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Figure B.7
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to the axis (ZZ ) is constant. It is this property which results in a parallel beam being radiated when a source is placed at the focus. It will be seen that when = 90 , r = SP = p. But SP = eSZ = 2f (since e = 1), and therefore, p = 2f. Thus, the radius as given by Eq. (B.4) can be written as
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(B.33)
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This is essentially the same as Eq. (6.27) of the text, where identity 1 + cos = 2cos2( /2). Thus is used to replace r, and = 180 , and use is made of the trigonometric
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(B.34)
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For the situation shown in Fig. B.8, where D is the diameter of a parabolic reflector, Eq. (B.34) gives
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(B.35)
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But from Fig. B.8 it is also seen that
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Figure B.8
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Figure B.9 where use is made of the double angle formula sin 0 = 2 sin( 0/2)cos( 0/2). Substituting for 0 in Eq. (B.34) and simplifying gives
(B.36)
This is Eq. (6.29) of the text. It is sometimes useful to be able to locate the focal point, knowing the diameter D and the depth d of the dish. From the property of the parabola, 0 = PQ = f + d. From Fig. B.8 it is also seen that . Substituting for 0 and simplifying yields
(B.37)
With the zero origin of the xy coordinate system at the vertex (point A) of the parabola, the line PQ becomes equal to x + f (Fig. B.9), and y = r2 (x f)2. These two results can be combined to give the equation for the parabola:
(B.38)
The Hyperbola
For the hyperbola, e > 1. The curve is sketched in Fig. B.10. Equation (B.4) still applies, but it will be seen that for cos = 1/e the radius goes to infinity; in other words, the hyperbolic curve does not close on itself in the way that the ellipse does but lies parallel to the radius at this value of . In constructing the ellipse, because e was less than unity, it was possible to find a point A to the right of S for which the ratio SA /A Z = e applied in addition to the ratio e = SA/AZ. With the hyperbola, because e > 1, a point A
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Figure B.10 to the left of S can be found for which e = SA /A Z in addition to e = SA/AZ. By setting 2a equal to the distance A A and making the midpoint of A A equal to the x, y coordinate origin O as shown in Fig. B.10, it is seen that 2OS = SA + SA . But SA = eA Z and SA = eZA. Hence,
Hence,
(B.39)
Also, SA SA = 2a, and hence
Therefore,
(B.40)
Point P on the curve can now be given in terms of the xy coordinates. Referring to Fig. B.11, S is at point ae, and
Also,
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Figure B.11 Combining these two equations and simplifying yields
(B.41)
where in this case
(B.42)
By plotting values it will be seen that a symmetrical curve results, as shown in Fig. B.12, and that there is a second focus at point S . An important property of the hyperbola is that the difference of the two focal distances is a constant. Referring to Fig. B.12, S P = ePQ and SP = ePQ.
Figure B.12
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Hence,
(B.43)
An application of this property is shown in Fig. 6.23a of the text which is redrawn in Fig. B.13. By placing the focus of the parabolic reflector at the focus S of the hyperbola and the primary source at the focus S , the total path length S P + PP is equal to 2a + SP + PP or 2a + SP . But, as shown previously in Fig. B.7, the focusing properties of the parabola rely on there being a constant path length SP + P P , and adding the constant 2a to SP does not destroy this property. The double-reflector arrangement can be analyzed in terms of an equivalent parabola. The equivalent parabola has the same diameter as the real parabola and is formed by the locus of points obtained at P which is the intersection of S P produced to P and P P which is parallel to the x axis, as shown in Fig. B.14. The focal distance of the equivalent parabola is shown as f e and of the real parabola as f. Looking from the focus S to the real parabola, one sees that
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