Solution4 - Physics 458 Problem Set 4 Solutions Graded...

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Physics 458- Problem Set 4 Solutions Graded Problems 5.3, 5.11, 5.14,5.22, 5.65 5.3) We are given that for the ellipsoid, 2( ) a b (Eq 1). We need to show that the optical path from F 1 to A to D (where D is on the screen but at the same distance from the axis as A) is independent of angle. 2 1 1 20 2 1 20 1 2 ( ) (2 ) 2 2 2 path O n n a x n v n b a n v n b a b a a a n a x a x a x (Eq. 2) We use the law of cosines and the trig relation for x, 2 2 2 (2 ) 4 ( ) ( ) b b Cos x Cos   (Eq. 3)
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Solving these two equations for and , we find (2 ) 1 ( ) a b a a b Cos (Eq. 4) So now we need to show that: 1 2 ( ) constant n a n a x a a Cos (Eq. 5) If we substitute Eq. 4 for this becomes, 1 2 ( ) n a b a Cos a b n (Eq. 6) For any ratio 1 2 n n , we can find values of a and b so that Eq. 6 is true and so the light from F
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