Tensile stresses in electrically charged wires and electrochemical electrodes

Elwrochimica

Acta.

1960, Vol. 3, pp. 169 to 174.

Pergamon

Prcsr

Ltd.

printed

in Northern

Ireland

TENSILE STRESSES IN ELECTRICALLY CHARGED WIRES AND ELECTROCHEMICAL ELECTRODES* P. J. GELLINGS We&spoor

N.V., Amsterdam,

Holland

Ah&act--In electrically charged bodies, such as e.g. metallic electrochemical electrodes, a state of stress exists because of the mutual repulsion of the charges present in such a body. These stresses are studied for the case of bodies in the form of rotational ellipsoids and an expression is given for the highest stress present therein. Application of this expression to the case of thin wires shows that these stresses and the thermodynamic and electrochemical effects thereof are so small that they can justifiably be neglected. R&mn&--Dans lea corps electriquement charges, tels que, par exemple, les tlectrodes electrochimiques metalliques, il existc un &at de tension par suite de la repulsion mutuelle entre lcs charges presentes dans de tels corps. Ces tensions sont ttudi&s pour le cas de corps en forme d’ellipsoides de revolution et une expression eat don& pour la tension la plus Qlevee presente dans de tels cas. L’application de cette expression au cas de fils minces montre qui ces tensions et leurs effets Clectrochimiques sont sufhsamment petites pour qu’il soit permis de les ntgliger. Zusammenfassung-In elektrisch geladenen Kiirpern, wie z.B. in elektrochemischen Metall-Elektroden, sind infolge der gegenseitigen Abstossung der Ladungen mechanische Spannungen vorhanden. Diese Spannungen werden fiir den Fall von rotationsellipsoiden Kiirpern studiert und es wird ein Ausdruck ftir die hiichste vorkommende Spannung angegeben. Seine Anwendung ftir den Fall von dtlnnen Drahten zeigt, dass die vorkommenden mechanischen Spannungen und die daraus resultierenden elektrochemischen und thermodynamischen Effekte so klein sind, dass ihre Vernachllssigung gerechtfertigt ist. INTRODUCTION

any electrically charged body there must exist a state of mechanical.stress because of the mutual repulsion of the charges present in the body. This is at once obvious if we imagine the body to be cut into two parts without disturbing the distribution of the charges. It is clear that the two parts will repel each other and that the tensile stress acting in the plane of the imaginary cut is the repulsive force between the parts divided by the surface area of that plane. Particularly in wires of small cross-sectional area this stress might be appreciable, as was remarked by, Floodl. In this paper an expression for this stress in terms of the charge and the dimensions of the body will be derived. Also, the effect of this stress on the thermodynamical state will be considered, mainly in view of the influence on electrode potentials. As the field quantities of cylindrical wires involve elliptic integrals we shall approximate the wire by a rotational, prolate ellipsoid. IN

THE ELECTRIC FIELD OF A WIREPo*.’ We shall start by considering a finite straight line of length 2c and uniform charge density Q (see Fig. 1). This gives for the potential 4 in the point P

* Manuscript

received 19 November

1959. 169

llA-_(Q

PP.)

P. J. GBLLINOS

170

Fro.

1. Straight line of length 2c with uniformly distribute4i charge g.

with E = dielectric constant of the me4lium surrounding the wire. The integratkm can at once be effected, leading to + _

Q

6 + 4 +

In

2cs

r1

(x - c)+ r, ’

(2)

with rls = (x + c)s + pa and rsa = (x - c)a + ps. In view of the following it iS convenient to introduce the elliptical co-ordinates U = +@I + rs)

fJ= a(% - @,

which gives r,=u+v

XC-

r,=Z4-U

uv C

p = f [(us - cqcs - ua)]?

The surfaces u = constant are confocal rotational (prolate) ellipsoids Mth the fixed foci F’(-c, 0) and Fs(+c, 0). Substituting in (2) gives d=

2Elnz

.

(3)

Thus the potential 4 is independent of v and the equipotential surfaces are the ellip soids u = constant. Since any equipotentia~ surface can be taken as the surface of a conductor this expression also gives the potential at outside points of a prolate ellipsoidal conductor with the same total charge Q and constant potential #a on the conductor surface u = a, (4) where 2u = major axis of the ellipsoid. For the field strength we have E = E, . e, + E,e, = -grad 4, (5) where e,, and e, are the orthogonal unit vectors for the co-ordinates u and v. In these co-ordinates we have

Tensile S~~BSW in elect&ally charged wiresand electrochemicalelectrodes

171

which leads to

2

E,=

&

[(u* -

u&-

E, = 0.

cs)p

The field lines are given by the system of hyperbolae confocal with the ellipsoids and orthogonal to them. If u> c then we have, because of 0 < u s c,

Q

E-u-

eu2

W

’

which is the field of a point charge Q in the origin. STRESSES IN AN ELECTRIC FIELD’*’ As was shown by Maxwell, the stresses in an electric field can be obtained as a system of surface forces. To obtain these the system, which is in electrostatic equilibrium, is divided in two parts: the enclosed part, 1 and the remainder 2. The total force exerted by 2 on 1 can then be expressed as F=

1Tdf

ss

(7)

where T is the force per cm* acting on the surface of 1.

2

surface

I

FIO.2. Force on surfaceelement. It can be shown that the field strength E bisects the angle between the normal II on the plane and the surface force T and that with II, the unit vector in the direction of the normal T=-$E.nJE-$E%..

(8)

If E and II coincide we have a pure tension on the plane, T=+;Cn,.

@a)

If E lies in the plane a pure pressure on that plane results, T=

-

f

EBn,.

(W

For any direction of E one obtains for the absolute value of the surface force ITI = $E2.

(f-w

P. J. GELLINCW

172 STRESS

IN A PROLATE

ELLIPSOID

From symmetry considerations it is clear that the highest stress will act in the equatorial plane (x = 0; D = 0) of the ellipsoid and we shall now derive an expression for this stress. To be able to apply the considerations of section 3 we add to this plane a half sphere with centrum in the origin and radius R and take the limit for .R-+ co. Equations (6a) and (8) give on the surface of the sphere

T= f E2 = Q2/8mR4. The surface element is in spherical co-ordinates df = R2 sin 8 de dy. Thus sin 8 d0 dq -+ 0

asR+oo.

This shows that the contribution of the half sphere to the force ,and we need only consider the contribution of the surface forces In this plane we have, after introducing the major axis 2a and the ellipsoid, Q Q E, = Eu(uz - cz)l/z = u 2 a, &&a” + cy E, = 0

u < a,

vanishes as R -+ m on the plane x = 0. minor axis 2b of the p>b p
(10) I.

As E, lies in the plane x = 0 T is a pressure on the plane, leading to a tension in the ellipsoid. As surface element we choose a ring of radius p and width dp, so that df = 27rp dp = nd(p2>. Further, so that we obtain

ITI = F=

Q2 8?rep2(p2+ c”)

7~4p2) “e2. s z, 8~r.s p2(p2+ 3) ’

which leads to P=E21n-

b2 + c2 Q2 ln c =8sc2 b2 ’ b2

(11)

The cross-sectional area of the ellipsoid in the plane x = 0 is nb2, which gives for the tensile stress a, ‘F Q2 h a2 (12) a,= -=8mb2c2 b2 ’ rb2 APPLICATION

TO THIN

WIRES

We shall now apply the above to thin wires wherein, because of the small crosssectional area, the expected stresses are highest. We shall approximate these wires of length 1 and thickness t < 1 by thin and long prolate ellipsoids with 2a = I 2.b = t,

Tensile stresses in electrically charged wires and electrochemical electrodes

173

and we may write c = (a9 - @l/2 = +!JleE~(l-&J, b2

a+c32a

a-c=2aa

We shall first calculate the magnitude of the charge Q on a wire with a = 5 cm, .b = 5 x 1O-3cm, which has a potential of 10 V with respect to a similar, but uncharged, wire. Then we obtain from equation (4)

which leads to Q = 2.2 x 10d2 E abs. e.s.u. Substituting this in (12) we obtain for the stress of a wire with this charge o, = 0.425 E dyn cm-2 Thus in air (E N 1) a, = O-425 dyn cm-2 and in water (E ‘u 80) a, = 34 dyn cm-2. As the externally applied mechanical stresses, for example in the study of stress corrosion or of mechanical properties, are usually of the order of at least lo6 dyn cm-2 we see that the internal stress due to the wire being charged can be completely neglected. The highest stress in a wire, of the same dimensions as the charged wire considered here, hanging vertically in the earth’s gravitational field without other forces than those of gravity is of the order of 105 dyn cm-2 depending upon the density of the material of the wire. THERMODYNAMIC

CONSIDERATIONS

In treatises on the theory of elasticity5 it is shown that the change in internal free energy density between the uniaxially, isothermally stressed and the unstressed state is AF=

&, m

where Em = Young’s modulus of the material of the wire. With Em = 1G2 dyn cm-2 as a representative value of Young’s modulus for a metal, we obtain with the stresses calculated in section 5 AF = 0.18~~ x lo-l2 erg cm-3, and thus in water AF = 1.15 x 1O-s erg cm-3. In this calculation we have assumed 0 to be constant over the length of the wire and the estimate of AF is thus certainly too high. If we assume the wire to be acting as an electrochemical electrode this change in internal energy leads to a change in electrode potential of the order of 1O-m V. It is clear that the thermodynamic consequences of the stresses due to the repulsion of the charges in the wire, can be justifiably neglected.

174

P. J. GEL~INOS CONCLUSION

Both the stresses due to the mutual repulsion of the charges in a charged body, and the thermodynamical changes in the system caused by these stresses, are so small that they may be neglected in all practical cases. Acknowfe&ement--The author wishes to thank the Board of Directors permission to publish this paper.

of N.V. We&spoor,

REFERENCES 1. 2. 3. 4. 5.

E. A. FLOOD, C'timf.J. Chem. 36,1332 (1958). E. WEBER, Electromagnetic Fields, Vol. I. Mapping of Fields. John Wiley, New York (1950). R. BECKER, Theorie akr Elektrizitiit Bd. 1, Teubner, Leipzig (1949). J. A. STRA-~~ON, Electromagnetic Theory, McGraw-Hi, New York (1941). I. ~OKOLNIKOFF, Mathematical Theory of Elasticity, McGraw-Hill, New York (1946).

for