Points A and B are 60.0m apart along an east-west line. At each of these points,
ID: 1299645 • Letter: P
Question
Points A and B are 60.0m apart along an east-west line. At each of these points, a radio transmitter is emitting a 14.0MHz signal horizontally. These transmitters are in phase with other and emit their beams uniformly in a horizontal plane. A receiver is taken 0.650km north of the AB line and initially placed at point C, directly opposite the midpoint of AB. The receiver can be moved only along an east-west direction but, due to its limited sensitivity, it must always remain within a range so that the intensity of the signal it receives from the transmitter is no less than 1/4 of its maximum value.
1) How far from point C (along an east-west line) can the receiver be moved and always be able to pick up the signal?
Explanation / Answer
lambda = c/f = 3*10^8/(14*10^6) = 21.428 m
I = I0/4 = I0*cos^2(A/2)
cos^2(A/2) = 1/4
cos(A/2) = (+or-)1/2
A/2 = pi/3 ; A = 2*pi/3 or
A/2 = 2*pi/3; A = 4*pi/3
A= (delta r)*2*pi/lambda
delta r = y*L/D
A = 2*pi*y*L/(D*lambda)
for A = 2*pi/3
2*pi/3 = 2*pi*y*L/(D*lambda) =
y = D*lambda/(3*L ) = 650*21.42/(3*60) = 77.35 m
for A = 4*pi/3
4*pi/3 = 2*pi*y*L/(D*lambda) =
y = 2*D*lambda/(3*L ) =2* 650*21.42/(3*60) = 154.70 m
min is 77.35 m
so,
the receiver can be moved 77.35 meters along the east-west line and still detect thesignal