A2. (a) Define the Poynting Vector S (2 marks) (b) Derive an expression for the

Question

A2. (a) Define the Poynting Vector S (2 marks) (b) Derive an expression for the time-averaged magnitude of the Poynting vector for a linearly polarized plane wave propagating in a medium with relative permittivity r and relative permeability (2 marks) (c) A spaceship in lunar orbit is ~ 3.8 × 108 m from the Earth's surface. It trans- mits plane waves isotropically at 1 GHz with a total power of 1 MW. A sub- marine just below the surface of a freshwater lake (E 81, ? 1) receives the signals from the spaceship. Determine, at the submarine i. the magnitude and direction of the time-averaged Poynting Vector ii. the speed of the wave; iii. the amplitude of the electric field iv. the amplitude of the magnetic field (You may assume that freshwater behaves as a lossless dielectric.) (4 marks) (Total: 8 marks)

Explanation / Answer

a. Poynting vector is defined as

S = E x H

E = electric field vector

H = magnetic fields' auxillary vector field

S = Poynting's Vector

b. For a linearly polarised plane wave

relative permittivity er

relative permeability = ur

e = permittivity fo free space

u = permeability of free space

hence E = Eo*sin(kx - wt)

B = (Eo/c)*sin(kx - wt)

hence

H = B/u*ur

hence

E x H = (Eo^2/u*ur*c)*sin^2(kx - wt)

time average value = (Eo^2)/2u*ur*c

c^2 = 1/u*e*ur*er

hence

<S> = Eo^2*sqrt(e*er/4(u*ur))

c. given

lunar orbit, h = 3.8*10^8 m from earth surface

Radius of earth =R = 6.4*10^6 m

hence

r = h + R is about h = 3.8*10^8 m

f = 1*10^9 Hz

Po = 1 *10^6 W

er = 81

ur = 1

a. magnitude of time averaged poynting vector = <s>

<S> = Eo^2*sqrt(e*er/4(u*ur))

now, intensity at thsi point <S> = Po/4*pi*h^2 = 2.095*10^-4 W/m^2

direction = along the radius of earth, towards it ( directin of motion of the EM wave)

b. speed of wave = c = 1/sqrt(e*er*u*ur) = 3.33*10^7 m/s

c. amplitud eof electric field = Eo

Eo^2*sqrt(e*er/4(u*ur)) = 2.095*10^-4 W/m^2

Eo = 1063.093 V/m

d. amplitud eof magnetic fieldc = Bo

Bo = Eo/c = 3.1924*10^-5 T

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