Question

[132 2 2 3 4 17 marks] Question 4 A plane wave is travelling in a vacuum in the +z-direction with wavenumber k and angular frequency . It is linearly polarised in the x-direction, and has electric field given by E(t, z) Eo Cos(kz - wt)f This wave is normally incident on a perfectly electrically conducting, semi-infinite slab in the region z > 0 and the resulting field in vacuum (z < 0) is a superposition of the incident and reflected fields. (a) What boundary condition must be satisfied by the total electric field at z = 0. Explain (b) Hence, show that the total electric field in the region (z < 0) is given by: Ē(t, 2) 2Eo sin(kz) sin(wt) (c) Show that the total magnetic field in the region (z 0) is given by: H(t, z) 2cos(kz) cos(wt) Zo where Zo is the impedance of free space. Write down an expression for the surface current K(t) on the boundary between the conductor (d) and vacuum (ie z = 0)? What is the time-averaged Poynting vector (S(z)) at all points in space? (e) Write down an expression for the time-averaged Maxwell's Stress Tensor, (T(z)) at all points (f) in space Using your answer to (f) write down an expression for the time-averaged net force per unit area (g) on the conductor. Comment on your answer Information: cos(a ± B) = cos(a) cos(B) sin(a) sin(ß)

Answer 1

Here, plame forms the intertnee between the dielectrie medium T characteri sed by lne permihvity , and and a permeadility eondueting medium with permihivily 62 and permeability condu rty So 6 is very lrge As h eonduetor is perfeatly onueing EH describes lhe ineidentwave ; E,, Hj deseribes the eieeld E2, H descibe the tramsmitted wave. Let Naye and ireetion k',, and K^ ane Ihe incident, refiected and trancmtted ve veetor Therefre ncident wave the Wave fields will be. Ckz-st) Refiected Wave Taramsmittel Wiav e; T have taken the complex form ot I eleetrc and magnetic field initialiy and uill replaee by th real fexm part at tat Aalds ane related to lhe amplitude of E Aetd by H E ond Ho- ,E. His Tequired by the faet that the veetor mustaim în the direetign of p**» Propagation tangential component of ele etric field The minus sign Lynting's aeross the boundauy enditiea Nequines thel Eo + El.- E 20 boundary eandition Satisfiefd by tue eleetoi field bounday dondition H- fald requines hat EZ (E-E b NO E20

We ean wse. u,2 * . equation N kinghe media to be non Then in terms of refrnetive index com be urtten as (n--) above n E, Eto) n E26 Tnevere Salving and We can urila e3 Eo Tu-u and 2ni nitn Eo nitn2 r12 is amplitude refiectior enefficient and t. is tme transmiHed amplitude tramsmitted coeffieient 4nside Tefracive index n^ic imaginay the metallie medium the п, з - Se For gord conduetrr 2. Sa = ?u So con be uritten aa Here 2n and ti- (n,tntini (nt n)tjn fr perfect conductor = 0, -. PH îndicalzs aphasa shifFEf 18o* due to reftechon from a perfeet condueting trams m Itted wave (E, H. =°). The ineident and refteeted Kill onstihut to form a So, SX Surtaee . Also there hrill be no stationany WAYe So in lhe eaion electn'e fied usill be - 2E. isin kz eiut - Taking Tel part we gef (t,2) 2Egsin (kz) Sin (wt) ( -Totel magnetie ffeld ikz Сио - - н] Here, 2. Re,) 2esowtj3 - 25cosk2)espyj Taking ree Pant

ondition H 9 canrile d)Fm he boundany Аx (нig- н maanetie tieleL in medium 1 and T perfect conductu H2 and H ane lhe Here, iis NTw H (,2)- 2 Es CogCkz) Cas (wt There fore 4,2) Her xH QEo (t,2) CSnterfr at foynting veetor 2 A Eo Sin kz cos kz sinwt eos Af K Theretore time ave vaged poyting vcetor 217