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The proportionality constant in Ohm's law is the electrical resistance. The resistance of conducting objects depends on their geometrical features such as cross-sectional area and length. Since the cross-sectional area of resistors usually varies

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A simpler derivation of the integral formula of electrical resistance
This content has been downloaded from IOPscience. Please scroll down to see the full text.Download details:IP Address: 202.108.50.70This content was downloaded on 07/10/2013 at 08:01Please note that terms and conditions apply.2009 Eur. J. Phys. 30 L47(http://iopscience.iop.org/0143-0807/30/4/L01)View the table of contents for this issue, or go to the journal homepage for more
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IOP P
UBLISHING
E
UROPEAN
J
OURNAL OF
P
HYSICS
Eur. J. Phys.
30
(2009) L47–L50
doi:10.1088/0143-0807/30/4/L01
LETTERS AND COMMENTS
A simpler derivation of the integralformula of electrical resistance
Miguel ´Angel Rodr´ıguez-Valverde and Mar´ıaTirado-Miranda
Grupo de F´ısica de Fluidos y Biocoloides, Departamento de F´ısica Aplicada,Facultad de Ciencias, Universidad de Granada, E-18071 Granada, SpainE-mail: marodri@ugr.es
Received 2 February 2009, in ﬁnal form 12 April 2009Published 8 May 2009Online at stacks.iop.org/EJP/30/L47
Abstract
The proportionality constant in Ohm’s law is the electrical resistance. Theresistance of conducting objects depends on their geometrical features suchas cross-sectional area and length. Since the cross-sectional area of resistorsusually varies as its length, the calculation of resistances is a non-trivial task.In this letter, a closed-form integral expression for resistance is just derivedfor resistors with parallel curved terminals. The derivation proposed here,equally applicable to the calculation of capacitances, uses the properties of an electric ﬁeld and the mathematical concept of parallel surfaces. This newteaching strategy can be useful for ﬁrst undergraduate courses of science andengineering studies.
1. Introduction
Ohm’s law [1, 2] is the main formula in electrical circuit theory. At the introductory level,
electrical resistance is macroscopically interpreted as a mechanical friction [3, 4]. However,
unlike friction coefﬁcients, the electrical resistance depends on the material (bulk properties)and the
geometry
(shape and size) [1, 2]. Thereby, geometry is a controlling factor in the
conducting wire design. But the calculation of resistances is a non-trivial task. In thispaper, the integral formula of electrical resistance [1, 5] is derived for variable cross-section
resistors with parallel curved terminals. This derivation is focused on ﬁxed resistors ratherthan potentiometers or rheostats. The current teaching strategy enables us to understand theinﬂuence of the resistor geometry on resistance with minor mathematical skills. The samestrategy can be accordingly extended to capacitors.
2. Review: formulae of electrical resistance
The electrical resistance of an ohmic resistor is calculated from its deﬁnition [1, 2] and
the mathematical problem of ﬁnding the electric ﬁeld inside the resistor, with appropriate
0143-0807/09/040047+04$30.00 c
2009 IOP Publishing Ltd Printed in the UK L47
L48 Letters and Comments
boundary conditions. However, this operation can be simpliﬁed in resistors with certaintypical geometries.Assuming a uniform current density, the well-known formula of electrical resistance[1, 2] is
R
=
ρAL,
(1)where
ρ
is the material resistivity,
L
is the resistor length measured along the current directionand
A
is the cross-sectional area. This well-studied relation proves the dependence of theelectrical resistance on the ohmic material and its geometry. However, expression (1) is just valid for linear objects with a constant cross-section area and resistivity. Instead, thecontinuous version of equation (1) reads as [1, 5]
R
=
ρ
L
0
1
A(l)
d
l,
(2)where the line integration over
l
is performed along the electrical current direction and thecross-section area can be variable,
A(l)
, even being curved. This integral expression is foundfrom the net effect of inﬁnitesimal resistors attached in series [6]. An important issue is that
within each inﬁnitesimal resistor, the current density magnitude must be uniformly spreadover its cross-section.The electric ﬁeld inside a dielectric material without free electric charges becomes verysimilar (except for constants) to the ﬁeld in a conducting material when both are linear,homogeneous and isotropic and the fringing-ﬁeld effects can be ignored (i.e. if the electricﬁeld is completely conﬁned or
A(l)
l
2
). Under these conditions, there is an analogybetween the electric displacement ﬁeld and the current density produced in capacitors andresistorsrespectively. Sincebothcapacitanceandelectricalresistancearegeometry-dependentquantities, like equation (2), an analogous equation can be proposed for capacitors with twoterminals separated by
L
:
C
=
ε
L
01
A(l)
d
l,
(3)where
ε
is the permittivity of the dielectric material and
A(l)
is the cross-sectional area ata distance
l
between the conducting plates. It is worth pointing out that the integral term inequation(3)isinthedenominatorbecausethereciprocalequivalentcapacitanceofinﬁnitesimal
capacitors attached in series is equal to the sum of the reciprocal inﬁnitesimal capacitances[6]. Further, the electrical to heat conduction analogy [7] allows us to apply equation (2) to
compute thermal resistances through conducting layers.The derivation of equation (2) is not usually found in textbooks. Instead, a brief thorough
rigorous derivation is presented here, without any inﬁnitesimal partitioning, using just themathematical condition of the resistor with parallel curved terminals.
3. Electrical resistance of objects with parallel curved terminals
Weassumeconstant-currentresistorswithparallelcurvedends,suchasforinstanceasphericalshell(seeﬁgure1). Asurfaceparalleltoanarbitrarysurfaceisformedbytranslatingtheinitialsurface along its normals by an equal distance everywhere on the surface [8]. Each terminal of
the resistor is an equipotential surface and further, inasmuch as both ends are parallel surfaces,an arbitrary cross section also becomes an equipotential surface. Thus, within a resistor withparallel curved terminals, the electric ﬁeld,
E
, follows the direction of the terminal normals.Since the parallel surfaces are equidistant by deﬁnition, the module of the electric ﬁeld will be
Letters and Comments L49
Figure 1.
Resistor with parallel sphere-based ends. The surface curvature is deliberatelyexaggerated for illustrative purposes.
(This ﬁgure is in colour only in the electronic version)
constant over any cross section of the resistor (but not along it). These remarks equally applyto the current density,
J
, for an isotropic ohmic resistor although with variable resistivity.The scenario described above leads us to write the current
I
passing through a resistor as
I
=
J(l)A(l).
(4)Moreover, from the Ohm law,
E(l)
=
ρ(l)
J(l)
, and equation (4), the voltage across the
terminals, is expressed as
V
1
−
V
2
=
I
l
2
l
1
ρ(l)A(l)
d
l,
(5)where
l
1
and
l
2
are the arc-length parameters of each terminal measured through its surfacenormals
(l
2
> l
1
)
. By substituting equation (5) into the deﬁnition of electrical resistance, wecome to
R
V
1
−
V
2
I
=
l
2
l
1
ρ(l)A(l)
d
l.
(6)Iftheresistorterminalswerenotparallel,thentheelectricalresistanceshouldbecomputedsolving the electric ﬁeld inside the resistor. This happens with truncated cones [5] andright-angled wedges. However, in some textbooks of ﬁrst undergraduate courses [1], these
geometries are incorrectly used for illustrating the calculation of resistances with equation (6).
Only Grifﬁths highlights this misunderstanding on p 334 of his great textbook [9].The integration of equation (6) requires certain tricky geometric resources which canbecome misleading for undergraduate students. Instead, the integral can actually be expressedin a closed form. For parallel surfaces separated by an unsigned distance
l
−
l
1
, differentialgeometry [8] provides the following result:
A(l)
=
A(l
1
)(
1 +
κ
↑
(l
−
l
1
))(
1 +
κ
→
(l
−
l
1
)),
(7)where
κ
↑
and
κ
→
are the signed principal curvatures of the surface located at
l
=
l
1
, whosearea is
A(l
1
)
. Although curvature is a point-to-point measurement,
κ
↑
and
κ
→
were assumedconstant in equation (7). A positive principal curvature indicates that the surface is concave at
the principal direction concerned, as illustrated in ﬁgure 2. If equation (7) is substituted into
equation (6), the electrical resistance is reduced to
R
=
1
A(l
1
)
l
2
l
1
ρ(l)(
1 +
κ
↑
(l
−
l
1
))(
1 +
κ
→
(l
−
l
1
))
d
l.
(8)If the resistor were homogeneous
(ρ
=
const
)
, the analytical primitive function of thisintegral will just depend on the speciﬁc signed values of
κ
↑
and
κ
→
, i.e. the geometry of
L50 Letters and Comments
(a) (b) (c)(d) (e) (f)
Figure 2.
Distinct geometries obtained as the values of curvature: (a)
κ
↑
=
κ
→
=
0, (b)
κ
↑
>
0
,κ
→
=
0, (c)
κ
↑
<
0
,κ
→
=
0, (d)
κ
↑
=
κ
→
>
0, (e)
κ
↑
=
κ
→
<
0 and (f)
κ
↑
=−
κ
→
>
0.
Table 1.
Dimensionless electrical resistance of homogeneous resistors with parallel curved endswith different curvatures. The terminals are placed at
l
1
and
l
2
(>l
1
)
, respectively.Terminal
κ
↑
κ
→
A(l
1
)(l
2
−
l
1
)ρ
R
Plane 0 0 1Spherical
κ
0
κ
011+
κ
0
(l
2
−
l
1
)
Cylindrical
κ
0
0
1
κ
0
(l
2
−
l
1
)
ln
(
1 +
κ
0
(l
2
−
l
1
))
Minimal
κ
0
−
κ
012
κ
0
(l
2
−
l
1
)
ln
1+
κ
0
(l
2
−
l
1
)
1
−
κ
0
(l
2
−
l
1
)
the resistor terminals: plane, sphere, cylinder or minimal surface (see ﬁgure 2 and table 1).
As expected, equation (1) is straightforwardly derived from equation (8) for plane ends (i.e.
axial-lead resistor). The results collected in table 1 are equally applicable to the reciprocaldimensionless capacitance,
εA(l
1
)(l
2
−
l
1
)C
, of capacitors with parallel curved ends, provided that thefringing-ﬁeld effects were neglected.
Acknowledgments
This work was supported by the ‘Ministerio Espa˜nol de Educaci´on y Ciencia’ (contract‘Ram´on y Cajal’ RYC-2005-000983), the European Social Fund (ESF) and the ‘Junta deAndalucia’ (project FQM-02517). The authors would like to thank the referee for his
/
heruseful comments.
References
[1] Serway R A and Jewett J W Jr 2006
Physics for Scientists and Engineers
6th edn (Philadelphia: Saunders)[2] Tipler P A and Mosca G 2004
Physics for Scientists and Engineers
5th edn (New York: Freeman)[3] Johnstone A H and Mughol A R 1978
Phys. Educ.
13
46–9[4] Viard J and Khantine-Langlois F 2001
Sci. Educ.
10
267–86[5] Romano J D and Price R H 1996
Am. J. Phys.
64
1150–3[6] Efthimiou C J and Llewellyn R A 2005
Phys. Teacher
43
366–70[7] Efthimiou C J and Llewellyn R A 2005
Eur. J. Phys.
26
441–56[8] Jinnai H, Koga T, Nishikawa Y, Hashimoto T and Hyde S T 1997
Phys. Rev. Lett .
78
2248–51[9] Grifﬁths D J 1999
Introduction to Electrodynamics
3rd edn (Englewood Cliffs, NJ: Prentice-Hall)

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