ambiguous. What is commonly referred
to as transmission line voltage is actually a special case of
voltage equal to the potential difference. This voltage can be
called transversal voltage.
Similarly, in direct current, time-harmonic and transient
electrical circuit analysis, voltage is unique and equal to
difference of nodal voltages, which are also called nodal
potentials.
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