A communications engineer is designing a system to transfer data using a microwa

Question

A communications engineer is designing a system to transfer data using a microwave beam from one city building to another city building 10 km away. (a) Define the Poynting vector of an electromagnetic wave and explain its meaning. (3 marks) (b) Use the Poynting vector to find the electric field amplitude of the microwave beam at the receiving site, if the average microwave power crossing unit area at the re ceiver is 10 mWm2 (4 marks) (c) Find the electric field amplitude at another building 20 km away from the transmit- ting dish, giving your reasoning. (3 marks)

Explanation / Answer

In this solution, i am using vector notation as vec like vec(a), which means vector a.

(a) The Poynting vector is defined as vec(S) = vec(E) x vec(H), this came out because of the consequence of the poynting theorem which states that the work done on the charges by the electromagnetic force is equal to the decrease in energy stored in the field, less the energy that flowed out through the surface. Also, as we know that, poynting vector is energy per unit time per unit area which can also be defined by using the relation B = u0H, where u0 is the permeability of free space, then the poynting vector S becomes

vec(S) = 1/u0(vec(E) x vec(B)).

(b) To find the amplitude of electric field, where S = 10 mW m-2 is given, since the signal is transmitting through an electromagnetic wave, for an electromagnetic wave we know the relation between the amplitudes of electric and magnetic fields E0 = cB0 or B0 = E0/c. In the formula of the poynting vector, in terms of the amplitudes of electric and magnetic fields

S = 1/u0(E0 x B0) = 1/u0(E0 x E0/c)

10 = 1/4 x pi x 10-7 (E02/c)

10 = 0.07 x 107 (E02 x 0.33 x 10-8)

10 = 0.0231 x 10-1 (E02)

10/10-1 = 0.0231 (E02)

E02 = 100/ 0.0231 = 4329.004

E0 = 65.795 = 65.80m ( since the amplitude is disturbance which does not have any direction so i am not indicating the direction)

(c) To find the amplitude of electric field at another building 20km away from the transmitting dish, since the distance is 20 km from the transmitting that means the wave have to repeat its period 20 times to reach the desired location, then total amplitude of this transmitting wave is 20, since the wave motion is periodic and its motion is being repeated at certain time intervals, so to reach the wave destination which is 20 km away the wave have to propagate 20 times to reach that building with large amplitude so the total amplitude is the sum of the all amplitudes in the motion of a wave, therefore, taking 20 as the total amplitude of the electromagnetic wave . Which means,

A2 = E02 +B02, where A is the total amplitude of the electromagnetic wave and E0 and B0 are the amplitudes of electric and magnetic fields respectively. So,

A2 = E02 + E02/c2 , since E0 = cB0 for an electromagnetic wave,

c2 x(20)2 = c2E02 + E02

400c2 = E02(c2 + 1)

400c2/(c2 + 1) = E02

E02 = 360

E02 = 60km

  

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