Lec25 - t = tOP O tOO = (tOP tOD (n1 ) + (tP O tDO ) (tV W...

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Δ t = t OPO ± - t OO ± = t OP + t PO ± - t OD ( n 1 ) - t DO ± - ( t VW ( n 2 ) - t VW ( n 1 )) =( t OP - t OD ( n 1 )) + ( t PO ± - t DO ± ) - ( t VW ( n 2 ) - t VW ( n 1 )) = n 1 h 2 2 s + n 1 h 2 2 s ± - ( n 2 - n 1 ) ± VW and we still need to evaluate the distance ± VW . To evaluate this distance, notice that it is the distance on the axis between two spheres with radii R 1 and R 2 (non–concentric!) that intersect. From ± PDC 2 , R 2 - ± DC 2 = h 2 2 R 2 . But R 2 - ± DC 2 is also the distance ± WD . Therefore, ± WD = h 2 2 R 2 . Similarly, from ± PDC 1 , R 1 - ± DC 1 = h 2 2 R 1 = ± VD . Next, ± VW = ± VD - ± WD = h 2 2 R 1 - h 2 2 R 2 . We now require that Fermat’s Principle holds, or Δ t = 0. Therefore, n 1 h 2 2 s + n 1 h 2 2 s ± =( n 2 - n 1 ) ² h 2 2 R 1 - h 2 2 R 2 ³ n 1 s + n 1 s ± =( n 2 - n 1 ) ² 1 R 1 - 1 R 2 ³ . Now take the limit s ± →∞ , which would give us an s value we identify with f . Therefore, n 1 f =( n 2 - n 1 ) ² 1 R 1 - 1 R 2 ³ , 58
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or 1 f = ± n 2 n 1 - 1 ²± 1 R 1 - 1 R 2 ² . We now defne n := n 2 /n 1 , so that 1 f =( n - 1) ± 1 R 1 - 1 R 2 ² . This last relation is sometimes called the lens maker’s Formula . (Notice, it is a Formula, not an equation.) IF we now take s →∞ we identiFy s ± with f ± . Then, n 1 f ± =( n 2 - n 1 ) ± 1 R 1 - 1 R 2 ² 1 f ± =( n - 1) ± 1 R 1 - 1 R 2 ² which is exactly what we Found For 1 /f . ThereFore, 1 f ± =( n - 1) ± 1 R 1 - 1 R 2 ² = 1 f Notice that we fnd that f = f ± . This is a particular case oF
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This note was uploaded on 07/19/2011 for the course PH 113 taught by Professor Liorburko during the Spring '09 term at University of Alabama - Huntsville.

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Lec25 - t = tOP O tOO = (tOP tOD (n1 ) + (tP O tDO ) (tV W...

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