Question

Que (2 In a media characterised by, o = 0, y = H, E = 2€, and E = 40sin(27x10t + kz)ay a) State the direction of wave propagation b) Find; E i) ii) D k B H (E iv) c) Determine if the E field satisfies Maxwell's 1st and 2nd equations. d) Determine the phasor form of this field.

Answer 1

The media is characterised by $\sigma$ =0, $\mu$ =$\mu$_o and $\epsilon$ =2$\epsilon$_o

The electric field is given as

E=40sin(2$\pi$*10⁶t+kz)a_y

a) Since the electric field has a phase given by kz, the direction of propagation along the Z-axis

b) i) The electric displacement is given as D=$\epsilon$E

D=80$\epsilon$_osin(2$\pi$*10⁶t+kz)a_y

ii) k=$\omega$/c

where $\omega$ =2$\pi$*10⁶ rad/s (given)

c=velocity of light=3*10⁸m/s

Thus, propagation vector k=2$\pi$*10⁶/3*10⁸=2.09*10^-2 /m

iii) k$\times$E=-$\omega$B

Thus,

B=k$\times$E/$\omega$=2.09*10^-2 a_z$\times$40sin(2$\pi$*10⁶t+kz)a_y/(2$\pi$*10⁶)=13.33*10^-8sin(2$\pi$*10⁶t+kz)a_x

iv) Magnetic intensity H=B/$\mu$=13.33*10^-8sin(2$\pi$*10⁶t+kz)a_x/(4$\pi$*10^-7)=0.11sin(2$\pi$*10⁶t+kz)a_x

c) First Maxwell's equation is $\bigtriangledown$ .E=0 or k.E=0

Since k and E are perpendicular to each other, Maxwell's 1st equation holds true.

Second Maxwell's equation is $\bigtriangledown$ .B=0 or k.B=0

Since k and B are perpendicular to each other, Maxwell's 2nd equation holds true.

d) In phasor form, the electric field can be written as E=E_oe^{j(kz-$\omega$t)}

^{where the term in the brackets in the exphonent gives the phase of the E-field}