8432-1-F Copy - THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory iNTEGRAL EQUATIONS APPROPRIATE TO CURRENT'S INDUCED ON A PERTURBATION OF A CONDUCTING SURFACE Final Report 18 October 1966 through 28 May 1967 Vaughan H. Weston PO 514-648171 8432-1-F = RL-2178 Contract With: Northrop Corporation Norafr Division HNoRwon, California HNit Is h to Hi, C^I Administered through: OFFICE OF RESEARCH AM INISTRATION * ANN ARBOR

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THE UNIVERSITY OF MICHIGAN 8432-1-F TABLE OF CONTENTS Page ABSTRACT I INTEGRAL EQUATION FOR A GENERAL PERTURBED SURFACE 1 II APPLICATION TO A PERTURBATION ON A SPHERE 7 III SPECIAL CLASSES OF PERTURBATION ON A SPHERE 11 IV SUMMARY 20 4.1 Geometry 20 4.2 Incident Field 20 4.3 Fundamental Integral Equations 21 4.4 The Far Field 23 APPENDIX A 25 APPENDIX B 30 REFERENCES 33 iii

THE UNIVERSITY OF MICHIGAN 8432-1-F ABSTRACT An integral equation useful for computations of currents induced on a perturbed conducting surface by an incident plane wave is developed, where the perturbation is of stub or protuberance type. However unlike the usual approach, the free space Green's function is not used. Instead a dyadic type Green's function whose tangential components vanish everwhere on the unperturbed surface is employed. In this way the resulting integral equation involves an integral taken only over the perturbed surface. An explicit expression is developed for the kernal of the integral equation for the case of the unperturbed surface being a sphere. A great simplication is achieved in the integral equation approach for a special class of perturbations on a sphere. For these cases, the vector field can be represented in terms of two scalars; and the resulting vector integral equation reduced to two sets of coupled scalar integral equations, taken only over the sufface of the perturbation. iv

THE UNIVERSITY OF MICHIGAN 8432-1-F I INTEGRAL EQUATION FOR A GENERAL PERTURBED SURFACE The integral equation for the currents on the perturbed portion of a general perturbed surface will be developed in this section. The surfaces under consideration will be perfectly conducting, and the perturbations that are considered, will be limited to protuberance types. Define the following: S: surface of unperturbed perfectly conducting shape 81: surface of perturbation or protuberance S: surface composed of S1 and the unperturbed portion of 0 Let (E, H) be the total Maxwelllan field (harmonic time dependence exp (-it) assume, whch are generated by a plane wave (E. H i) incident upon the perfectly conducting shape S. The incident field will be given explicitly by the relation i A +ikz E xe (1) In the far field (R- oo), the total field E E+ E E will have the form A kz e ikR E v x e + f ( ) R and (2 0 A +ikz ~ A H,,e + R x f(~.) ~ 0 'I

THE UNIVERSITY OF MICHIGAN 8432-1-F Let (E, H) be the total Maxwellian fields which are generated by a magnetic dipole at R, in the presence of the unperturbed surface 8. The orientation -o 0 of the dipole will be given by the unit vector m. The component of this field, due to the source only, has the form ikr E Vx - m, where r |R-R- | (3) r- - and in the far field the total field (E, _i) has the form ikR E' R f (,; R Employing the well-known Lorentz lemma, the following integral relation connecting the two fields, is obtained i -- A 0 e4 a~o) Since nx E vanishe on, and ni x E vanishes on S the above Integral reduces to S +S:+ n ExHdo [ExH- x do (6) ofiniit adu, n ~ asml shreeclsngte iol ouc a 2

THE UNIVERSITY 8432-1-F OF MICHIGAN a (Small Sphere Centered at R ) — o n S Perturbation S Unperturbed Surface FIG. 1: GEOMETRY OF SURFACE 3

THE UNIVERSITY OF MICHIGAN 8432-1-F The integral taken over the sphere of infinite radius can be evaluated as follows. Explicitly it is given by the relation 2 r im JJ dec10 [cosjfe -f sinJ Rsain(l- cosO) exp ikR(l+cose)J R —r0o 0 4i2 r A -, -4 - f(; R.) (7) The integral taken over the small sphere of vanishing radius r - I R - Ro -oo lim ' [E x E xH r2 dO (8) r — Oj 4

THE UNIVERSITY OF MICHIGAN 8432-1-F where E is given by Eq. (3). For r mall it can be shown that i - (r x -) E _ ___~ E2 -r LL Hi [3r (m _ r)- m] r 3 r in which case expression (8) reduces to the form f iExm 1 lim (rx m)xH+i d f. r-4O 0Ex To evaluate the integral local spherical polar coordinates (r, 0', ' ) are used, with the z-axis being in the direction of the vector m. The fields at a point (r, ', p' ) on the surface are expanded out in a Taylor type series about the origin. A (A It follows that (r x m)x H -(Hb m) + (r * m)(H r). -(H m) sin2' + sin ' coo ' (H cosol/+'+, s') x y where the components of H are those given at the origin ie: H H(R). -- — 0 Simlary A Exm sin' r. -- (cosC' E - sin' E ) r r y x aE aE QE sin 'cos'l [sin, 'os '0 +sinesn, ',o+o ' x 8y 2z aE aE aE sin x + sio'sinF' LX +Xc x ] - sin sin' sinO' cos0' + sine' sin' — + coso' ft dxlty z 5

THE UNIVERSITY OF MICHIGAN 8432-1-F The above integral reduces to 3 /)E 8E \ 0 0 2wp ax y r; i (,in')3 d0'd' { - (H. m) + iw o \With these results the integral relation (6) reduces to <( - mX |J n *(- B; )X H(Rt)ds' -f (e- r; R). (10) S1 It is evident that the above can be expressed in the form H(R m H (Ro + E(R; R [n x H(R') d (11) 81 for R outside surface S. H (Ro) is the field generated at the point R by the -o - -0 -o plane wave [given by Eq. (1)], in the presence of the unperturbed perfectly conducting surface S. The above equation indicates that if the solutions are available o0 for the unperturbed surface (ie: H and E can be prescribed), then the field H at a point R generated by a plane wave incident upon the perturbed surfaces is -o given in terms of the current on the perturbed section. An integral equation for this current is obtained by letting R approach the surface of the perturbation for tangential orientations of the dipole m. It can be shown that when R is on S1, the following equation is obtained 0 H(Ro)-m 2H (Ro).mt+2 E(R; R; mt).[n H(R')] do (12) 0 t -m-o mt Sm i where is tangential to the surface. where m is tangential to the surface. t 6

THE UNIVERSITY OF MICHIGAN 8432-1-F APPLICATION TO A PERTURBATION ON A SPHERE The special case will be considered, where the unperturbed surface 8 is a sphere of radius a, From Stratton p.569, the value of H (R) corresponding to the total field -o generated at the point R by a plane wave incident on the perfectly conducting -O sphere can be obtained immediately as follows. n(n+l) n l01 where ln(P) a (14) n h(1)(p) n n [P Jn(pI' b -a... (15) n Lp Wh(l)(p] with p ka. To obtain an expression for the kernel of the integral Eq. (12), the electric field generated by the dipole at R is decomposed into two parts -o E i Es E-E +E (16) Hi where E is the field of the dipole in free space, given by ikr ^i e E V x - r and E is the scattered field due to the sphere. From Appendix A it is shown t i that E can be expressed in the form 7

THE UNIVERSITY OF 8432 -1-F MICHIGAN XV x VxR@ + Vx Rx - - - -a (17) where ^i V a m* V x R r X -m- V xV xR f -o -o - I (18) and r ik L 2n+l) _ n(n-l) nml (kR)h )(kR) ) R P(cos'y) n 0^) R<R 0 R>R o (19) with R R RRcos'. The cattred field can be written in a imilar manner a follow: The scattered field can be written in a.imilar manner as follows: E5 xVx Vx R~+Vx R (20) The boundary condition on "the total field eani ~e xpreased km1 tii6 f6rm R a. x +x R a (21) (22) a R ^-* i+ ^-* a 0 aR VI 41 R a. It is immediately evident that ikm.V xV xR a h (kR) h )(kRP (co Y) o o 0 - ^n(n+) n n n on (23) 8

THE UNIVERSITY OF MICHIGAN 8432-1-F a -Ikm-V xR 2 bh h( (kR )P (eos) (24) - o -o n(n+l) n n n on nul where the constants an and bn are given by Eqs. (14) and (15) respectively. The kernel of the Integral Eq. (12) has the explicit form - -o E(RRo) o Vx R(+e) +Vx Xn b X) (25) 0 where the scalar functions are given by Eqs. (18), (19), (23) and (24). Explicit forms have now been found for H (R) and E (R; R ) associated with the integral equation for the currents n x H on the perturbation. To complete the analysis the appropriate explicit expression for the far ~ A where (h e 0 f untio nil byE(.1 8 )) (27) field should be obtained from Eq. (11). In this case the value of I(R, Ro) for Rs needed. It can be shown that ikR -icos 0 E (R. Ro) v - ~R ikR x m, (26) ikR X (m a)ke (2 n+1) (-i) n+ )(kR) )(co (27) R v -lnm b -i hn (kR)) (cos) (28) R — too o n o n=1 0 Rn(n+l) 0 n= 1 where A acos + a cos ] o sin 80e a ae n (29) - sin y 9

THE UNIVERSITY OF MICHIGAN 8432-1-F F a cosy C os Y1 -o sine a0 o0 8 0 L 0 0o - J (30) sin y cos y = cos 0 cos 0e + sin sin 0 cos (0 - 00) When R -> 00, E(R, Ro) is essentially the total field produced at R by a plane wave incident on the conducting sphere R = a. To obtain the far scattered field expansion (11) is used with R —> 0 and O A A the vector m taken to be in the 8 or 0 directions. The far scattered field - -o o is comprised of two parts, the scattered field produced by a plane wave on perfectly conducting sphere given by the term H~ (R ) * m, and the scattered -ofield due to the perturbation H (R ) m where H (R ) m = - ' E (R', R;m) [x H(R')] dS' -p -o -0 i where _(R', R;m) for R - co is foundfromEqs. (25), (26), (27)and (28). 0 0 10

THE UNIVERSITY OF MICHIGAN 8432-1-F III SPECIAL CLASSES OF PERTURBATION ON A SPHERE Great simplication of the integral equation approach is achieved when the perturbation on the sphere belongs to a special class. This special class of perturbation is characterized by surfaces which are comprised of spherical caps R = constant and conical sections f (0, 0) = constant. Two examples of such surfaces are (1) the conical stub: with cap R = b, and conical side 06 0 0 (2) the flat plate: with cap R =b, and sides 0 = O, = ar - 0, 0= 0 and 0 = -0 For these types of perturbations, the total field (E, H) generated by the plane wave (E, Hi) incident on the perturbed sphere, can be represented in terms of two independent scalars as follows: 1 H a Vx R4 + j Vx Vx Rx'i (31) -i v E = V_ x V x R+ + V x R. (32) Using the relations V x V x RL Rk 2R + V( aR) V x Rx _VxxR it follows the boundary condition n x E = 0, applied to a spherical cap R = constant, yields the conditions 11

THE UNIVERSITY OF MICHIGAN 8432-1-F A ARiP A aR Rx V;R ~- = ~0 and Vx - R(R Vx) = 0. Since both the operators Rx V and [- R(R * V imply differentiation along the surface, the above two conditions can be expressed in reduced form as follows a R 0, = 0, and X =O for R = constant aR In a similar manner it can be shown that the boundary condition n x E = 0 applied to a conical portion f (0, 0) = c reduces to the conditions t = 0 and =0 an a where - implies the normal derivative. Thus, since the incident field can be decomposed in a similar form as Eq. (31), namely H V xRR + - V x V x RX1 (33) it follows that the vector problem can be reduced to the following two scalar problems: (1) find = s + i such that (a) is is the given component of the incident field, (b);s satisfies the scalar Helmoltz equation and represents, an outgoing wave (c) with boundary conditions aR - 0 for R = constant aRIR and (34) = 0 for f(0, ) = constant 12

THE UNIVERSITY OF MICHIGAN 8432-1-F (2) to find X X +X such that (a) xi is the given component of the incident field, (b) x satisfies the scalar Helmoltz equation and the outgoing radiation condition, (c) with the boundary conditions X 0 for R = constant, o (35) a 0 for f(0, ) = constant The above decomposition of the vector field in two scalar quantities will result in a simplication of the appropriate integral equation, the development of which follows. Define the following: S: the surface of the unperturbed perfectly conducting sphere R = a S1: the surface of the perturbation whcih will be comprised of two parts, (a) C the sides given by the cone f (0,0) = constant, a < R < b, and (b) T the cap given by R = b, S: the surface comprised of S and the unperturbed portion of S. The scalar quantities X and t associated with the total field (E; H) generated by the plane wave incident on S, must satisfy the following boundary conditions ( )a = 0, =0 for surface S (37) 8Rb ( R-)b' 0, X=0 for cap T O.X = 0 0 for sides C *=0 n, 13

THE UNIVERSITY OF MICHIGAN 8432-1-F In addition, the field (E~ H ) (which is the total field generated by the plane incident on the unperturbed surface So(R = a)) and associated scalar quantitites X and ~ will be introduced. It then follows that aR 0 R for R =a (38) X~ = 0 To derive the appropriate integral equation for @, the Green's function G1 (R, Ro) must be introduced, where 2 2 V G1+k G1 - -4w6 (R-Ro) (39) and which satisfies the boundary condition aRG1 — R 0 for R a. (40) The Green's function has the explicit form ikr G (R,R) r - ik 7 (2n+1)b h1) (kR)h( )(kR )P (cosy) - -o r nn n o n n~ O(41) where r R - R|, RR cosy = R R, andthe constant b is givenby Eq. (15). Integrating the relationship 2 2 V21 -G 1V 2 -4, 6(R-Ro) one obtains n, (R) 41 - Gi VY) dS (42) 0 4 S+S o0 where S is an infinite sphere, and n is the inward normal to the enclosed volume. 14

THE UNIVERSITY OF MICHIGAN 8432-1-F Similarly it can be shown that O(Ro) = 41r + 0 00 n- (/~ VG -G VYJ~)dS. 1 1 (43) The integral over the surface S vanishes since G1 and @ satisfy the same boundary conditions on this surface. On combining Eqs. (42) and (43) one obtains -o,<(KL - G )dS a R(R o) + )"a G G1 1 + 4- s 4ir 's O0 n [( - o~) VG1 - G1V(_ - 0)] dS The integral over the infinite sphere vanishes, since Jp - 1 does not contain the incident field, inwhichcaseboth G and i- o satisfy the, outgoing radiation condition. Hence the above reduces to i ( -1 d. ' (Ro) '~O(R)+ J (T-P~- G + - d In a ar manner it can be shown that In a similar manner it can be shown that (44) X(R ) = X~(Ro) + R 1 an2 ax (x- 2 -o a —.)dS lx -a 2 81 (45) where the Green's function G2(R, R ) is given by the relation ikr G2(_R, Ro ) = _ k _A (2n+1) a h(1) (kR) h (1)(kR) Pn(cosr) r 11n n (46) 15

THE UNIVERSITY OF MICHIGAN 8432-1-F with the constant a given by Eq. (14). The above integral representation can be written as follows where use is made of boundary conditions (37) I O1 dS + O 0 4 1 an dS b (, )-1 G 2 1 - d X(R) (B X - dS - dS (47) X +4 or a n 4 2 an where n is the outward normal. A description of the surfaces are given in Fig. 2. The fundamental problem remains to find and X on the side C n an and ~ and X on the cap T, for when these are known, the total fields X an and f can be determined. The integral equations for those quantities can be developed by letting )R approach the surface, and taking the appropriate limit. It thus follows that for R on the surface T ( -)=2 (R) 2 G1 a d-S + 2-) ( adS' (48) 8x(R) 2Ox~(Ro) 2G O 8n an X dS 2r d (49) and for R on the surface C - O a(R,) 2O () 1 Of 8 G o G G dS 19G 1 dS a n0 (a n ~ 2 1 J an a n d 2 r T an [an b] ann IOn (50) _n'0.-n X<o =2X~(B) + Sx1 OdS _ - 0G dS. (51) - - r 2T 16

THE UNIVERSITY OF MICHIGAN Cap of Perturbation T (R =b) Side of Perturbation (Conical Surface) C s/ —'- Unperturbed Surface SO (Sphere Radius a) FIG. 2: GEOMETRY OF SURFACE PERTURBATION. 17

THE UNIVERSITY OF MICHIGAN 8432-1-F It is of particular importance to briefly investigate the behavior of the scalar functions and their normal derivatives in the immediate vicinity of the edge. The top edge formed by the intersection of the conical side C and cap T, has an exterior wedge angle of 37r/2 radians. From Maue (1949) it is seen that the tangential component of the magnetic field which is parallel to the edge, H II, is finite whereas the other tangential component which is perpendicular to the edge, H1, must behave like s / where s is the distance from the edge. In the local region of the wedge the leading term of X and kr are solutions of the two-dimensional Laplace's equation, with solutions of the form + v + v s cos V, s" sin ', log s where '0 is the angle measured from the top face T. Applying the boundary' conditions Eq. (37) on the wedge faces 0' = 0 and 0' = 3n/2, it follows that v = 1/3, 2/3,.... -1/3 However using relation (31) and the conditions that H11 is finite and H I s /, it can be shown that the lowest admissable solution is given by v = 5/3. This implies that X,, ax, and - are finite at the edge. The far field expression (for R — oo) can be easily obtained. Using the O relations, ikR O G1(, Ro), eR Fl(o0, 0o;, (52) ^ V^ o^ -(52) and ikR G2(R, Ro)- F 2(o, R), (53) 0 R 2 0 0 18

THE UNIVERSITY OF MICHIGAN where F = e-ikRcos - (2n+ 1) bn(- i) h(1)(kR) P (cos 7), (54) n =0 F2 = e CO R o - (2n+ 1)a (- i)n h) (kR) Pn(cos 7), (55) n=0 it follows that ikR e R *(-o - *(o) ~ R f1( 0o ) o ikR 0 4(-Ro) ' (fRo) R0) o where F f(dS+ L J dS (56) ikR fp o o) o' -. ' R(58) ikR The total scattered field is obtained by adding to expression ( 58) the scattered field produced by the plane wave incident upon the unperturbed conducting sphere R =a. 19

THE UNIVERSITY OF MICHIGAN 8432-1-F IV SUMMARY 4. 1 Geometry: The perfectly-conducting surface consists of a stub on a sphere of radius a, with the surface of the stub described by a conical side C and cap T, where C: T: e F.(), R b, a R b 0 and 0 contained in the cone C tub: C; 0=0, a<R< b, 0<(<2r 0(0(0, Examples (1) conical s T; R=b, 0<e<o 0<0p<2X (2) flat plate: C; surfaces = o, QUr-0o. 0 O. 0" O T; Rb. 4.2 Incident Field: The incident field is a plane wave. (harmonic time dependence), of polarization e, and direction of incident given by -k 1 where k lkl(sin 1 cos 1 n, ssinin1, cosO 1) In particular, the incident field is described by the relation E eexp(-iklR)..A A where e e 9 +e i e - 1- 1 21. 20

THE UNIVERSITY OF MICHIGAN 8432-1-F 4. 3 Fundamental Integral Equations: There are four unknown quantities to be found, P and which are associated with the surface T, and and X an an which are associated with the surface C The resultant four integral equations decouple into two sets of two The two integral equations involving a and a bn are as follows, for R on T, ". ' / " O -0 0(h 2^^0LR + fI ds (-Ro) 2 ~(Ro) - G1 (R R d2s C T and R on C, -o t(R) 2qR f aG G 01 0 ( GI+~) dB an an 2 an an 2j an- aR R C. T The Green's function G (R, R ) has the precise form ikr G (R Ro) e - ik (2n+ l+)b h()(kRh(1)(kR )P(cosy) 1 -' o r n n 0 n n=0 and Ite derivative is given by", ' aG a -n o VOGl(R Ro) a n. o -o where n In the above expression forthe Green's function, the following relations are used, 21

THE UNIVERSITY OF 8432-1-F MICHIGAN r- R-R o. -s -O cos -y= cos 8 cos O +B sin0 sin 0 cos ( -0 ) O O O' - [Pn( I b —: - n~ hil^)p \), [fi WPhrb3 with p ka. The function g/ (R ) is given by the relation (see Appendix B) 0 iO(Ro) a o (*_ e), n=l (-i) (2n+1) p)(cosY)[(kRo)-b h(kR )] n(n+l) cR n n 0J where cosyl cos 8cos + sine sin0 cos o-1 +( ) a *1 [-cosOe sine1 + sinO cos01 cos(0o-01/ sin ' +T 1 [sinOasBin(lt 00)] ins l The two integral equations involving X and _ are as follows Cn for R on T -o ax(Ro) 8n o 2ax~(R) ) o 1 aR 2 O Ja fxI c aG2 aR ) o aG 2P d 2r aR an T and for R on C - 1 X(Ro) 2X (Ro) + -o -o 2w C dG2 21 Aan 2ds -2r T G x ds. 2 an 22

THE UNIVERSITY OF MICHIGAN 8432 -1-F The Green's function G2(R, R ) is given explicitly by the relation 2-o ikr G2 (R. R ) e-, -ik ~_ (2n+l)a h( )(kR)h(l)(kRo)P (cosC) 2 r n onn n n(Co n=0 where i (P) n h(1) with p z ka. The known function X (R.) is given by the relation (from Appendix B) X(R ) i ( xk) () (n+ )P(co1) -o_ n (n+1) n 01 J.(kk~,)- a h(,) (okR & 4.4 The Far Field: The far scattered field is composed of two parts H H + H OS where Hos is the scattered field due to the perfectly-conducting sphere R=a, and HP is the scattered field due to the stub. Since the expression for H0 is wellknown, only the addition term HP will be given here. ikR Ro) e 1 aA ^o a. 0s 0 ' 0 ikR ikR +ik +t o so (0o Po) o o o 23

THE UNIVERSITY OF 8432-1-F MICHIGAN where l (O' 00) -4, C an d 1 T 4Tr T F1 + '1rI % -R+ — da f2 (0o' 0o) C 8F2 x -da F R ds 2 AR with -ikR~oosy nxO (2n+1)b (-H) h )(kR)P (coay) n n n F2^ e-kRos-. (2n+1)a (-i) n=0 h(1)(kR) P (cos-y) n n 24

THE UNIVERSITY OF MICHIGAN 8432-1-F APPENDIX A It is the purpose of this appendix to prove the relation ikr Vx( t)= Vx Vx R(t. V x R ) + V x R(t- V xV xR RT) _ r -o -o. o -o -o where (A. 1) " 7 (n+ ) - (kR) h( (kR) P (cos ) nn+) n R<R o In the above relation, t is a unit constant vector, r = R- Ro, y is the angle between the two vectors R and R, and the subscript on the operator V indicates differentiation with respect to the variables (R,.o,0 ). The vector field can be expressed in terms of the orthogonal set of spherical vector wave functions M k mn k N e mn " Vx R % e mn _ VxVxRm - - - n o 9 where emn (kR) Picos ) os mO mn a ( sin m0 as follows ikr Vx (e t) = V x x Vx (e t) _- r - (RBa vv) + k V x V x (RebV) = V x V x (R7a&v) + k V x (RBbI &) The summation over V indicates summation over n = 1,2,3,... m = 0, 1, 2,..., n, and for both even e and odd o functions. 25

THE UNIVERSITY OF MICHIGAN 8432-1-F From the orthogonal properties of the vector wave functions, integrated over the surface of a sphere, it follows that ikr (VXR ) S = a (VxR VxR )dS (A.2) - -r - S (!X v). ( Vx- t eve _ S S a sr \ R r dS V v) -r -bx dS (A.3) From Stratton ( p. 418) it is seen that In order to compute the left-hand side of Eqs. (A. 1) and (A. 2) the following lemmas are needed. Lemma I: if S is a closed spherical surface of radius R, then ikr ikr | Vxx R 'R dS = V xR dS (A. 5) S S Proof: The left-hand side of (A. 5) can be written in the form Since S is a closed surface and R is the normal to the surface, then the first integral vanishes leaving the second integral 1 ikr 1r xikr 5b x -xRdS =.V 7xR dS+SbY --- xr.dS r ) Rds- r -0 r

THE UNIVERSITY OF MICHIGAN 8432-1-F ikr The second integral vanishes since V ( e ) is parallel to r. This leaves -O r v e x R dS = v x Rds namely, the right-hand side of Eq. (A. 5) Lemma II: ikr i kr Vx(R) ' Vx t dS = t V V R x — dS} (A 6) - r - -S Proof: The left-hand side can be written in the form ikr r ikr ikr t e = t V e x (VO xR) = t V x Vj xR dS - -o r.- - and using lemma I, this reduces to r p *ikr t.V xV x R r I dS} -0 -o -o -- r Lemma III: S (VxRt), (VxVx r-t)dS =k t V x{RO j* er dS (A.7) - r -( ) r Proof: The left-hand side can be written in the form 2 ikr i kr 2 ikr e. k 2 VxR. t dS + (v x. Vo).dS S[k2 + ( V x R dS r - - r and in using lemma I, this reduces to 27

THE UNIVERSITY OF MICHIGAN 8432-1-F 2 ikr ' - 0 -0 r 2 ikr k t tV x R - -o {o J r dS} The results of the above lemmas can now be used to evaluate the unknown quantities a and b. It follows that v v ikr 3a - tV xV xR P --- dS v v - -o v r i kr vb =kt V x R 0 - e dS. v V - -o -o S v r It follows that ~a% % () = t. V x V x (R,) (A. 8) 'v ' -0 '-0 ' - v k1bvIv(R) t * Ox(R ) (A.9) where -o(R) ikr A -v QV (R') r dS' (A.10) and r =R' - R. I - -o I Simplification is achieved when the coordinate system is rotated lining the z-axis up with the.direction R. In this case -o 28

THE UNIVERSITYOFMCIA OF MICHIGAN 8432-1-F ir (R.. R 0 7 ) = (2n + 1) qe (. ) eon,Y n (n+ 1) 4 [in(kR')1 2 ikr e r sinoy'd7y' dol = 1k n..i.1 2n (n+ 1) eon (2n+'l h1)h '(kR )JP (Cos'Y )1 2sinyt doy' 1 k (2n+ 1) j (kR) h~ (kR ) P(coo'Y) n(n+ 1) n n 0 n (A. 11) It follow's immediately from (A. 8), (A. 9) and (A. 1 1) that (A. 1 s proved. 29

THE UNIVERSITY OF MICHIGAN 8432-1-F APPENDIX B 0 0 The problem is to find explicit expressions for 4i and X associated with the field (E H0) generated by a plane wave Incident upon the perfectlyconducting sphere R = a, by the relations H =Vx R~ + -VxVx Rx k ( (B.1) 0 1 0 ) -i E~ =- Vx VxR/ +VxRx From Section II, it was shown that that the total field generated by a magnetic dipole at R ikr,~i e E Vx - m - - r in the presence of a perfectly conducting sphere of radius R = a, could be written in the form - E - k Vx Vx R + Vx Rx where, -i ~ k m* V x R r -1/ m V xV xR wr 00 -o - 0 - 0 with nk (n+ i) n (cos ) hn) (kR) i(kR) - h ) (kR). n n n 30

THE UNIVERSITY OF MICHIGAN 8432-1-F A Let R -co and rm J R in which case E becomes a plane wave 0. -0 - Ei[ik eRO -ikR cos y A e mx R — o In this case we have X l ~ ikR -O n(-i) n=l (2n + 1) Pn(osy) [(kR) n (n+'1) n n -ah( (kR)] a nn (B.2) r ik1 ikR lo o,~a - 6u" I, 0k m*V () n+ 1) P(cos (kR) o~(-n=nn(n+ 1) -b h) (kR). n a (B.3) The term V P (cos 7) can be written in the form -o n 8 cos 7 P(cos) o 0ae o A a cos 1Y sine 8a 0 O. a Pn(cos ) Ra cos y 0 1 (1) - a Pn (cos y) R n 0 o where o 0 [cos sin o + sin e co os s ( - 00)] /siny + i [sin sin (0 - 0) / sin ] Normalize E so that it represents a plane wave of unit amplitude, i.e. E eexp (-ik R) E' "' 0 R)" (B.4) (B. 5) 31

TH E UNIVER SITY OF MICHIAN - 8432-1-.;;-:: F wher (s ine cos80. CA 8, cA Bin O) 0 * 0 0- 0 and A Then It follows that the total field generated by the plane wave incident on a U( (-..).).hI '( )] e -(o e no e+ 2. 0 'A 1 ' n-!. n*;. 32

THE 'UNIVERSITY OF MICHIGAN 8432-1-F RE FERENCES Maue, A.W. (1949), "On the Formulation of a General Diffraction Problem Through an Integral Equation", Zcit f. Physik 126 pp. 601-618. Stratton, J. (1941), Electromagnetic Theory, McGraw-Hill. 33