Abstract
Case II. Signal at an Angle to Antenna, Zero Loss Antenna. So far we have considered only signal waves traveling in the direction of the antenna. To calculate the directive properties of the antenna, we must find the effect of a signal of the same wave length and intensity coming at an angle to the antenna. In the first place there will be a difference in the electromotive force E <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> induced in a unit length of wire. In Fig. 30a we imagine ourselves looking down at two horizontal wires, each of unit length, situated in the midst of a signal wave whose magnetic lines are shown as dotted lines in the figure. Only the magnetic lines in a very thin layer immediately over the plane of the wires are shown. These magnetic lines have a horizontal movement indicated by the arrow, and a downward movement resulting from the forward tilt of the wave front. The horizontal movement causes no cutting of the conductor. Owing to the downward movement, the wires will be cut by all of the lines which cross them in the figure. The numbers of magnetic lines which cross wires No. 1 and No. 2 are in the ratio 1 to cos θ, and the induced electromotive force per unit length will be in the same ratio. Another point of view is illustrated in Fig. 30b, in which we imagine ourselves as looking at the ends of the magnetic lines, which now appear as dots. Here the same two wires are shown, and the actual direction of motion of the magnetic lines is indicated. Only those lines within the region a will cut wire No. 2 which appears in this projection as A′ − B′, while all those in region b will cut No. 1, and the ratio of a to b is again cos θ. We shall therefore multiply by cos θ to take account of the difference in induced electromotive force. That is to say if a signal coming in the direction of the antenna induces E <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> volts per kilometer, a signal of the same intensity coming from an angle θ to the antenna will induce E <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> cos θ volts per kilometer.
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