Solutions to selected problems from Exercise 5

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Knut Ellingsen
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1 Soluions o seleced poblems om Execise 5 Po. Rakhesh Singh 1

2 Execise 5.1 Since j E E ˆ ˆ x + y e he diecion o wave popagaion k x ˆ + y ˆ x+ y he coesponding magneic ield can be obained om Maxwell s cul equaion

3 Execise 5.1 j E E ˆ ˆ x y e + x+ y xˆ yˆ ˆ E E E jµ H H jµ jµ x y e x+ y x+ y j j xˆ yˆ ˆ jµ x y x+ y x+ y x+ y x+ y j j j j E e e e e + + x+ y x+ y x+ y x+ y E j j j j j j j e + jµ jµ η e e ˆ E e ˆ E e ˆ 3

4 Execise 5.1 ime-aveage powe low pe uni aea ( yˆ xˆ ) x+ y x+ y 1 * 1 j j E S Re Re ( ˆ ˆ avg E H + E x y) e Ee ˆ + + η η 4

5 Execise 5. Fo any uncion o be a soluion o he wave equaion, i mus saisy he ollowing equaion ε µ 5 ε µ ε µ

6 Execise 5. ε µ ε µ 6 ε µ ε µ

7 Execise 5. 1 µ ε µ v p c ε This wave has phase velociy equal o speed o ligh Hence, any uncion () is a soluion o he wave equaion 7

8 Execise 5.3 Aleady solved in he class 8

9 Execise 5.4 ( ˆ ˆ) j a E E jx + e Noe ha xy om a igh handed sysem, puing in ime dependence jy j E E ( + jx ˆ ˆ ) e e Taking he eal pa o he wave π E E ( y) + y + xˆ cos ˆ cos y 9

10 Execise 5.4 Obseving a y E E cos ( ˆ sin( ) xˆ ) x Time Time LHCP 1

11 Execise 5.4 { ˆ ˆ } j b E E + j x j e Obviously EP Puing ime dependence j an j j E E an 3 e + e xˆ 1 ˆ 5 e e Taking eal pa E E 1 cos y ˆ + 5 cos y y j ( ( 6.6 ) xˆ ) + y 11

12 Execise 5.4 Obseving a y E ( 1 cos( 71.6 ) ˆ 5 cos( 6.6 ) xˆ ) + + E + x Time Time RHEP 1

13 Execise 5.4 j ( c) H H ( xˆ jyˆ ) e + In ha case, yx (-) oms a igh handed sysem Puing in ime dependence and aking eal pa H π H + yˆ + xˆ cos cos( + ) Obseving a H H ( sin( ) yˆ xˆ cos( ) ) + 13

14 Execise 5.4 H H ( sin( ) yˆ xˆ cos( ) ) + x Time Time y LHCP 14

15 Execise 5.5 equaion o he ellipse o polaiaion E y 3cos π ( ), E 5cos + 5sin( ) E y E + cos + sin ( ) Maximum ield magniude equals 5 V/m 15

16 Execise 5.5 yx oms a igh handed sysem E ( ) yˆ 5sin( )ˆ 3cos Time Time y LHEP 16

17 Execise 5.9 σ ε (a) Loss angen o a medium is given by A equency 1 kh, loss angen is appoximaely equal o 899 which is geae han 1 and hence he solid gound acs as a good conduco A equency 1 MH, loss angen is appoximaely equal o.899 which is close o.1 and hence he solid gound may ac as a bad conduco o good dielecic 17

18 Execise 5.9 Fo good conduco, α πµσ and good dielecic α σ µ ε The aveage Poyning veco educes a he ae o inside he solid gound e α 18

19 Execise 5.9 Hence, o deph ino he gound whee he aveage powe o he EM wave has been educed by 1% o ha o he suace is given by e Fo 1kH, (. ) α ln 1.1 α ln (.1 ) α he equied deph is appoximaely m wheeas o 1MH, i is appoximaely m 19

20 Execise 5.9 (c) In ode o image an objec a a deph o 5 m, o oundip o he ada signals we need a disance o 1m Fom he knowledge o he above pa o he secion (1 MH won be able o ge ha a), we may assume ha he solid gound should behave as a good conduco o he poblem a hand

21 Execise 5.9 Then o a powe eleced back om he objec which has dopped o jus 1% o he oiginal powe, we have, e α (.1) 4π µσ ln(.1).1 π µσ ln { } 1 { ln(.1) } 6. kh 86 4πµσ 1

22 Execise 5.9 Hence he wavelengh a his equency is appoximaely km, heeoe, we may no be able o image a any small objec a his deph

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