Revista Brasileira de Ensino de Fısica, vol. 38, nº 3, e3315 (2016)www.scielo.br/rbefDOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2016-0065

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The problem of the electrical discharge in an internallywired capacitor

O problema da descarga eletrica um capacitor internamente conectado

F. B. Rizzato∗

Instituto de Fısica, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil

Recebido em 18 de marco de 2016. Revisado em 26 de abril de 2016. Aceito em 27 de abril de 2016

The present work investigates the dynamics of electrical discharges in a capacitor where both platesare internally connected by a conducting wire. A simplifying assumption used to solve the problem isto consider a uniform distribution of the surface charges on the plates. This implies uniform electricfields as well, and allows to calculate the current in the wire as if it were flowing along a simple RCcircuit. We show that although the assumption is good if the conductivity of the wire is small, it fails ifthe conductivity becomes larger than a critical value. The device’s behavior beyond this critical valueis then analyzed and discussed. The relevance of the problem is discussed in connection to solid-statedevices, as well as in problem solving contexts.Keywords: Electrical discharges, Vacuum and solid-state electronics, Electromagnetic waves.

O presente trabalho investiga a dinamica de descargas eletricas em um capacitor onde ambas as placasestao internamente conectadas por um fio condutor. Uma suposicao simplificadora usada para resolver oproblema e a de considerar uma distribuicao uniforme de cargas superficiais nas placas. Isto implica emcampos eletricos tambem uniformes, e permite calcular a corrente no fio como se estivesse fluindo aolongo de um circuito RC simples. Mostramos que embora a suposicao seja boa se a condutividade do fioe pequena, falha se a condutividade ultrapassa um valor crıtico. O comportamento do dispositivo alemdeste valor crıtico e entao analisado e discutido. A relevancia do problema e discutida em conexao adispositivos de estado solido, bem como em contextos relacionados a resolucao de problemas.Palavras-chave: Descargas eletricas, Eletronica de vacuo e de estado solido, Ondas eletromagneticas.

1. Introduction

One problem occasionally discussed in electromag-netism classes is that of the electrical dischargein an internally wired capacitor [1–3]. That is, acapacitor whose plates are connected directly oneto another by a resistive wire, and not indirectlyconnected across an external electronic circuit asis the usual situation. The setting is not only ofrelevance in problem solving contexts, but also insolid-state device electronics, where leakage currentsmay flow from one plate to the other along faultyinner channels of solid-state planar capacitors [4].

∗Endereco de correspondencia: rizzato@if.ufrgs.br.

The discharge dynamics is of peculiar interest be-cause it involves simultaneous fluxes of electric anddisplacement field currents inside the capacitor, asopposed to the usual model of isolated plates whichadmits only displacement currents if the circuit isset to operate under nonstationary regimes [5]. Thesimultaneous presence of electric and displacementcurrents in the same volume provides one convenientsetting to discuss the Ampere’s law, in its form cor-rected by Maxwell, since the total current flowinginside the capacitor must satisfy continuity, the verycondition that motivated Maxwell’s analysis. Fur-thermore, adequate manipulation of electric anddisplacement currents allows the evaluation of themagnetic field inside the capacitor, which is often

Copyright by Sociedade Brasileira de Fısica. Printed in Brazil.

e3315-2 The problem of the electrical discharge in an internally wired capacitor

the primary question posed in theoretical discus-sions and which will be used in the present analysisto pose proper boundary conditions.

The usual approach to accomplish the task ofcalculating the fields inside the capacitor is to as-sume circular geometry, uniformly distributed sur-face charges over both plates at all times, and aresistive wire connecting both geometric centers ofthe plates. Under the condition of uniform electricfields, which is critical for reliability of solid-statedevices, and with help of the capacitive relationshipbetween electric field and surface charges, one cancalculate: (i) the current flowing along the wire, (ii)the resulting charge depletion over the plates alongwith the associated variation of the electric field and,ultimately, (iii) the magnetic field as obtained fromthe Ampere’s law [1,2].

The issue with this kind of approach is preciselythe assumptions on uniformity. As pointed out byFeynman, in an analysis excluding the wire and theassociated electric current, if there is dynamical fluxof magnetic field, there must be space dependenceof the electric field because Faraday-Lenz’s law de-mands so [6, 7]. In other words, the assumptions ofa uniform electric field must be examined with care.

Feynman goes on and obtains a solution for theunwired case in terms of Bessel’s functions [8]. Hissolution is fully self-consistent, admits boundaryconditions, and is in fact what motivates the presentanalysis. More to the point, we wish to examinewhat happens with a discharge in the internallywired capacitor when we allow for space-time effects.In particular, we would like to examine how good isthe usual assumption of a thorough uniform electricfield inside the device during the discharge.

The paper is organized as follows: in §2 we intro-duce the model to be studied; in §3 an analyticaland numerical investigation is carried out; and in§4 we draw our final conclusions.

2. The model

The model we shall employ, as represented in Fig.1, is almost self-explanatory. Two perfectly conduct-ing circular plates, both of radius a, are connectedthrough a cylindrical solid structure of radius b < aand length L. The cylindrical structure is a materialof uniform conductivity g which is defined in termsof the electric current density j and electric fieldE as j = gE. The conducting region is somewhat

Figure 1: Relevant ingredients and geometry of the inter-nally wired capacitor. Both plates have radius a and thewire has radius b. The role of surface S bounded by curveC is discussed in the text.

equivalent to lossy dielectric regions in electronicdevices [9].

The evolution law of the electromagnetic field inthe volume between the plates follows from the timedependent Maxwell’s equations

∂B∂t

= −∇×E, (1)

∇×B = µ0j + 1c2∂E∂t, (2)

with B as the magnetic field, µ0 as the vacuumpermeability, and c as the speed of light. The pair ofequations (1) and (2) can be combined into a waveequation with a current term

∇2E− 1c2∂2E∂t2− µ0

∂j∂t

= 0, (3)

if one uses ∇·B = 0 and assumes no charge build-upbetween the capacitor plates so that ∇ · E = 0 aswell.

In the following we use the small gap conditionL� a, as roughly suggested by the aspect ratio ofFig. 1. Under this circumstance we neglect fringeeffects and also consider the case of uniform fieldsalong the axis, noting that in this case the lengthL has no functional bearing on the analysis to beperformed. However, we allow for full azimuthallysymmetric dependence on the radial coordinate rof a cylindrical set of coordinates, which preciselyembodies the kind of nonuniformity to be studiedin this work.

Taking the polarization of the fields into account,the electric and magnetic fields are repectively rep-resented by

E(r, t) = E(r, t) z (4)

Revista Brasileira de Ensino de Fısica, vol. 38, nº 3, e3315, 2016 DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2016-0065

Rizzato e3315-3

andB(r, t) = B(r, t) θ, (5)

with the hatted quantities as the polarization versorsfor the respective fields. The underlying adopted ge-ometry is consistent with the divergence-less electricfield and promptly leads to the following equationfor the scalar field E(r, t) in the form

1r

∂

∂r

(r∂E(r, t)∂r

)− ∂2E(r, t)

∂t2

−H(b− r)g∂E(r, t)∂t

= 0. (6)

In Eq. (6), the Heaviside step function H(x ≥ 0) =1, H(x < 0) = 0 controls the presence of the electriccurrent between the plates; current can only flowin the region r < b. In addition, space, time andthe conductivity have all been made respectivelydimensionless with r/a→ r, ct/a→ t, and µ0gca→g.

3. Solutions for the wave equation

3.1. Analytical approach

We look for solutions of the wave equation (6) in anormal mode separable form [10],

E(r, t) = E0e−iωtψω(r). (7)

The normal mode frequency ω and the spatial eigen-function ψω should be sought-for with help of thedifferential equation and its boundary conditionswhich we now discuss. We start off with the bound-ary conditions, and analyze the differential equationand the appropriate continuity conditions immedi-ately afterwards.

Solutions must be finite at any radial position,particularly at the origin, so

ψω(r → 0) <∞. (8)

Furthermore, we recall that fringe effects are ab-sent and avoid the inherent complications of anopen boundary problem assuming that most of theelectric field lines reside between both plates. Un-der this approximation, the total current (electricplus displacement) flux crossing the open surface Sbounded by circular curve C of radius r = 1 in Fig.(1) vanishes. Likewise, the magnetic field vanishesat r = 1 since its line integral along C equals the

total current flux across S from Ampere’s law. Fromexpressions (1) and (4) the magnetic field is seento be proportional to the simple derivative of theelectric field E(r, t). Therefore, from expression (7)our boundary condition at r = 1 reads

∂ψω

∂r

∣∣∣∣r=1

= 0. (9)

Let us then resolve the temporal part of Eq. (6)and write down the differential equation in its spatialform both in the conducting and vacuum regions.One has

1r

∂

∂r

(r∂ψω

∂r

)+

κ2

in ψω = 0 for r ≤ b,with κ2

in ≡ ω2 + igω,

κ2out ψω = 0 for r > b,

with κ2out ≡ ω2,

where we recall that both magnetic and electric fieldsare continuous at r = b. In other words, both ψω andits space derivative must be continuous across thewire radius. We note that the vacuum wave vectorκout coincides with the frequency ω.

Both equations contained in the above set (10) arein the form of azimuthally symmetric Bessel’s differ-ential equations, whose solutions can be formed byappropriate linear combinations of Bessel’s functionsof first and second kind, J0 and Y0 respectively [11].

In the conduction region, where r can go to zero,one must pick the J0 function, which is finite at theorigin. In the vacuum the complete solution mustbe formed as a linear combination of the J0 and Y0functions. One thus has

ψω(r) ={J0(κinr) if r ≤ b,α J0(κoutr) + β Y0(κoutr) if r > b,

(10)where ψω is conveniently scaled that ψω(r = 0) = 1,and where we see that the Bessel’s functions dependon complex arguments.

The boundary condition (9) along with the deriva-tive properties of the Bessel’s functions, J ′0(x) =−J1(x); Y ′0(x) = −Y1(x) [8], yields

β = −α J1(κout)Y1(κout)

, (11)

and combination of relations (10) and (11) with thecontinuity of function ψω and its derivative at r = bfinally generates relation

DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2016-0065 Revista Brasileira de Ensino de Fısica, vol. 38, nº 3, e3315, 2016

e3315-4 The problem of the electrical discharge in an internally wired capacitor

J0(κin b)J1(κin b)

= κin

κout

Y1(κout)J0(κout b)− J1(κout)Y0(κout b)Y1(κout)J1(κout b)− J1(κout)Y1(κout b)

, (12)

from which one can determine the spectrum of allowed values of ω.

3.2. Numerical analysis of the frequencyspectrum

We have two free parameters in the dimensionlesssystem: the conductivity g and the radius b of theinner, conductive region. In order to emulate a thinconducting wire, we shall take b = 0.1 unless other-wise stated. Given the fixed radius b, we shall thenexamine the imaginary and real parts of the complexfrequency ω = ω(g) as functions of g.

If we took a uniform medium, characterized byg = 0, solutions would be given in the form of singleBessel functions of the first kind J0(κnr) throughoutthe entire volume of the capacitor, with κn as one ofthe roots of expression J ′0(κnr)|r=1 = −κnJ1(κn) =0. One particular solution would be the one with nospatial dependence, κ0 = 0. The dispersion relationω = κ for this g = 0 case would imply a vanishingfrequency ω as well, which places this particular set-ting in a fully stationary regime where the capacitorplates remain uniformly and invariantly charged.

We may start our numerical investigation at thispoint, since our central question is precisely on howfar does the discharge remain approximately homo-geneous as the conductivity increases. Given theinitial root κ0 = ω(g = 0) = 0 we will iterativelycalculate the corrected frequency ω = ω(g) as gdeparts from zero. We shall also extend the tech-nique to the adjacent smallest nontrivial real rootκn=1 > 0 to have a more panoramic view of thesystem behaviour even when the zero conductivitycase is neither homogeneous nor stationary.

Figure 2 reveals the behaviour of the imaginaryand real parts of the complex frequency ω(g) asfunctions of g (only the root with positive real partis depicted in each panel).

As mentioned, panel (a) is the one of central inter-est in our discussion, since it describes the frequencyas it rises from the fully stationary and uniform caseω(g = 0) = 0. One indeed sees that at relativelysmall values of g the real part of the frequency staysglued the zero frequency axis, while its modularimaginary part rises very slowly away from the axis.

Figure 2: Real and imaginary parts of the eigen-frequenciesof the discharge modes. In panel (a) with start from ω(g =0) = 0 and in panel (b) from the lowest nontrivial rootof J1(κ) = 0, which is approximately ω(g = 0) = 3.831.The vertical line in (a) locates the respective gtransition asestimated by Eq. (13).

This kind of behaviour goes on until a sharply de-fined point where the real part starts to increasewhile the modular imaginary part starts to decrease.This sharply defined point may appear unexpectedat a first look, but a closer examination shows thatit must exist. Indeed, when g lies within its range ofsmall values g � 1, there is little difference betweenthe conducting region r ≤ b and vacuum regionr > b [12–14]. The electric field is thus mostly uni-

Revista Brasileira de Ensino de Fısica, vol. 38, nº 3, e3315, 2016 DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2016-0065

Rizzato e3315-5

form and slowly varying, and the current basicallygrows as the conductivity increases. The decay timescale for the electrical discharge acquires a large butfinite value and the modular imaginary part of thenegative imaginary frequency, which can be thoughtas the inverse of the discharge time scale, increasesslightly. But this kind of behaviour cannot go on in-definitely. If g becomes too large, skin depth effectsexpel the electric field from the conducting region,dissipation ceases [15], and the system re-morphsinto a resonant cavity which extends from r = b tor = 1 and supports a real resonant frequency. Thisis why, as seen in panel (a), the real part of thefrequency saturates at its terminal value and theimaginary part approaches a final vanishing value.

Estimates can be even obtained for the transitionpoint. First of all one should notice that for smallconductivities, the time scale of the electrical dis-charge can be evaluated as the characteristic timeτ of a RC circuit composed by the capacitor andthe resistive wire connecting the plates: one has|ω| ≡ 1/τ ∼ 1/RC ∼ gb2 with R and C respectivelyas the resistance and capacitance of the RC circuit.Secondly, one can estimate the already mentionedskin depth characteristic length δs associated withthe penetration of the electric field lines into theconducting portion of the circuit: δs ∼ 1/

√g|ω| [10].

For very small values of either g or |ω|, the skindepth is large and the field freely diffuses into thewire. For large values of the combination g |ω|, onthe other hand, δs diminishes to a point where it be-comes smaller than b. This is where the field beginsto be expelled from the wire as commented before.The critical g can be estimated as one takes δs ∼ band makes use of |ω| ≡ 1/τ above. One obtains

gtransition ∼ 1/b2, (13)which well approximates the transition in the caseb = 0.1 seen in panel (a) of Fig. 2 - gtransition isindicated by the vertical black line. We have furthertested expression (13) for b = 0.01 and once againnice agreement is obtained. We point out that thetransition seen in Fig. 1(a) is similar to the one seenin RLC circuits, whenever a dominant RC modejumps to an LC dominant mode.

To make a comparison with an inherently nonuni-form case, in panel (b) we examine the behaviourevolving from the first nontrivial oscillatory modeassociated with the g = 0 case. More precisely, wepick κ1 with J1(κ1) = 0 and analyse what happensas g increases.

Figure 3: Space-time history for the eigenmodes of theinternally wired capacitor: g = 0.01 in panel (a) and g = 104

in panel (b). For small conductivities panel (a) shows thatthe discharge follows the decay pattern of a simple RCcircuit, while for large conductivities panel (b) shows thatthe system behaves as a weakly dissipative resonant cavity.

The result is depicted in the curves of panel (b)which essentially shows the same kind of profile thanpanel (a), with the exception of the expected offsetassociated with the nonzero frequency at g = 0;ω(g = 0) = 3.831.

To complement the present analysis, Fig. 3 of-fers two panels displaying the space-time history ofthe real part of the electric field in two conductiveregimes: small conductivity g = 0.01 in (a) and largeconductivity g = 104 in (b), both roots arising fromthe κ0 = 0 case.

While the first panel, panel (a), shows the nearlyuniform decay of the electric field towards its vanish-ing final value, the second panel, panel (b), revealswhat is best described as a very slow decay of aneigenmode of a high quality resonant cavity affectedby a truly small dissipation rate. In particular, wepoint out how much smaller is the normalized fieldE/E0 inside the conducting wire, within r < b, ascompared with the field without. We also point outthat since the electric field is flat inside the wire,the associated magnetic field appears for the firsttime only at its surface.

DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2016-0065 Revista Brasileira de Ensino de Fısica, vol. 38, nº 3, e3315, 2016

e3315-6 The problem of the electrical discharge in an internally wired capacitor

4. Conclusions

In the present work we performed a fully self-consistentelectromagnetic analysis of a device used to give stu-dents a firm grasp on the role of displacement cur-rents in Maxwell’s theory. The model is built-up onthe concept of an internally wired capacitor wherethe plates are directly connected one to another asseen in Fig. 1.

The model is also suggested to be of relevanceto leaky solid-state devices. Reference [4] indeedshows that conductive regions are likely to appearin the central regions of capacitive structures, whereinsulation between the plates is susceptible to struc-tural defects. We note that under the extremes ofan avalanche breakdown one would have a highlyconductive path between the plates.

Given the azimuthally symmetric geometry of theproblem, the task is to calculate the electromagneticfields generated at an arbitrary radius inside thecapacitor. To do that along an initially pedagogicalway, one first uses the simplifying hypothesis ofa fully uniform electric field between the plates.The current is calculated as one takes the wiredcapacitor as a simple RC circuit, and the magneticfield is obtained with help of Ampere’s law, withthe symmetric source provided by the electric anddisplacement currents.

The present work is concerned with the validityof this approach. We examine in more detail howfar can we take the uniformity of the electric fieldas a good approximation. The conclusion is thatthe approach is accurate while the conductivity gis small enough that the electric field can diffusefreely into the conducting wire. However, as theconductivity increases and electric field lines areexpelled from the conductor, the system becomesincreasingly inhomogeneous. This is demonstratedby panel (a) of Fig. 2 which shows the clear birthof an inhomogeneous and oscillatory mode as onegoes past the transition value for g. A brief analysisof inherently nonuniform modes is also performed.In a way, as the conductivity grows what is seenis a transition between a dominant RC mode to adominant LC mode of a RLC series circuit.

The present work does not look at certain features,as for instance nonuniformities along the capacitoraxis, and finite conductivity of the capacitor plates.Despite these limitations, our model provides a wayto determine the range of values for g where thedischarge is uniform across the axis, and what can

be expected when one goes beyond the transitionpoint.

Acknowledgments

This work is supported by CNPq and FAPERGS,Brazil, and by the Air Force Office of Scientific Re-search (AFOSR), USA, under the grant FA9550-12-1-0438. The author thanks fruitful discussions withRenato Pakter and the suggestions of an anonymousreviewer.

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Revista Brasileira de Ensino de Fısica, vol. 38, nº 3, e3315, 2016 DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2016-0065