A satellite in geostationary orbit is used to transmit data via electromagnetic

Question

A satellite in geostationary orbit is used to transmit data via electromagnetic radiation.  The satellite is at a height of 35000.0 km above the surface of the Earth, and we will assume that it has an isotropic power output of 2.75 kW (in practice, satellite antennas transmit signals that are less powerful but more directional).  The satellite dish which will receive the signal sent from the satellite is located on the surface of the Earth directly below the satellite.

(a) What is the intensity of the signal from the satellite at the location of the antenna?
     W/m2

(b) What is the energy density of the electromagnetic radiation from the satellite which is being received by the antenna?
     J/m3

(c) The satellite dish detects the variation in the radiated electric field from the satellite.  What is the amplitude of the radiated electric field vector of the satellite broadcast at the location of the dish?
     N/C


Imagine that the satellite described in the problem introduction is used to transmit television signals. You have a satellite TV reception antenna consisting of a circular dish of radius R which focuses (thereby concentrating) the electromagnetic energy incident from the satellite onto a receiver which has a surface area of 4.20 cm2.

(d) For satisfactory reception, an electric field vector amplitude of 0.220 mV/m must be acheived at the receiver.  What must be the intensity of the received radiation at the receiver?
     W/m2

(e) How large are the required area and radius R of the dish in order to achieve satisfactory reception from this satellite?  HINT: for simplicity, assume that the dish reflects all of the incident radiated energy onto the receiver, and that there are no losses associated with the reception process. The dish has a parabolic curvature, but the radius R refers to the projection of the dish into the plane perpendicular to the direction of the incoming signal.
area of the dish in square cm:      cm2
radius of the dish in cm:      cm

Explanation / Answer

Intensity I = power/areA = Emax^2/2u0C

I = 2.75*10^3/ 4*3.14* 35000*35000*10^6

I = 1.78 *10^-13 W/m^2

b. Emax^2 = I * 2uoC = 1.78*10^-13 * 2*4pi*10^-7 * 3*10^8

Emax =1.15*10^-5 V/m or N/c

energy density u = 0.5 e0E^2 = 0.5*8.85*10^-12 * 1.15^2 *10^-10

u =5.85 *10^-22 J/m^3

c.Emax^2 = I * 2uoC = 1.78*10^-13 * 2*4pi*10^-7 * 3*10^8

Emax =1.15*10^-5 V/m or N/c

d.intensity I = 0.22*0.22*10^-6/2 * 4pi*10^-7 *3*10^8

I =6.42*10^-11 W/m^2

E. I = Emax^2/2 uoC = power/area

area = power * 2uoC/Emax^2

area = 2.75* 1000* 2*4* 3.14*10^-7 *30/0.22*0.22*10^-6

area 4piR^2 = 42.81 *10^12 m^2

Radius R = 42.818*10^12/4pi = 1846.61 Km

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