The Physics of Rainbows

Transcription

1 The Physics of Rainbows Prof. Chad A. Middleton CMU Physics Seminar September 13, 2012 Charles A. Bennett, Principles of Physical Optics, 1 st ed., pps Jearl D. Walker, Multiple rainbows from single drops of water and other liquids, American Journal of Physics Vol. 44, No. 5, May John A. Adam, The mathematical physics of rainbows and glories, Physics Reports 356 (2002),

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3 Quiz: The Physics of Rainbows 1. For the primary rainbow, what is the exterior (largest observed angle) color? a) Violet b) Red c) Yellow d) Green 2. For the primary rainbow, what is the interior (smallest observed angle) color? a) Violet b) Red c) Yellow d) Green 3. For the secondary rainbow, what is the exterior color? a) Violet b) Red c) Yellow d) Green 4. For the secondary rainbow, what is the interior color? a) Violet b) Red c) Yellow d) Green 5. Is the region between the primary and secondary rainbows dark or bright? 6. Is the interior region of the primary rainbow dark or bright?

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5 What path should the lifeguard take to minimize her transit time? Lifeguard v i > v t v i v t Drowning swimmer

6 Smallest time interval? v i > v t Shortest distance = smallest time? v i v t

7 Smallest time interval? Shortest H 2 O distance = smallest time? v i > v t v i v t

8 Smallest time interval? v i > v t v i v t Shortest land distance = smallest time?

9 Smallest time interval? Path of smallest time! v i > v t v i v t

10 Smallest time interval? v i > v t v i v t

11 Smallest time interval? v i > v t a v i c v t b

12 c x Smallest time interval? x v i > v t p a2 + x 2 a v i c v t b p b2 +(c x) 2

13 Smallest time interval? x v i > v t p a2 + x 2 a v i To calculate the total time t(x) =t i + t t b p b2 +(c x) 2 v t = p a2 + x 2 v i + p b2 +(c x) 2 v t c x

14 Plot of t(x) vs x t x Time vs Position x The total time is t(x) =t i + t t p a2 + x = 2 + v i p b2 +(c x) 2 v t a, b, c =1 v i =0.8 v t =0.6

15 Smallest time interval? x v i > v t p a2 + x 2 a i v i To calculate the path of smallest time d t(x) =0 yields 1 v i dx x p a2 + x = 1 (c x) p 2 v t b2 +(c x) 2 b c p b2 +(c x) 2 x v t

16 Smallest time interval? n i < n t i n i The path of smallest time n t n i sin i = n t sin n c where v

17 Fermat s Principle: The actual path between two points taken by a beam of light is the one that is traversed in the least time. When light enters a new medium, it s path obeys Snell s Law: n i sin i = n t sin

18 Geometrical setup Sunlight Rays of sunlight are nearly parallel to each other. Suspended H 2 O droplets are nearly spherical due to surface tension. d d is the deviation angle. is the observed angle, measured from the anti-solar direction.

19 Geometrical setup Sunlight At certain observed angles, a particular color will dominate. This angle forms a cone around the anti-solar direction. All raindrops that lie on this cone can contribute to the rainbow => The rainbow is circular!

20 Primary Rainbow: Light is incident at angle i i 1 st refraction transmitted angle given by Snell s Law sin =sin 1 i n where n n t n i n air ' 1.00 n H2O ' 1.33

21 Primary Rainbow: i 1 st refraction, by geometry, same angle

22 Primary Rainbow: i 1 st refraction, 2 nd reflection Law of Reflection, same angle

23 Primary Rainbow: i 1 st refraction, 2 nd reflection Law of Reflection, same angle by geometry, same angle

24 Primary Rainbow: i 1 st refraction, 2 nd reflection, 3 rd refraction by Snell s Law, same i i

25 Calculate the net deviation angle, d i ( i ) d =( i )+ i

26 Calculate the net deviation angle, d i ( i ) (180 2 ) d =( i )+ (180 2 )+ i

27 Calculate the net deviation angle, d i ( i ) (180 2 ) d =( i )+ (180 2 )+ ( i ) i ( i )

28 The net deviation angle, d, is d = i 4 or d ( i ) = i 4 sin 1 sin i n where we used Snell s Law. d = 180 d observed angle

29 An infinite number of parallel sunbeams hit the spherical raindrop, so which ones do we see? Notice: The Cartesian Ray is the ray that has the minimum deviation angle.

30 Plot of θ d vs sin -1 θ i 180 Deviation angle vs sin 1 Θ i The backscattered rays cluster at the minimum deviation angle, yielding an enhanced brightness Θd sin 1 Θ i The net deviation angle is d = i 4 sin 1 sin i n

31 Minimum deviation angle 180 Deviation angle vs sin 1 Θ i 170 Θd To calculate the minimum deviation angle sin 1 Θ i d d =0 d i sin i = r 4 n 2 3 yields

32 Dispersion Different frequencies of light have different indices of refraction. n t,r =1.331 n t,v =1.344 Observed angle vs index of refraction Red r = 42.4 v = 40.5 =1.9 Α Violet Notice: The outside of the primary rainbow is red, whereas the inside is violet! n r! r! 4 n (n) = 2 sin n +4sin n 2

33 Secondary Rainbow d =( i )+ (180 (180 2 )+ 2 )+ ( i ) (180 2 ) i ( i ) (180 2 ) i ( i )

34 The net deviation angle, d, is d = i or d ( i ) = i 6 6 sin 1 sin i n where we again used Snell s Law. d = d 180 observed angle

35 Minimum deviation angle 360 Deviation angle vs sin 1 Θ i Θd To calculate the minimum deviation angle sin 1 Θ i d d =0 d i sin i = yields r 9 n 2 8

36 Secondary Rainbow n t,r =1.331 n t,v = Observed angle vs index of refraction Violet Α 53 r = 50.3 v = 53.8 = Red r n! r! 9 n (n) = sin n 6 sin n 2 Notice: The outside of the secondary rainbow is violet, whereas the inside is red!

37 Why is the interior region of the primary rainbow bright? 40 Observed angle vs Incident angle 30 For the primary rainbow one has backscattering for all angles in the regime: 0 apple apple 42.4 Α deg Θ i rad Normal incidence Primary Grazing incidence Your eye receives backscattered light of all wavelengths from the interior of the primary rainbow => Bright white light!

38 Why is the region between the primary and secondary rainbows dark? For the primary rainbow one has backscattering when 0 apple apple 42.4 For the secondary rainbow one has backscattering when 50.3 apple apple 180 One has ZERO scattering from one or two reflections when 42.4 < < 50.3 => Alexander s dark band! Α deg Α deg Observed angle vs Incident angle Primary Θ i rad Observed angle vs Incident angle Secondary Θ i rad

39 Is a 3 rd (or l th ) rainbow theoretically possible? After allowing for l internal reflections, the net deviation angle is d ( i )=l(180 )+2 i 2(l + 1) sin 1 sin i n To calculate the minimum deviation angle d d d i =0 yields sin i = s (l + 1) 2 n 2 l(l + 2) The first 13 rainbows of water have been observed from a drop suspended in a spectrometer!* *Jearl D. Walker, Multiple rainbows from single drops of water and other liquids, American Journal of Physics Vol. 44, No. 5, May 1976.

40 Why are two rainbows sometimes visible in the sky, but one never sees a third (or fourth)? For the tertiary rainbow (l = 3) r = v = =6.8 So why don t you see it? 1. Each successive Cartesian ray is at a greater incident angle, therefore a reduction in intercepting cross-sectional area. 2. Larger for each successive rainbow. 3. Loss of light due at each successive reflection.

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