On the electric (magnetic) fields associated with isolated corners of large conductors

Journal of Electrostatics, 22 (1989) 245-256

245

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

ON THE ELECTRIC (MAGNETIC) F I E L D S A S S O C I A T E D WITH I S O L A T E D C O R N E R S OF L A R G E C O N D U C T O R S

D. OWEN

Department of Mathematics, The University of the West Indies, St. Augustine (Trinidad, W.I.) (Received April 18, 1988; accepted in revised form January 19, 1989 )

Summary Two-dimensional electric fields associated with isolated corners of general shapes which form part of large conductors are investigated using conformal mapping. As examples we consider four conductors, three of which are of similar geometric shapes to the conductors previously treated by Langton and Smith. All of our results are in agreement with those obtained by the method used by Langton and Smith.

1. Introduction Langton [ 1 ] and Langton and Smith [ 2 ] examined the electric (magnetic) fields associated with corners of large two-dimensional conductors. In all the shapes they considered, the conductor consisted of two semi-infinite planes at right angles to each other with the vertex replaced by a curved section. This curved section was always part of a circular cylinder and the surfaces were equipotentials of value + Vo units. To perform the mappings these authors used the Schwarz-Christoffel transformation together with a complex inversion. In Langton's work [ 1 ] there are at least four errors: equations (10), ( 11 ) and the second and third equations in Section 8. W h e n equation (10) is corrected we see that the intensity tends to zero as we move away from the corner. This agrees with the results obtained using the technique of the present paper. In this paper we look at problems of similar geometric shapes to those examined by Langton and Langton and Smith. However, the curved sections will be of more general shapes. The technique used is straight-forward and particularly useful because, for example, we can deduce, analytically, the general behaviour of the field intensity along the corresponding surfaces of the conductors. This is carried out for one case in the appendix. Also, and very significantly, we are able to obtain a single general transformation which not only can be used to map all corners considered but also to examine, for example, semi-infinite slabs or slots or other shapes quite easily. All of our results agree with those of Refs. [ 1 ] and [2 ] after correction of the errors in Ref. [ 1 ].

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© 1989 Elsevier Science Publishers B.V.

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It should be pointed out that the formula used in this work is actually a special case of those obtained by Bloor [3] and King [4]. Bloor, however, does not give a derivation of his formula and in any case we feel that it is very useful to know how to derive a particular transformation directly rather than using Bloor's general formula. A derivation is given by King. In Section 2 we shall give the general formula (derived in Ref. [5 ] ) which maps the upper half-plane onto the region in which our field lies. In Section 2 we derive a general expression for the field intensity, and consider some of its implications. In Section 4 we consider four examples, deriving the corresponding curvature functions as needed. We obtain expressions for KE/A, the dimensionless electric field intensity, along and away from the surfaces of the conductors. Three of these conductors have the same geometric shapes as three of those considered by Langton and Langton and Smith. 2. The general mapping formula The shape of the general corner we consider is shown in Fig. 1. The arrows indicate the direction in which we move along the curve. We always move in a direction such that the region in which the field lies is on our left. For example, if we follow the double arrows we move from A to D and the region on our right is the conductor i.e. region I, with the field in region II, whereas if we follow the single arrows we move from D to A and this means that the conductor is on our right, i.e. region II, and the field in region I. Cade and Owen [5 ] derived a general formula for similar shapes (see Section 4 of that paper) and we shall quote the general formula corresponding to the case where the conductor occupies region I. This is t2

~tt=Keia2(t-tl)l-°'/~(t--t2)-°2/Xexp[lf fl(u)du u-t 1] tl

B

1I

~,t o

~o

~ A

A Fig. 1. G e n e r a l s h a p e o f a corner.

Fig. 2. G e n e r a l s h a p e of t h e c o r n e r to be e x a m i n e d .

247

where K > 0 and the integral is interpreted as t2

1 ~ fl(u)du lim .o+ ~ u - t - i e

J

tl

The factor exp (~- lffl ( u ) / ( u - t) du ) is a shape factor which determines the shape of the curved section of the conductor. We call fl(u) the curvature function. Now without loss of generality we shall take the real axis to lie along OA so 01 -- 0 and we choose tl = 0 and t2-1. This gives, with 02 -= 0, 1

dz iO" " e'~ [-1 ~=Ke t(t-1)-/exp[~f

fl(u)du]l -~_~ J

(1)

o

where K > 0. This corresponds to the shape in Fig. 2. From an examination of arg (dz/dt), 0 < t < 1, we see that if we do not want vertices at O or B we take fl(0) --~ and fl(1) =0 otherwise we have vertices with internal angles al at O and G2 at B, where al=2~--fl(0)

and

a2-=n-O+fl(1)

(2)

The special case 0= 0 corresponds to a semi-infinite slab and will be considered elsewhere. The cases 0= ½~ and ~n will be dealt with in this paper where we extend the work of Langton and Smith. These conductors are shown in Fig. 3. (The other special case 0= ~, f l - ~ is that of a semi-infinite plane surface and is trivial.) If we set 0=~, fl=n+flo(t), flo~O we get 1

expL f flo(u)du~j

_ io -~=Ke dz

El

(3)

0

which can map the shape in Fig. 4. Shapes of this type were also considered by

~i

i_

]I

~ o

I o

A

(o) Fig. 3. T w o special corners considered. (a) 0 = ½~; ( b ) 0 = -32~.

(b)

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Owen in Ref. [6 ]. Essentially this transformation was also derived by Shercliff [7] in his work on molten metal columns as well as by King [4]. Conductors with other values of 0 can also be considered. For example, if 0=- ~ + 0, OX ½u, we get the conductor shown in Fig. 5. If O= 2~, fl= 2~ + flo we get the transformation corresponding to a semi-infinite slot. These shapes will be considered elsewhere. If we wished to consider the shape in Fig. 6, which corresponds directly to Fig. 1 (a) of Ref. [2], we could use the transformation formula t2

~dz= K (

t-tl)-'/2exp[l f fl(u)du~ u - t _] tl

where K > 0 and tl and t2 correspond to B and O respectively and fl (t) describes the arc BO. This formula will not be used in this paper. It may be derived using the technique of Ref. [5 ]. 3. The expression for the electric intensity, E, and implications

The surfaces of all of these conductors are equipotentials of value + V0. Now the mapping of the complex potential, W, plane ( W= U+ i V) onto the upper half t-plane is given by o

A

I B

B

Tr

0 ]I D

Fig. 4. General shape corresponding to eqn. (3).

I

tl

Fig. 6. Shape considered in Ref. [ 2 ] Fig. 1 ( a ).

Fig. 5. Shape with 0 = ~ + ~ , 0 < O < ~ .

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W= -At+iVo where A > 0. Hence the expression for the electric intensity, E, is

E = IdW/dzl =A Idt/dzl

(4)

i.e. 1

K

-AE = l t - l ( t - 1 ) ° / = l

fl(u)du~ u-t ]l

exp(-lf 0

This shows t h a t as I t[ -, oo, K E / A --, I t l o/=- 1, which says t h a t (i) if 0 > ~, t h e n E--,oo; (ii) if 0 < x , t h e n E-*0; and (iii) if 0=~, t h e n E-~constant. If in addition, fl= ~ in this latter case t h e n E = constant everywhere. In fact it is noted t h a t by the m e t h o d used to derive this transformation the behaviour of E at infinity is determined by the Schwarz-Christoffel theory. For example, in Fig. 7 (a), it does not matter what shape the cross-section B has; the behaviour of the field E, as we move very far away from B along the straight line segments BA and BC, is the same as in the case of the simple vertex in Fig. 7 (b), i.e. E - , 0 , which is physically acceptable. Similarly, in the

c

I" Conductor

1"r

! 1

I

h (a)

Conductor

.= (b)

Fig. 7.(a) Conductor in region I with general corner at the origin. (b) Simple conductor in region I with a vertex at the origin.

Condu~or

Tr Conductor =

(a)

A (b)

Fig. 8. Conductor in region II with general corner at origin. (h) Simple conductor in region II with vertex at origin.

250

case of the conductor in Fig. 8 (a), the behaviour of E, as we move very far away from region B, is the same as if we had the simple vertex in Fig. 8 {b) i.e. E--, ~ . This result is also reasonable, since it is accepted t h a t there is an affinity for charge to move to the "outside" of a conductor, i.e. the charge will move away from the "inside" region B and towards the regions C and A. This also tells us t h a t if we are interested in a region sufficiently far away from a vertex region it really does not matter what shape t h a t vertex region has.

4. Examples The conductors we consider are shown in Fig. 9. In the following examples we shall consider the dimensionless electric field intensity, KE/A, instead of E.

4.1. Conductor in Fig. 9(a) For this shape we use eqn. ( 1 ) with 0= ~/2. The first quantity to be obtained is the curvature function fl (t), ti < t < t2 where we choose tl = 0 and t2 = 1. Now

Conductor

Conductor

t=l B

t=O

~

=1

~h

o

t=O

h

0

(a)

(b)

Conductor ---

A

Conductor

;

u

0

A

L=I

(C) Fig. 9. Shapesof conductors examined.

(d)

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arg (dz/dt) = fl (t), 0 < t < 1, and since we do not want a vertex at O (t = 0 ) we must have f l ( 0 ) = ~ . At B ( t - - l ) we must have f l ( 1 ) = 0 to give an internal angle of ½x. We shall use a polynomial expression for fl(t) because it is easy to integrate analytically. Also arg (dz/dt) = fl (t), 0 < t < 1, is a decreasing function as is seen from Fig. 9 (a). W i t h this information we choose

fl(t)=~(1--t2N+l),

N = 0,1,2,...

where N is introduced because it allows us to examine an infinite sequence, {CN}, of such surfaces. If 0 < t < l and N-~oo, we see that f l ( t ) - , x , i.e. for 0 < t < 1, arg(dz/dt) ~ ~ for large N and the corresponding shape has a very small protrusion BO. We shall consider the shape with N= O. It is easy to get expressions for arbitrary N. We obtained the following for KE/A along the surface DBOA:

KE

~e(t-1)l/2t -1 ( l - l / t ) (t-l) -- l:(l-t)~/2t -~ ( l - t ) -(~-t) A (l+u)l/2u -1 ( l + l / u ) -(l+u)

t>l O
t=-u,u>O

Away from the surface, where t is complex,

KAE_ t - l ( t - 1 ) l / 2 e x p [ 1 - ( 1 - t ) l o g ( ~ t ) l l The variation along the surface of this conductor is shown in Fig. 10. This result agrees with that for the corresponding shape treated by Langton [ 1 ], after correcting the error in that paper. The general shape here was also obtained analytically by considering the derivative of KE/A. If we used instead, j ~ = ~ ( 1 - - t ) 2N+1 we see that the first shape (N=O) is identical with the first shape of the previous fl-function but the limiting shapes are different. We expect the new limiting shape to have a long thin protrusion BO. We thus get different sequences of surfaces each having its defining curvature function, fl(t), satisfying the same conditions.

4.20,

o

0.32

i.o

,.4

t

Fig. 10. Variation of KE/A along the surface in Fig. 9 (a) corresponding to fl(t) = g ( 1 - t ).

252

4.2. Conductor in Fig. 9(b) The second example also has 0 = ½u in eqn. (1). As in section 4.1 we can show that a suitable simple fl, which satisfies fl(0) = ½~ and fl(1 ) = u with fl an increasing function in (0,1), is fl(t) = ½u ( 1 + t eN+' ), N= 0,1,2,... We also note that since 0 < t < 1, fl(t)-. ½~ as N - , ~ and we expect the cavity to become a long shallow groove along one side. Considering the case with N= 0 we get for KE/A, along the surface DBOA,

~e -1/2 (t-1)a/2t-1 ( l - l / t ) -(1+t)/2 KE_ ~e -1/2 t (t-1/2) (l--t) -t/2 A

[.e -~/2 ( l + u ) l / 2 u -1 (1-{-1/u) -(l-u)~2

t>l 0O

Away from the surface, where t is complex,

KE_ t_~(t_l)l/2exp[ - ~1- ~ 1 ( l + t ) log (~--~)1 d The variation along the surface is shown in Fig. 11. This shape was also obtained analytically by considering the derivative of KE/A. Conductors of this shape were not treated by Langton.

4.3. Conductor in Fig. 9(c)

The third example has 0 = ~3 in eqn. (1). Here we want the internal angles at O and B to be ~x at both vertices so from eqn. (2) we choose fl(t) to satisfy 2 ~ - f l ( O ) = ~3 and ~ = f l ( 1 ) - ½~ i.e. fl(0) = ½~ and fl(1) =2n. Also from Fig. 9(c) we see that fl(t) increases from ½x at t=O to 2~ at t=l. We thus choose for0
fl(t)=½~(l+3t2N+l),

N = 0,1,2 ....

For large N we expect to get a distorted shape like Fig. 12. Considering the surface corresponding to N= 0 we get the following for KE/ A along the surface DBOA:

K___~E

A

~ 2 0.48

Fig. 11. V a r i a t i o n of

1.0

t

KE/A along t h e surface in Fig. 9 ( b ) c o r r e s p o n d i n g to fl (t) = ½~ ( 1 + t).

Fig. 12. L i m i t i n g shape of surface w i t h fl ( t ) = ½x ( 1 + 3 teN +, ) as N - ~ ~ .

253

KE_ A

~e -1"~ ( t - 1 ) 3 / 2 t - 1 ( 1 - 1 / t ) -(1+3t)/2 ~ e - l 5 t - 0 - 3 t ) / 2 (1-- t) -(3t/2-1) [ e -1"5 (l+u)3/2u -1 (1+1/u) -(1-3")/2

t>l 0
t=--u,u>O

Away from t h e surface, where t is complex,

n -

l(t-1)3/2exp(-~-½(l+3t)l°g

T h e variation along the surface is shown in Fig. 13. T h i s shape was also obt a i n e d analytically by considering the derivative of K E / A .

4.4. Conductor in Fig. 9(d) T h e final sample also has 0= ~3 in eqn. (1). Here we have the same basic corner as in Section 4.3 b u t there is now an external bulge. F r o m eqn. (2) we see t h a t we m u s t satisfy f l ( O ) = ~ , f l ( 1 ) = ~, with fl decreasing in (0,1). A suitable f u n c t i o n is

fl(t) = ~' n ( 3 - t 2N+1~j,

N=0,1,2,...

In this case we see t h a t for large N, f i x ~ a n d hence we expect the limiting

K_~E A

0.16 -o.3o

o

0.5

,.o

,.25

t

Fig. 13. Variation of KE/A along the surface in Fig. 9 (c) corresponding to fl( t ) = ½:~( 1+ 3t).

KE

0.5

Fig. 14. V a r i a t i o n of

i.O

t

KE/A along t h e surface in Fig. 9 (d) c o r r e s p o n d i n g to ~ ( t ) = ½~ ( 3 - t ).

254

shape to have a long thin bulge along one side. Using N = 0 we obtained the following expression for KE/A along the surface DBOA: ~e 1/2 (t_l)3/2t -1 ( l - l / t ) -(3-t)/2 t>l K E _ ~e 1/2 t (1-t)/2 ( l - t ) t/2 0 O Away from the surface, where t is complex,

KEA - t - i ( t - 1 ) a / 2 e x p ( ½ - ½ ( 3 - t )

log(~)

The variation along the surface is shown in Fig. 14 and is obtained analytically in the appendix. This result agrees with that of Langton and Smith (Fig. 13 of Ref. [2] ). Concluding remarks

Using a modification of the Schwarz-Christoffel formula it was shown how to examine the electric (magnetic) field associated with corners of rather general shapes which form part of very large conductors. These corners may consist of cavities or bulges and may or may not have vertices. We considered four examples, three of which were of the same geometric shape as conductors previously examined by Langton and Langton and Smith using a different conformal mapping method. All of our results agree with those obtained by the method of Langton and Langton and Smith for similar shaped conductors after certain errors are corrected in Ref. [ 1 ]. An advantage of our method is that we are able, in all cases, to obtain analytically the general shapes of the field intensity over the surfaces of the conductors and this was shown for one case in the appendix. It is to be noted that the second, third and fourth examples will not necessarily be symmetrical. If we wanted symmetrical shapes we could consider the "half" shapes in Fig. 15(a), (b) and (c) respectively. The dotted lines represent field lines. Shapes which are of similar geometric

\ / /

\ \

\

/

\

(a)

(b)

Fig. 15. Shapes which may be used to consider symmetrical conductors.

(c)

255

shapes to the others considered by Langton and Smith can easily be obtained by properly choosing the curvature functions. Acknowledgements

The author is grateful to Dr. R. Cade for discussions which helped to clarify certain aspects of conformal mappings of curves with vertices. References 1 N.H. Langton, The electric or magnetic fields external to a series of corners, J. Electrostatics, 2 (1976/1977) 351-366. 2 N.H. Langton and M. Smith, Electric fields inside a series of curved corners, Proc. IEE, 124 (3) (1977) 277-284. 3 M. Bloor, Large amplitude surface waves, J. Fluid Mech., 84 (1) (1978) 167-179. 4 A.C. King, Ph.D. thesis, University of Leeds, 1988. 5 R. Cade and D. Owen, Charge density, vertices and high curvature in two-dimensional electrostatics, J. Electrostatics, 17 (1985) 125-136. 6 D. Owen, An extension of the Schwarz-Christoffel theory with applications to two-dimensional ideal fluid flow hydrodynamics, Z. Angew. Math. Mech., 64 (1984) 91-99. 7 J.A. Shercliff, Magnetic shaping of molten metal columns, Proc. R. Soc. London, Ser. A, 375 (1981) 455-473.

Appendix

Here we consider the conductor in Fig. 9 (d) and determine analytically the shape of the corresponding dimensionless intensity, F = - K E / A . t>l

In this interval we can write F = e x p [½ + ~ l o g ( t - 1) - l o g t - ½( 3 - t) l o g ( l - 1 / t ) ] where F' = FG and 5 G= ~ [ 3 ( t - - 1 ) ~ - l o g ( 1 - l / t ) ]

Now G' <0 for all t> 1 and since G o +oo as t-~l ÷ and G--,0 and t o +oo, it is clear that G> 0 for t> 1. Hence F' > 0 for t> 1, i.e., F is an increasing function in this region. O
Here we write F=exP[½+½ ( l - t ) l o g t + ½ t l o g ( 1 - t ) ]

256 where again F' = FG with

7 N o w F ' = 0 when G=O i.e. when

[1-t~ 2(t-½)

i This equation has a solution t = ½ and by sketching both log[ ( 1 - t ) / t ] and 2 ( t - ½) / t ( 1 - t) we see t h a t this is the only solution. H ence F has at most one turning point. By examining F" (t) we see t h a t F" (½) < 0 i.e. t = ½ is a local maximum. Also F--,0 as t o O + and F - * 0 as t - , 1.

t 0 . T h e n we get F-exP[½+~3 log(l+u)-logu-½(3+u)

log(l+l/u)]

and d F / d u = F G where 2u+1 G=u(l+u)

log(l+ l/u)

Now G tends to + o c as u-~0 +, G tends to zero as u--, + ~ and d G / d u < O . Hence G > 0 and d F / d u > 0 for u > 0, which gives d F / d t = - d F / d u < 0 for t < 0, i.e. F decreases as t increases from - ao to zero or F--, + ~ as t - , - oo. T h e overall shape is t h a t of Fig. 14.