sity at the center of the receiver, Ko, is to be equal to or less than a specified constant and the fence distance, RF, is to be determined from the fact that the permissible power density there is decided upon in advance. The basic formula used to begin this analysis is that which relates the complex field strength, E0(x0, y0),of the receiver to the complex field strength, E^, yj, of the transmitter. The formula (2) is where k = 2tt/A. The importance of this equation for our problem resides in the fact that the power density is proportional to the square of the magnitude of the field strength. The constant of proportionality cancels out from both sides. In our case we have circular symmetry, so that when polar coordinates are used, only one integral involving a Bessel function remains. The field strength, of course, must be properly phased so that the undesired exponential factor is removed. If we let r equal the running radius in the plane of the transmitter and p the corresponding radius in the plane of the receiver, then after some manipulation we find that Here Po(p.r) is the magnitude of the power density (watts/meter2) in the receiving plane and pM the power density in the transmitting plane. The factor involving Jo is a Bessel function of the first kind and order zero. When r = Ru the as yet undetermined final radius of the transmitter, then the upper limit in equation (4) is simply replaced by R^ An indefinite limit is used in the equation because r is the independent variable in our formulation. Thus r in this formulation takes the place of time, the customary independent variable in optimal control problems. Two more equations are now needed and they derive from the fact that the power radiated by the transmitter or collected by the receiver is simply the integral of the appropriate power density over the area involved. Thus, we write
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