Question

1.) (a) State Maxwell’s equation for the curl of the magnetic and the electric field in free space. State the meaning of all the terms in the equations and identify the displacement current density. Using Maxwell’s equations, derive the wave equations for B. Show that the wave equations admit plane waves for the electric and magnetic fields in free space of the form ? = ??? ?(??−??) , ? = ??? ?(??−??) where ?? and ?? are constant vectors with real components if ? 2 = ? 2?0?0. Show that E and B are transverse waves propagating with speed ? = 3 × 108 ms-1 . (b) Show that the wave number of an electromagnetic wave plane wave is complex in a material with conductivity ?. Assume the wave is propagating along the y-axis. Determine the real and imaginary parts of the wave number in terms of ?, ?, ? and ?.

Answer 1

ANSWER :
Maxwell equation' 7.€ = Sv leo 7.B-0 T XE =- OB at J X B = MJ + I de 22 it. In free space 3 x 8 = 2.0B -> Electric field criculate around changing dt magnetic Hur L d 1 عار Mхв. (2 dt (2 JXB Ho&o. de 1 Š x B Mo . JE dee dt B = Mgh :: B=&e Displacement curient Density Tx = dB Magnetoc feld circulate around change in electric flux [ taking card. Both side) Ō xl 5xE)=-o co xo) 7 x (x 7.) -ved (222 dt ŽXE = -dB at at . ) . It dt Io (Nocharge)

(re VE- d²E (2. Jt2 Lwave equation for electric field similarly. T XB. t.de с2 dt 1 (2 dt 2 (xe) x (FXB) - :: ² B = / dB (2 ota Eate Ickz-wt) ickz-wt) B = Bo e | 1 de Bom de (2 dz2 dt. substitute the value o from given (ik) Cik) I (W) (-iw). .(2 7. K² = +62 (2 k² w2 (2 c²= loco who & ne Too

More well equation in medium- Š xê dB It ut T X = Mo I + Moeo de taking cool on both side of equation @ anda we will get TE illo de d² + ME dt dte ot 2 - Es ė AB= no de M & ² B dt time E? ickz -wt) ilikz-wt) Boe V²C = ke otake de - i wt - I -iw dt B = at Substituting @ and 8 in eqs and © -K? - PW ura w²&o lo K²= wheeft P W Mo bromegn @ we can see that kis complex in calculating medium .. ka ID Mew²tilo Ew k=kq +114 kt = Real port + The Cu kina imaginary pat

(9 > k ² (kt - K2) +27 k4 k_ comparing @ and (12 Ckit _K2) = new² akt low (18 2 l 2 Squaring eg 4K Ake How (atb)² Carble tuab. (k; ²+k?). (K2 K²) + 4e a we addeng. and 46 & 0² (1t SH (El Substracting 15 and 12 (12 172 kt 22 ME K Ew 2