LETTERS AND COMMENTS: A simpler derivation of the integral formula of electrical resistance

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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
  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: 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 HomeSearchCollectionsJournalsAboutContact usMy IOPscience  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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