# Chapter 1 Electromagnetic Theory Â² Maxwell's Equation

• Aug 23, 2005
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1 = D

2 ! B =0

3 ! = B E t " = J + H D t # D

4 \$C/m2 % E

5 ! \$V /m% B

6 ! \$T V s/m2% H

7 !

8 \$A/m)

9 \$C/m3 % J

10 \$A/m2 % &

11 '

12 D = 4 B =0 = 1 B E c t 4 1 D H = c J + c t D(statcoul/cm2 ) E(statvolt/cm) B(gauss) H(oersted) (statcoulomb) J(statamp)

13 !

14 q0 (

15 ! E

16 F = q0 E

17 E D B

18 H D B ! = 0 E D + P = 0 H B +M 0 = (1/36) 109 \$#)% 0 = 4 107\$*)% P

19 M

20 +

21 P M !

22 P M

23 !

24 P M

25 ,

26 *

27 D B

28 E H

29 -

30 J

31 E .

32 / = f (E) D J = g(E) = h(H) B

33 0 "

34 E H

35 1 D = 0 E

36 = 0 E D + P = 0 E + 0 E = E = 0 (1 + )E 2

37 \$ %

38 3

39 D = 0 n2 E = E n= = r 0

40 J = E 2

41 M = 0

42 = 0 H B

43 4

44 2 &

45 \$ % 2

46 '

47 2

48 2 = 0 J = 0 5 2 = 0 6 7

49 2

50 ! 2 E =0 B =0 = B E t = E B t D = E B = H

51 D = 0 E + (2) E E + (3) E + E E

52 5

53 7

54 B !

55 E !

56 1

57 ,

58 E \$

59 % 4

60 B = ( B ) = ( B) ( E) t t = t ( tE )

61 ( E) E 2 E = ( E) = 0 2 2 E = E2 t #

62 E

63 2 2 2 B = B2 t

64 1 2 f (r, t) 2 f (r, t) = v 2 t2 v8 \$ ./% f (r, t)8 \$%

65 ! 4

66 v = 1/

67 c = 1/00 = 2.998 108 m/s

68 0 1 c v= = 0 0 r n

69 6

70 kr) r , t) = E0 ei(t E( k

71 *

72 !

73 7

74 *

75 E( r , t)

76 E( r, t)

77 9

78 r) 2 E = E0 2 ei(tk = E0 [ikei(tkr) ] = k 2 E0 ei(tkr) = k 2 E 2E = 2 E t2 k =

79 "

80 E =0 [E0 ei(tkr) ] = E0 ei(tkr) = ik E0 ei(tkr) = 0 k E0 = 0 !

81 \$

82 B !% #

83 = B E t ik E = i B

84 : k E = B k E B

85 B E 9

86 2 z E k B y x "

87 B !

88 |k| 0 n |B| = |E| = |E| = |E| c

89 7 E B 7 !

90 !

91 B H 2 |E| |E| = = Z |H| |B|

92 Z

93 ()

94 Z0 = = 377 0 0

95 9

96 ;

97 S 9

98 !

99 ! S H =E

100 9

101 J/(m2 sec)

102 7

103 !

104 < U = 12 (D E H) +B

105 B = nc E 1 B2 1 n2 1 r 0 0 2 U = (E 2 + ) = ( + 2 )E 2 = ( + )E 2 2 c 2

106 (J/m3) #

107 0 U

108 ! U = E 2

109 9

110 2 Wmech U dA + = S t t s Sin Sout A U Wmech 9

111 ; \$Sout Sin% A

112 ,

113 V

114 ! \$U %

115 Wmech

116 9

117 |S| = v U v

118 U

119 7

120 \$

121 9

122 S

123 J

124 U

125 %

126 ; S .

127 7 /

128 S I

129 7

130 ;

131 \$W/m2 % = I k = I n S k

132 1015 1011 1015

133 = =E S H = (E0 H 0 ) cos2 (t k r) # k E = B H = 1 k E k = = E0 ( 1 k E0 ) cos2 (t k r) = S 2 E0 cos2 (t k r) n -

134 \$

135 n%

136 \$ .

137 /

138 % -

139 7 ; 1 t0 +T 2 I S = T t0 0 E cos2 (t k r) dt n 1 2 t0 +T = T E 0 t0 cos2 (t k r) dt n > = t k r

140 cos2 = 12 (1 + cos 2) (t0 +T ) (t0 +T ) 1 2 2 1 2 E cos d = E (1 + cos 2)d T 0 t0 2T 0 t0 1 2 1 (t0 +T ) = E0 (T + cos 2d2) 2T 2 t0 1 2 1 = E0 {1 + [sin 2(t0 + T k r) 2 2T sin 2(t0 k r)]}

141 \$T 109 f 1 ?%

142 \$ 3.5 1015 *? [email protected]

143 60' % T = 3.5 105 1 T 1 11

144 I S = 1 2 2 E 0 = 12 ( Z1 )E02

145 I = 1 ( 12 E02) = v U \$ 12

146 7

147 % #

148 U = 12 E02

149 A

150 ;

151 7

152 >? F = qE + qv B +

153 >?

154 J = qv >? 7

155 \$)%

156 F1 = E + J B -

157 J

158 E = /0 , B = 0 J + 0 0 E/t E F1 = 0 ( E) + 1 ( B) 0 E B B 0 t -

159 = E B (E B) B + E t t t + 1 ( B) 1B F1 + 0 (E B) = 0 ( E) E ( E) 0 E B ( B) t 0 0 B = 0

160 E = B/t

161 V

162 B)dV Ftotal + 0 (E = [r.h.s.]dV t v v -

163 dPmech dPf ield + = [r.h.s.]dV dt dt v B)dV 1 Pf ield = 0 (E = 2 SdV v c v

164 !

165 ! 2 g = S/c

166 &B 4

167 r.h.s.

168 T dV T ds v s 1 0 2 B 2 T = 0 E E + B B I( E + ) 0 2 20 T

169 . /

170 2nd r.h.s.

171 V 4

172 V

173 |P | |g |V f orce P = = t = t area A A

174 t

175 ct

176 V = Act | g |Act S t P = = |g | c = A c C

177 P = S /c = I/c

178 #

179 1.34 103J/m2 s #

180 4.46 106 N/m2

181 105 N/m2

182 D

183 ! E H

184 0 exp i(k r t) =E E 0 exp i(k r t) =H H

185 E 0 H 0

186 ? ! E H

187 !

188 ?

189 !

190 ?

191 && H E H E k 7

192 ?

193 ?

194 ;

195 #

196 ?

197 ,

198 ./

199 !

200 ?

201 , 1

202 >

203 ? ./

204 ,

205 D

206 ?

207 E0

208 ? 7

209 , 2

210 !

211 = E0 [x exp i(kz t) + y exp i(kz t )] E 2

212 ei/2 = i = E0 (x y) exp i(kz t) E

213 E

214 "

215 1

216 !

217 &' H E H E k

218 ?

219 !

220 !

221 ? H E H E k

222 !

223 0 = (xE0 iyE ) exp i(kz t) E 0

224 "

225 ?

226 ?

227 ? !

228 !

229 \$

230 ?% "

231 D

232 ?

233 ? -

234 ! E

235 E 1 E 2 E 1

236 ?

237 &0 (Incident wave) E2 E Transmitted axis (Transmitted wave) of polarizer E 1 E

238 7

239 !

240 E1 = E cos "

241 I1

242 !

243 I1 = I cos2

244 #

245 ?

246 ?

247 ?

248 ( cos2 12 >

249 ?

250 ?

251 ?

252 P

253 !

254 ? Ipol P = Ipol + Iunpol

255 ?

256 Imax = Ipol + 12 Iunpol Imin = 12 Iunpol Imax Imin P = Imax + Imin Imax Imin

257 ? "

258 2

259 \$

260 % D

261 ?

262 ?

263 "

264 &5 Fast Fast Slow Slow /2 Fast Fast Slow Slow /2 Quarter waveplate Half waveplate

265 ?

266 0 = xE0x + yE0y E E0x E0y "

267 7

268 i E0x = |E0x |e x E0y = |E0y |eiy

269 !

270 E0x |E0x |eix = E0y |E0y |eiy

271 ? E

272 E

273 ?

274 ?

275 E #

276 ?

277 ?

278 ? 1 1 1 + =2 i i 0

279 ''

280 >

281 B A A

282 B E

283 a b c d

284 a b A A = c d B B

285 &6

286 E

287 2 atotal btotal an bn a1 b1 = ctotal dtotal cn d n c1 d 1 E

288 7

289 ?

290 ? 1 0 0 0 >

291 9

292 ?

293 0 0 0 1 1 1

294 45 1 2 1 1 #

295 1 0 0 i F7 #

296 ? 1 0 0 i 1 i #

297 45 1 2 i 1 *7 1 0 0 1 ei 0

298 0 ei eix 0 G

299 0 eiy G

300 1 2 1 i i 1 D

301 ? 1 i > 1 2 i 1

302 &: ! "

303 # 4

304 , E E i t ki kt Bt Bi i, i t, t #

305 7

306 ;

307 " 7

308 ,

309 #

310 ;

311 # !

312 " Incident k i r ni kr i Reflected x kt Transmitted t y nt z D

313 ,

314 ;

315 7

316 2

317 &< exp(iki r it)

318 exp(ikr r it) ; exp(ikt r it)

319 \$%

320 t

321 = k r t

322 ki r|z=0 = kr r|z=0 = kt r|z=0 " z = 0 r = xx + y y ki = (ni )(k i x + k i y + k i z) c x y z kr = (ni )(k r x + k r y + k r z) c x y z kt = (nt )(k r x + k r y + k r z) c x y z x y z

323 x y z

324 (kx , ky, kz)

325 ;

326 = i r t

327 z = 0 ni (kxi x + kyi y) = ni (kxr x + kyr y) = nt (kxt x + kyt y) "

328 x y ni kxi = ni kxr = nt kxt ni kyi = ni kyr = nt kyt kxi = kxr kyi = kyr kxi = ( nnti )kxt kyi = ( nnti )kyt

329 ;

330 x y

331 4

332 !

333 ki

334 ki

335 y ki kt x kr plane of incident

336 &= #

337 kx ky kz

338 > !

339 x z

340 4 z ki kr Incident i r Reflected n i x n t Transmitted t kt

341 (kx , ky, kz) = i r t kxi = sin i kyi = 0 kzi = cos i kxr = sin r kyr = 0 kzr = cos r kxt = sin t kyr = 0 kzt = cos t

342 kxi = kxr !

343 ;

344 i = r " ni kxi = nt kxt

345 ni sin i = nt sin t #

346 4

347 7

348 &A

349 D ds = 0

350 s s B d s=0

351 dl = B E ds

352 t s H dl = D ds t s #

353 !

354 "

355 dA i dh t n n

356 i t1 ds = ndA

357 i ds = ndA

358 t !

359 i n = D D t n i n = B B t n

360 . D B

361 / - D i B i D t B t

362 !

363 i n

364 -

365 !

366 dA i dh t t n t

367 " dh 0 ; B = s ds = 0 B D = s D ds = 01 i t = E E t t i t = H H t t

368 'B

369 .

370 E H

371 / E i H i E t H t

372 !

373 !

374 i n

375 \$

376 " \$

377 ;

378 ,

379 *

380 ;

381 ;

382 \$C

383 ?

384 E !

385 ?

386 %2 &

387 ?

388 E

389 \$ - H

390 ?

391 % ' 9

392 ?

393 E \$

394 ?

395 % #

396 ?

397 !

398 Ei kr Bi ki Er i Br ni y(y) x(x) nt t Bt Et kt z(z)

399 %! "#\$

400 E " #\$

401 E

402 '&

403 2 i = yEi exp i(t ki r) E ki = ni (x sin i + z cos i ) c

404 B i = 1 ki E i i = ni (x cos i + z sin i )Ei exp i(t ki r) B c ; 2 kr = ni (x sin i z cos i ) c r = yEr exp i(t ki r) E r = ni (x cos i + z sin i )Er exp i(t ki r) B c

405 2 kt = nt (x sin t + z cos t ) c t = yEt exp i(t kt r) E t = nt (x cos t + z sin t )Et exp i(t kt r) B c -

406 \$ r = 0% # 7

407 !

408 ?

409 D !

410 ! #

411 - \$z% B 2 ni nt sin i (Ei + Er ) = sin t Et c c

412 \$x% H 2 ni nt cos i (Ei Er ) = cos t Et i c t c +

413 !

414 Ei + Er = Et

415 Et

416 ni nt cos i (Ei Er ) = cos t (Ei + Er ) i t "

417 !

418 !"

419 Er ( nii cos i nt t cos t ) rs = ni nt Ei ( i cos i + t cos t )

420 '' C

421 nt = ni sin i / sin t

422 ni ni sin i cos i (Ei Er ) = cos t (Ei + Er ) i t sin t t tan t (Ei Er ) = i tan i (Ei + Er )

423 Er (i tan i + t tan t ) rs = Ei (i tan i + t tan t )

424 Ei + Er = Et !

425 Et 2t tan t ts = Ei (i tan i + t tan t ) # !

426 C

427 i

428 t

429 !

430 7

431 i t 1 4

432 ! I

433 rs = sin(i t )/ sin(i + t ) ts = 2 sin t cos i / sin(i + t ) # 9

434 ?

435 !

436 , & i 0'

437 '0 Ei Bi Er kr ki Br i ni y(y) x(x) nt t Et Bt kt z(z) #

438 i tan i + t tan t rp = i tan i + t tan t 2i sin i tp = cos t (i tan i + t tan t )

439 i t 1 rp = tan(i t )/ tan(i + t ) tp = 2 cos i sin t /[sin(i + t ) cos(i t )] %

440 &

441 '

442 Ai Ai Sr Si ni i i Ao nt t S n t At

443 '5 # I

444 7

445 ! E ! E

446 *

447 ;

448 ?

449 I

450 nt

451 I

452 ni

453 *

454 !

455 I

456 ./

457 7

458 .

459 /

460 A W

461 ? J

462 ,

463 ?

464 + !

465 A0 Ai = Ar = A0 cos i At = A0 cos t

466 9

467 S \$

468 %

469 1 W

470 A

471 W = S A = i r t

472 S = 12 E2

473 0 = n0 #

474 Wi = ni 2 0 2 E A o i 0 cos i # ; Wr = ni 2 0 2 E A o r 0 cos i #

475 Wt = nt 2 0 2 E A o t 0 cos t \$

476 9

477 ?

478 % Wr E2 R = r2 = |r|2 Wi Ei Wt nt cos t Ei2 nt cos t 2 T = 2 = |t| Wi ni cos i Er ni cos i !

479 &

480 #

481 ! ;

482 2

483 '6 n =1, n =1.5 i r 0.2 1.0 Brewster Angle 0.0 0.8 P Glancing incident Power Reflectivity (R) -0.2 Field Reflectivity (r) S 0.6 -0.4 S 0.4 -0.6 P 0.2 -0.8 Brewster Angle -1.0 0.0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 incident angle incident angle #

484 2 ni=1.5, nr=1 1.0 1.0 0.8 0.8 Total Internal Total Internal Reflection Reflection 0.6 Power Reflectivity (R) Field Reflectivity (r) 0.6 0.4 0.4 S 0.2 P Brewster Angle S 0.2 0.0 Brewster Angle Critical Angle P Critical Angle -0.2 0.0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 incident angle incident angle *

485 = 0

486 ! !

487 9

488 ?

489 ;

490 9

491 ?

492 < 10 ;

493 I

494 = 0

495 I

496 sin(i + t ) = sin i cos t + cos i sin t

497 ': 2 sin t cos i ts = sin i cos t + cos i sin t +

498 > sin i = nn t i sin t 2 cos i ts = nt ni cos t + cos i "

499 i = 0

500 t = 0 cos i = cos t = 1 t= 2ni ni +nt T = ( n2n+n )2

501 i i t +

502 r= ni nt ni +nt R = ( nn n +n )2

503 i i t t

504 ?

505 tp rp

506 5K

507 ; 7

508 L&6

509 &B

510 ;

511 R = 0.04

512 ;

513 7;

514 \$*

515 7;

516 %

517 ;

518 ?

519 ?

520 .+ / #

521 ;

522 I

523 ?

524 rp = tan( ) i tan( + ) i 0 t t i + t = 90

525 nt sin i sin i = = ni sin(90 i ) cos i + !

526 + B B = tan1 ( nnti ) + 9

527 ?

528 ni > nt nt > ni

529 '

530 ;

531 ./

532 . /

533 ni > nt \$#

534 % # ni t = sin1 ( sin i ) nt 4 ni > nt

535 nn i sin i > 1 > ! c

536 nn sin c = 1 t i t c = sin1 ( nnti ) "

537 t = 90

538 4

539 Et eikr = eikt (x sin t +z cos t ) 4

540 ni sin i sin t = sin i = nt sin c sin i 2 cos t = 1 sin2 t = 1 ( ) sin c - i > c

541 #

542 sin i 2 cos t = i ( ) 1 i c < < sin c 2 sin i 2 = ( ) 1 sin c

543 Et eikt (x sin t +z cos t ) 1+2 Et = ekt z eikt x

544 \$

545 x

546 %

547 z

548 \$

549 %

550 H

551 '=

552 z

553 1

554 ; !

555 !

556 9

557 # ! E H

558 \$

559 7 9

560 %

561 = 1 Re(E I n = S ) H 2 # ;

562 kt E t 1 I = St n = Re[(Et H t ) n] 2 1 t ( 1 kt E t )] n} = Re{[E 2 t 1 = Re[Et2 (kt n)] 2t

563 kt n = kt cos t = ikt

564 St n = 0

565 4 !

566 z = 1/ z

567 1/e

568 1

569 1 1 = kt 4

570 7

571 !

572 ;

573 +

574 ;

575 7

576 ;

577 Variable attenuator Prism coupler

578 'A ! ;

579 I

580 9

581 ?

582 cos t = i sin t = 1 + 2 #

583 ?

584 sin(i t ) rs = sin(i + t ) sin i cos t cos i sin t = sin i cos t + cos i sin t 1 + 2 cos i i sin i = = eis 1 + 2 cos i + i sin i # 9

585 ?

586 !

587 sin cos = 12 {sin( ) + sin( + )} tan(i t ) rp = tan(i + t ) sin(i t ) cos(i + t ) = sin(i + t ) cos(i t ) sin 2t + sin 2i = sin 2t + sin 2i sin i cos i sin t cos t = sin i cos i + sin t cos t sin i cos i i 1 + 2 = = eip sin i cos i + i 1 + 2

588 ;

589 &

590 Rs = Rp = 1 !

591 ;

592 4

593 !

594 ;

595 ! 9

596 ?

597 sin i s = 2 tan1 1 + 2 cos i 1 + 2 p = 2 tan1 sin i cos i

598 0B 180 160 140 P Phase change () 120 100 S 80 60 40 20 0 40 50 60 70 80 90 Incident angle ()

599 ;

600 1

601 .

602 /

603 ;

604 4

605 90

606 ;

607 & ni nt

608 ! ;

609 9

610 ?

611 D 4

612 90

613 !

614 7

615 ;

616 !

617 ;

618 ;

619 7

620 #

621 & \$

622 &M%

623 7

624 (

625 ;

626 C 7

627 \$

628 i

629 & ' 90%

630 7 ; .

631 /

632 7

633 0&

634 .

635 / D = E B = H

636 !

637 4

638 J = E \$

639 % -

640 E = 0 B = 0 B E = t E B = E + t

641 2E E 2 E = + t2 t

642 !

643 .7

644 / 4 !

645 7

646 #

647 r, t) = E( E( r)eit

648 i 2 E( r) + 2 ( )E(r) = 0

649 (

650 - ! i =

651 2 E( r) + 2 E( r) = 0 4

652 1

653 2 E( r) + k 2 E( r) = 0 i n k = ( )= c

654 0' n = n(1 i)

655 ! ?

656 z

657 r) = E E( 0 exp(in z) exp( z ) 0 exp(ikz) = E c d

658 d = c/n

659 .

660 / n k

661 c2 n2 = [ 2 2 + ( )2 + ] 2 2 2 c2 n = [ 2 2 + ( )2 ] 2 #

662 n22 c2 2 2 d

663 1/

664 10N

665 4

666 #

667 ;

668 &%

669 '%

670 #

671 ;

672 # ! ;

673 t

674 ni sin t = sin i nt ;

675 I

676 ;

677 ?

678 ? ;

679 4

680 ;

681 (n1)2 +(n)2 R = |r|2 = n+1 n1 n 1 n +1 = (n+1)2 +(n)2 4n = 1 (n+1)2 +(n)2

682 00 -

683 n = in (in 1)(in 1) R= =1 (in + 1)(in + 1) . ;/

684 \$k = ( i )% "

685 ;

686 \$AB A6K%