B Bxi Byj coskztBBxiByjcoskzt where Bx 28 X 106 T By 47 X

B? =(Bxi? +Byj? )cos(kz+?t)BBxiByjcoskzt

where B_x = 2.8 X 10^-6 T, B_y = 4.7 X 10^-6 T, and i-hat and j-hat are the unit vectors in the +x and +y directions, respectively.

A plane monochromatic electromagnetic wave with wavelength = 4.7 cm, propagates through a vacuum. Its magnetic field is described by B = (Bri + Byl) cos(kz + (at) where Bx -2.8 x 10-6 T, By -4.7X 106 T, and i-hat and j-hat are the unit vectors in the x and *y directions, respectively. where By-2.8 x 10 *TBy-47x1o*, and i-hat and j-hat are the unit vectors in 1) What is f, the frequency of this wave? GHz Submit 2) What is I, the intensity of this wave? WIm2 Submit 3) What is Sz, the z-component of the Poynting vector at (x = 0, y = 0, z = 0) at t = 0? W/m2 Submit 4) what is Ex, the x-component of the electric field at (x = 0, y = 0, z = 0) at t = 0? V/m Submit 5) Compare the sign and magnitude of Sz, the z-component of the Poynting vector at (x-y-z-t-0) of the wave described above to the sign and magnitude of SIz, the z-component of the Poynting vector at (x-y-z-t 0) of another plane monochromatic electromagnetic wave propagating through vaccum described by: where Bllx = 4.7 X 10-o T, Blly = 2.8 X 10- respectively. T, and i-hat and j-hat are the unit vectors in the +x and +y directions, Sllz 0 and magnitude(Sllz) * magnitude(Sz) Sliz 0 and magnitude(Sllz) - magnitude(Sz) a Sliz > 0 and magnitude(Sllz) = magnitude(5) Submit

Solution

a)

frequency

f=c/lambda =3*10⁸/0.047

f=6.383 GHz

b)

Net magnetic field

B=sqrt[B_x²+B_y²]=5.46*10^-6 T

Electric field

E=cB =(5.46*10^-6)(3*10⁸) =1638 V/m

Intensity

I=(1/2)ce_oE²=(1/2)(3*10⁸)(8.85*10^-12)*1638²

I=3562 W/m²

c)

Z-component of poynting vectoe

S_z=ce_oE² =(3*10⁸)(8.85*10^-12)*1638²

S_z=7123.5 W/m²

d)

o=90-tan^-1(4.7/2.8) =30.8^o

X-component of electric field

E_x=1638Cos30.8 =1407.2 V/m