- Email: [email protected]
Refinement of the theory of formation of rainbow with incorporation of the refined unambiguous angles of incidence, reflection, and refraction
Optik 157 (2018) 644–650
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Refinement of the theory of formation of rainbow with incorporation of the refined unambiguous angles of incidence, reflection, and refraction Pramode Ranjan Bhattacharjee ∗ Retired Principal, Kabi Nazrul Mahavidyalaya, Sonamura, Tripura 799131, India
a r t i c l e
i n f o
Article history: Received 8 October 2017 Accepted 19 November 2017 Keywords: Rainbow Primary rainbow Secondary rainbow Reflection Total internal reflection Refraction Dispersion
a b s t r a c t This paper is concerned with one of the most common atmospheric optical phenomenon − the rainbow. In order to get rid of the ambiguity present in the traditional theory of formation of rainbow which is based on the traditional ambiguous angles of incidence, reflection, and refraction, a novel unambiguous theory of formation of rainbow which makes use of the refined unambiguous angles of incidence, reflection, and refraction has been offered. As a result, the present contribution will enhance and sophisticate the relevant optical physics literature there by enriching the same as well. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction Launching of the new world of geometrical optics [2], which is exclusively based on the refined unambiguous angles of incidence, reflection, and refraction, demands that all theoretical discussions or derivations of the traditional geometrical optics must proceed in the light of the refined unambiguous angles of incidence, reflection, and refraction instead of making use of the traditional ambiguous angles of incidence, reflection, and refraction to enhance and sophisticate the relevant optical physics literature as well as to bring preciseness in the relevant field of study. With that point in mind, the deviation problems in ray optics and the dispersive power of a prism have been considered in detail in [9] and [10] respectively in the light of the refined unambiguous angles of incidence, reflection, and refraction to obtain novel interesting results. This paper considers the theory of formation of rainbow, which has been dealt with in the traditional literature [3–8] on the basis of the long-running ambiguous angles of incidence, reflection, and refraction, as a result of which such a theoretical treatment is not at all free from ambiguity as well. With a view to getting rid of the ambiguity present in the traditional theory of formation of rainbow, an unambiguous theory of formation of rainbow has been offered in this paper with the incorporation of the refined unambiguous angles of incidence, reflection, and refraction for the development of: (i) novel expression for the net deviation suffered by an incident ray in emerging out of a raindrop, and (ii) novel expression for the primary angle of incidence for the minimum deviation to occur, in terms of the refractive index of water and the total number of total internal reflections taking place inside the raindrop. As a result, the present contribution will bring preciseness and sophistication in the relevant field of the optical physics literature there by enhancing and enriching the same as well.
∗ Correspondence to: 5 Mantri Bari Road, P.O. Agartala, Tripura 799 001, India. E-mail address: [email protected] https://doi.org/10.1016/j.ijleo.2017.11.106 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
P.R. Bhattacharjee / Optik 157 (2018) 644–650
645
2. Definitions and preliminaries From the view point of the interest of readership, the following three definitions of the refined unambiguous angles of incidence, reflection, and refraction are being reproduced below from [2]. Refined unambiguous angle of incidence (i): The angle of incidence (i) is the smaller of the angles between the vectors i and n subject to the condition that /2 < i ≤ , so long as the case considered is a reflection (or a refraction of light as it passes from a rarer to a denser medium). If however it is a case of refraction as light passes from a denser medium to a rarer medium, the angle i must be bounded by the relation 0 ≤ i < /2. Refined unambiguous angle of reflection (r): The angle of reflection (r) is the smaller of the angles between the vectors r and n subject to the condition that 0 ≤ r < /2. Refined unambiguous angle of refraction (R): The angle of refraction (R) is the smaller of the angles between the vectors n and R subject to the condition that, /2 < R ≤ when the ray of light passes from a rarer medium to a denser medium, or 0 ≤ R < /2 when the ray of light passes from a denser medium to a rarer medium. Furthermore, in order to enhance the readability of the paper the two laws reported in [1] are also being reproduced below. The generalized vectorial law of reflection: If i and r represent unit vectors along the directions of incident ray and reflected ray respectively and if n represents unit vector along the direction of the positive unit normal to the reflector at the point of incidence then n×i = n×r The generalized vectorial law of refraction: If i and R represent unit vectors along the directions of the incident ray and refracted ray of particular colour respectively and if n represents unit vector along the direction of the positive unit normal to the surface of separation at the point of incidence then n×i = (n×R) where = Refractive index of the second optical medium with respect to the first optical medium for the particular colour of light under consideration. 3. The novel unambiguous theory of formation of rainbow Let us consider Fig. 1, in which the path of a ray of light incident on spherical raindrop and undergoing one total internal reflection is shown. O is the centre of the raindrop. The ray SA is incident on the surface of the raindrop at the point A. Then it gets refracted inside the drop along AB, undergoes total internal reflection at the point B and moves along BC, the points B and C lying on the surface of the raindrop. The positive unit normal vectors to the surface of the raindrop at the points A, B, and C are n1 , n2 , and n3 respectively. At the point C, the ray BC after undergoing refraction emerges along CD. The refined unambiguous angles of incidence and refraction at the point A are taken as i and r respectively. Similarly the refined unambiguous angles of incidence and reflection at the point B are respectively r1 and r2 . Furthermore, at the point C, refined unambiguous angles of incidence and refraction are respectively r1 and i1 . Now, the deviation due to refraction at the point A = r − i. Again the deviation due to total internal reflection at the point B = r2 − r1 Furthermore, the deviation again due to refraction at the point C = i1 − r1 . Thus the net deviation (ó) suffered by the incident ray in emerging out of the raindrop is given by, ó = (r − i) + (r2 − r1 ) + (i1 − r1 ) or, ó = − i + i1 + r − 2r 1 + r 2
(1)
Now, it can be readily seen from Fig. 1 that, we have, r1 = − r, r2 = − r1 = − ( − r) = r, and from the principle of reversibility of light for refraction at the point A, it also follows that i1 = − i. Hence from the relation (1), we have, ó = −i + ( − i) + r − 2 ( − r) + r = − − 2i + 4r Thus the expression for the net deviation (ó) suffered by the incident ray in emerging out of the raindrop is given by, ó = − − 2i + 4r
(2)
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P.R. Bhattacharjee / Optik 157 (2018) 644–650
Fig. 1. Diagram showing the path of a ray of light entering into a raindrop and emerging out of it after suffering only one total internal reflection inside the drop.
If instead of only one total internal reflection, there are ‘n’ number of total internal reflections, the deviation due to reflections will be n(r2 − r1 ). Hence the net deviation (ó) suffered by the incident ray as it emerges out of the raindrop will be given by, ó = (r − i) + n (r2 − r1 ) + (i1 − r1 ) Making use of the relations, r1 = − r, r2 = − r1 = − ( − r) = r, and i1 = − i, which have been mentioned earlier, the above expression for the net deviation becomes
ó = (r − i) + n r − ( − r) + ( − i) − ( − r) or, ó = −n − 2i + 2r (n + 1)
(3)
The relation (3) which gives the expression for the net deviation suffered by an incident ray when there exist ‘n’ number of total internal reflections inside the raindrop is novel and in addition to ‘n’, it involves the refined unambiguous angles of incidence and refraction corresponding to the primary incident ray. Now,
and
dó dr = −2 + 2 (n + 1) di di
d2 r d2 ı = 2 (n + 1) 2 2 di di
P.R. Bhattacharjee / Optik 157 (2018) 644–650
So,
dó di
= 0 implies that,
Again,
d2 ó di2
dr di
=
1 n+1
= 2 (n + 1) 1 at dr = n+1
d2 r di2
di
647
(4) 1 at dr = n+1 di
Now, for refraction at the point A, we have from the generalized vectorial law of refraction, sin i = sin r
(5)
Hence, cos i = cos r
or,
dr di
(6)
dr cos i = cos r di
Then,
− cos r sin i + cos i sin r dr d2 r di = 2 di 2 cos2 r
Now,
d2 r di2
=
− cos r sin i + cos i sin r
1 n+1
2 cos2 r
1 at dr = n+1 di
or,
d2 r di2
= 1 at dr = n+1
−n cos r sin i + sin (r − i) (n + 1) 2 cos2 r
di
Now, since here,
2
< i ≤ , and
2
< r ≤ , the angle (r − i) will always lie in the first quadrant. Hence it follows from the
above expression that, the quantity
d2 r di2
1 at dr = n+1
is always positive.
di
Thus it follows from the relation (4) that, the quantity,
d2 ó di2
is always positive.
1 at dr = n+1 di
1 Thus the net deviation ó is minimum when dr = n+1 · di It then follows from the relation (6) that, when the net deviation is minimum we must have,
cos i = cos r
or, cos i =
1 n+1
− n+1
This is because since
2
1 − sin2 r
< r ≤ , cos r is negative and hence cos r = −
− or, cos i = n+1
1−
or, cos i = −
2 − 1 n2 + 2n
sin2 i 2
or, (n + 1)2 cos2 i = 2 − 1 − cos2 i
1 − sin2 r
(7)
[Since < i ≤ , cos i is negative] 2 This relation (7) is also novel and it involves the refined unambiguous angle of incidence. If only one total internal reflection takes place inside the raindrop, we have, n = 1 and hence under this condition the relation (7) reduces to
cos i = −
2 − 1 3
(8)
Again if two total internal reflections occur inside the raindrop, we have, n = 2 and hence under this condition the relation (7) reduces to
cosi = −
2 − 1 8
(9)
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Thus if the value of of water for a particular colour of incident light as well as the total number (n) of total internal reflection taking place inside the raindrop are known, the value of the refined unambiguous angle of incidence (i) can be calculated from the relation (7). Knowing the value of ‘i’, the corresponding value of the refined unambiguous angle of refraction ‘r’ can be calculated from the relation (5). Since such values of ‘i’ and ‘r’ correspond to the condition of minimum deviation, the value of minimum deviation can then be calculated by using the relation (3) by making use of these values of ‘i’ and ‘r’. Now, let us first consider the case of primary rainbow for which n = 1. Then considering red light and assuming that red = 1.329 for water droplet, we then obtain from the relation (7), cos i = −0.50538 = − cos 59.643420 = cos(1800 − 59.643420 ) or,i = 120.35658◦ Hence from the relation (5), we have,
sin r = 0.64928 = sin 1800 − 40.487550
or,r = 139.51245◦ Putting these values of ‘i’ and ‘r’ in the relation (3) and remembering that here n = 1, we then obtain the value of minimum deviation (ómin ) for a ray of red colour as ómin = 137.33664◦ = 137◦ , (nearly) = 180◦ − 43◦ Now, considering violet light and taking vio = 1.349, we obtain from the relation (7) for the case of only one total internal reflection (i.e. for n = 1), cos i = −0.52275 = cos(1800 − 58.483140 ) or,i = 121.51686◦ Hence from the relation (5) we obtain,
sin r = 0.63194 = sin 1800 − 39.193360
or,r = 140.80664◦ Using these values of ‘i’ and ‘r’ and remembering that in this case, n = 1, we obtain from the relation (3), the value of minimum deviation (ómin ) for a ray of violet colour as ómin = 140.19284◦ = 140◦ , (nearly) = 180◦ − 40◦ Now let us consider the case of secondary rainbow for which n = 2. Then considering red light and taking red = 1.329, we have in this case from the relation (7), cos i = −0.30948 = cos(1800 − 71.971880 ) or,i = 108.02812◦ Hence from the relation (5), we get,
sin r = 0.71550 = sin 1800 − 46.684510
or,r = 134.31549◦ Putting these values of ‘i’ and ‘r’ in the relation (3) and remembering that here n = 2, we then obtain the value of minimum deviation (ómin ) for a ray of red colour as ómin = 229.8367◦ = 230◦ , (nearly) = 180◦ + 50◦ Finally considering violet light and taking vio = 1.349, we obtain from the relation (7) for the case of two total internal reflections (i.e. for n = 2), cos i = −0.32012 = cos(1800 − 71.329980 ) or,i = 108.67002◦
P.R. Bhattacharjee / Optik 157 (2018) 644–650
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Fig. 2. Diagram for explaining the formation of rainbow when only one total internal reflection takes place inside the raindrop.
Hence from the relation (5) we obtain,
sin r = 0.70228 = sin 1800 − 44.610350
or,r = 135.38965◦ Using these values of ‘i’ and ‘r’ and remembering that in this case, n = 2, we obtain from the relation (3), the value of minimum deviation (ómin ) for a ray of violet colour as ómin = 234.99788◦ = 235◦ , (nearly) = 180◦ + 55◦ 4. Explanation of the formation of rainbow When light rays from the sun fall on water droplets in the atmosphere, each of the incident rays of white light will get dispersed into its constituent colours. We have seen above that due to only one total internal reflection inside the raindrop, the red components suffer a deviation having minimum value equal to (180◦ − 43◦ ). If we consider a cone having semivertical angle 43◦ with the water droplet at its vertex and its axis coinciding with the direction of the sun rays as shown in Fig. 2, then the red rays suffering minimum deviation will all lie on the curved surface of this cone. All other red rays which will suffer deviations greater than the minimum deviation will lie inside the curved surface of the above cone. Now that the rate of change of deviation with angle of incidence is very small in the neighbourhood of minimum deviation, there must be more concentration of the red rays close to the curved surface of the cone as a result of which the intensity of red rays will be maximum in the direction of minimum deviation. Again we have seen earlier by considering one total internal reflection that, for violet components, the minimum deviation is (180◦ − 40◦ ) . Hence, as before it can be said that the intensity of violet rays will be maximum close to the curved surface of a cone of semi-vertical angle equal to 400 . In between the curved surfaces of the two aforesaid cones of semi-vertical angles 430 and 400 , intermediate spectral colours will be found to concentrate. Now, considering two total internal reflections we have found earlier that for red rays, the value of minimum deviation is (180◦ + 50◦ ), and also for violet rays, the minimum deviation has been found to be (180◦ + 55◦ ). Thus as shown in Fig. 3, in this case, concentration of red rays will be found close to the curved surface of the cone of semi-vertical angle 500 and violet rays will be found to be concentrated close to the curved surface of the cone of semi-vertical angle 550 . Red rays suffering deviation greater than the minimum deviation, will lie outside the curved surface of the cone of semi-vertical angle 500 . In a similar manner violet rays in this case will be found to be concentrated close to the curved surface of the cone of semi-vertical angle 550 . Violet and other rays suffering deviation greater than the minimum deviation, will lie outside the curved surface of the cone of semi-vertical angle 550 . All other intermediate colours will lie in the conical space in between the cones of semi-vertical angles 500 and 550 . But there will exist no ray within the cone of semi-vertical angle 500 . 5. Conclusion A novel unambiguous theory of the formation of rainbow has been offered in this paper. The theory presented is unambiguous because of the fact that unlike the traditional theory of formation of rainbow [3–8], which makes use of the traditional ambiguous angles of incidence, reflection, and refraction [1], it makes use of the refined unambiguous angles of incidence,
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P.R. Bhattacharjee / Optik 157 (2018) 644–650
Fig. 3. Diagram for explaining the formation of rainbow when there is two total internal reflections inside the raindrop.
reflection, and refraction [2]. The incorporation of the refined unambiguous angles of incidence, reflection, and refraction makes the theory novel as well, there by leading to novel expressions: (i) for the net deviation of an incident ray in coming out of a raindrop after suffering ‘n’ number of total internal reflections inside the raindrop, and (ii) for the primary angle of incidence corresponding to minimum deviation in terms of the refractive index of water and the total number of total internal reflections taking place inside the raindrop. On account of preciseness and sophistication involved in the definitions of the refined unambiguous angles of incidence, reflection, and refraction, the development of such a theory of formation of rainbow with the incorporation of these three refined unambiguous angles, will enhance the relevant optical physics literature there by enriching the same as well. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
P.R. Bhattacharjee, The generalized vectorial laws of reflection and refraction, Eur. J. Phys. 26 (5) (2005) 901–911. P.R. Bhattacharjee, Launching of the new world of geometrical optics, Optik 126 (24) (2015) 5134–5138. R.A. Herman, A Treatise on Geometrical Optics, The University Press, Cambridge, 1900, pp. 291–293. P.N. Tverskoi, Physics of the atmosphere, a course in Meteorology, 1965, 470–472, NASA Technical Document, Identifier: nasa techdoc 19660014173. Mathematics of rainbows – Feature column, retrieved on 03 October 2017 from: www.ams.org/samplings/feature-column/fcarc-rainbows. L.V. Tarasov, A.N. Tarasova, Discussions on Refraction of Light, Mir Publishers, Moscow, 1984, pp. 119–122. R.S. Heath, A Treatise on Geometrical Optics, The University Press, Cambridge, 1887, pp. 340–347. M. Nelcon, P. Parker, Advanced Level Physics, Heinemann Educational Books Ltd., London, 1958, pp. 475–476. P.R. Bhattacharjee, Deviation problems in ray optics in the light of the refined unambiguous definitions of angles of incidence, reflection and refraction, Optik 125 (16) (2014) 4257–4261. [10] P.R. Bhattacharjee, Dealing with the dispersive power of prism on the basis of the refined unambiguous angles of incidence and refraction, Optik 127 (4) (2016) 2177–2182.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Refinement of the theory of formation of rainbow with incorporation of the refined unambiguous angles of incidence, reflection, and refraction Pramode Ranjan Bhattacharjee ∗ Retired Principal, Kabi Nazrul Mahavidyalaya, Sonamura, Tripura 799131, India
a r t i c l e
i n f o
Article history: Received 8 October 2017 Accepted 19 November 2017 Keywords: Rainbow Primary rainbow Secondary rainbow Reflection Total internal reflection Refraction Dispersion
a b s t r a c t This paper is concerned with one of the most common atmospheric optical phenomenon − the rainbow. In order to get rid of the ambiguity present in the traditional theory of formation of rainbow which is based on the traditional ambiguous angles of incidence, reflection, and refraction, a novel unambiguous theory of formation of rainbow which makes use of the refined unambiguous angles of incidence, reflection, and refraction has been offered. As a result, the present contribution will enhance and sophisticate the relevant optical physics literature there by enriching the same as well. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction Launching of the new world of geometrical optics [2], which is exclusively based on the refined unambiguous angles of incidence, reflection, and refraction, demands that all theoretical discussions or derivations of the traditional geometrical optics must proceed in the light of the refined unambiguous angles of incidence, reflection, and refraction instead of making use of the traditional ambiguous angles of incidence, reflection, and refraction to enhance and sophisticate the relevant optical physics literature as well as to bring preciseness in the relevant field of study. With that point in mind, the deviation problems in ray optics and the dispersive power of a prism have been considered in detail in [9] and [10] respectively in the light of the refined unambiguous angles of incidence, reflection, and refraction to obtain novel interesting results. This paper considers the theory of formation of rainbow, which has been dealt with in the traditional literature [3–8] on the basis of the long-running ambiguous angles of incidence, reflection, and refraction, as a result of which such a theoretical treatment is not at all free from ambiguity as well. With a view to getting rid of the ambiguity present in the traditional theory of formation of rainbow, an unambiguous theory of formation of rainbow has been offered in this paper with the incorporation of the refined unambiguous angles of incidence, reflection, and refraction for the development of: (i) novel expression for the net deviation suffered by an incident ray in emerging out of a raindrop, and (ii) novel expression for the primary angle of incidence for the minimum deviation to occur, in terms of the refractive index of water and the total number of total internal reflections taking place inside the raindrop. As a result, the present contribution will bring preciseness and sophistication in the relevant field of the optical physics literature there by enhancing and enriching the same as well.
∗ Correspondence to: 5 Mantri Bari Road, P.O. Agartala, Tripura 799 001, India. E-mail address: [email protected] https://doi.org/10.1016/j.ijleo.2017.11.106 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
P.R. Bhattacharjee / Optik 157 (2018) 644–650
645
2. Definitions and preliminaries From the view point of the interest of readership, the following three definitions of the refined unambiguous angles of incidence, reflection, and refraction are being reproduced below from [2]. Refined unambiguous angle of incidence (i): The angle of incidence (i) is the smaller of the angles between the vectors i and n subject to the condition that /2 < i ≤ , so long as the case considered is a reflection (or a refraction of light as it passes from a rarer to a denser medium). If however it is a case of refraction as light passes from a denser medium to a rarer medium, the angle i must be bounded by the relation 0 ≤ i < /2. Refined unambiguous angle of reflection (r): The angle of reflection (r) is the smaller of the angles between the vectors r and n subject to the condition that 0 ≤ r < /2. Refined unambiguous angle of refraction (R): The angle of refraction (R) is the smaller of the angles between the vectors n and R subject to the condition that, /2 < R ≤ when the ray of light passes from a rarer medium to a denser medium, or 0 ≤ R < /2 when the ray of light passes from a denser medium to a rarer medium. Furthermore, in order to enhance the readability of the paper the two laws reported in [1] are also being reproduced below. The generalized vectorial law of reflection: If i and r represent unit vectors along the directions of incident ray and reflected ray respectively and if n represents unit vector along the direction of the positive unit normal to the reflector at the point of incidence then n×i = n×r The generalized vectorial law of refraction: If i and R represent unit vectors along the directions of the incident ray and refracted ray of particular colour respectively and if n represents unit vector along the direction of the positive unit normal to the surface of separation at the point of incidence then n×i = (n×R) where = Refractive index of the second optical medium with respect to the first optical medium for the particular colour of light under consideration. 3. The novel unambiguous theory of formation of rainbow Let us consider Fig. 1, in which the path of a ray of light incident on spherical raindrop and undergoing one total internal reflection is shown. O is the centre of the raindrop. The ray SA is incident on the surface of the raindrop at the point A. Then it gets refracted inside the drop along AB, undergoes total internal reflection at the point B and moves along BC, the points B and C lying on the surface of the raindrop. The positive unit normal vectors to the surface of the raindrop at the points A, B, and C are n1 , n2 , and n3 respectively. At the point C, the ray BC after undergoing refraction emerges along CD. The refined unambiguous angles of incidence and refraction at the point A are taken as i and r respectively. Similarly the refined unambiguous angles of incidence and reflection at the point B are respectively r1 and r2 . Furthermore, at the point C, refined unambiguous angles of incidence and refraction are respectively r1 and i1 . Now, the deviation due to refraction at the point A = r − i. Again the deviation due to total internal reflection at the point B = r2 − r1 Furthermore, the deviation again due to refraction at the point C = i1 − r1 . Thus the net deviation (ó) suffered by the incident ray in emerging out of the raindrop is given by, ó = (r − i) + (r2 − r1 ) + (i1 − r1 ) or, ó = − i + i1 + r − 2r 1 + r 2
(1)
Now, it can be readily seen from Fig. 1 that, we have, r1 = − r, r2 = − r1 = − ( − r) = r, and from the principle of reversibility of light for refraction at the point A, it also follows that i1 = − i. Hence from the relation (1), we have, ó = −i + ( − i) + r − 2 ( − r) + r = − − 2i + 4r Thus the expression for the net deviation (ó) suffered by the incident ray in emerging out of the raindrop is given by, ó = − − 2i + 4r
(2)
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P.R. Bhattacharjee / Optik 157 (2018) 644–650
Fig. 1. Diagram showing the path of a ray of light entering into a raindrop and emerging out of it after suffering only one total internal reflection inside the drop.
If instead of only one total internal reflection, there are ‘n’ number of total internal reflections, the deviation due to reflections will be n(r2 − r1 ). Hence the net deviation (ó) suffered by the incident ray as it emerges out of the raindrop will be given by, ó = (r − i) + n (r2 − r1 ) + (i1 − r1 ) Making use of the relations, r1 = − r, r2 = − r1 = − ( − r) = r, and i1 = − i, which have been mentioned earlier, the above expression for the net deviation becomes
ó = (r − i) + n r − ( − r) + ( − i) − ( − r) or, ó = −n − 2i + 2r (n + 1)
(3)
The relation (3) which gives the expression for the net deviation suffered by an incident ray when there exist ‘n’ number of total internal reflections inside the raindrop is novel and in addition to ‘n’, it involves the refined unambiguous angles of incidence and refraction corresponding to the primary incident ray. Now,
and
dó dr = −2 + 2 (n + 1) di di
d2 r d2 ı = 2 (n + 1) 2 2 di di
P.R. Bhattacharjee / Optik 157 (2018) 644–650
So,
dó di
= 0 implies that,
Again,
d2 ó di2
dr di
=
1 n+1
= 2 (n + 1) 1 at dr = n+1
d2 r di2
di
647
(4) 1 at dr = n+1 di
Now, for refraction at the point A, we have from the generalized vectorial law of refraction, sin i = sin r
(5)
Hence, cos i = cos r
or,
dr di
(6)
dr cos i = cos r di
Then,
− cos r sin i + cos i sin r dr d2 r di = 2 di 2 cos2 r
Now,
d2 r di2
=
− cos r sin i + cos i sin r
1 n+1
2 cos2 r
1 at dr = n+1 di
or,
d2 r di2
= 1 at dr = n+1
−n cos r sin i + sin (r − i) (n + 1) 2 cos2 r
di
Now, since here,
2
< i ≤ , and
2
< r ≤ , the angle (r − i) will always lie in the first quadrant. Hence it follows from the
above expression that, the quantity
d2 r di2
1 at dr = n+1
is always positive.
di
Thus it follows from the relation (4) that, the quantity,
d2 ó di2
is always positive.
1 at dr = n+1 di
1 Thus the net deviation ó is minimum when dr = n+1 · di It then follows from the relation (6) that, when the net deviation is minimum we must have,
cos i = cos r
or, cos i =
1 n+1
− n+1
This is because since
2
1 − sin2 r
< r ≤ , cos r is negative and hence cos r = −
− or, cos i = n+1
1−
or, cos i = −
2 − 1 n2 + 2n
sin2 i 2
or, (n + 1)2 cos2 i = 2 − 1 − cos2 i
1 − sin2 r
(7)
[Since < i ≤ , cos i is negative] 2 This relation (7) is also novel and it involves the refined unambiguous angle of incidence. If only one total internal reflection takes place inside the raindrop, we have, n = 1 and hence under this condition the relation (7) reduces to
cos i = −
2 − 1 3
(8)
Again if two total internal reflections occur inside the raindrop, we have, n = 2 and hence under this condition the relation (7) reduces to
cosi = −
2 − 1 8
(9)
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P.R. Bhattacharjee / Optik 157 (2018) 644–650
Thus if the value of of water for a particular colour of incident light as well as the total number (n) of total internal reflection taking place inside the raindrop are known, the value of the refined unambiguous angle of incidence (i) can be calculated from the relation (7). Knowing the value of ‘i’, the corresponding value of the refined unambiguous angle of refraction ‘r’ can be calculated from the relation (5). Since such values of ‘i’ and ‘r’ correspond to the condition of minimum deviation, the value of minimum deviation can then be calculated by using the relation (3) by making use of these values of ‘i’ and ‘r’. Now, let us first consider the case of primary rainbow for which n = 1. Then considering red light and assuming that red = 1.329 for water droplet, we then obtain from the relation (7), cos i = −0.50538 = − cos 59.643420 = cos(1800 − 59.643420 ) or,i = 120.35658◦ Hence from the relation (5), we have,
sin r = 0.64928 = sin 1800 − 40.487550
or,r = 139.51245◦ Putting these values of ‘i’ and ‘r’ in the relation (3) and remembering that here n = 1, we then obtain the value of minimum deviation (ómin ) for a ray of red colour as ómin = 137.33664◦ = 137◦ , (nearly) = 180◦ − 43◦ Now, considering violet light and taking vio = 1.349, we obtain from the relation (7) for the case of only one total internal reflection (i.e. for n = 1), cos i = −0.52275 = cos(1800 − 58.483140 ) or,i = 121.51686◦ Hence from the relation (5) we obtain,
sin r = 0.63194 = sin 1800 − 39.193360
or,r = 140.80664◦ Using these values of ‘i’ and ‘r’ and remembering that in this case, n = 1, we obtain from the relation (3), the value of minimum deviation (ómin ) for a ray of violet colour as ómin = 140.19284◦ = 140◦ , (nearly) = 180◦ − 40◦ Now let us consider the case of secondary rainbow for which n = 2. Then considering red light and taking red = 1.329, we have in this case from the relation (7), cos i = −0.30948 = cos(1800 − 71.971880 ) or,i = 108.02812◦ Hence from the relation (5), we get,
sin r = 0.71550 = sin 1800 − 46.684510
or,r = 134.31549◦ Putting these values of ‘i’ and ‘r’ in the relation (3) and remembering that here n = 2, we then obtain the value of minimum deviation (ómin ) for a ray of red colour as ómin = 229.8367◦ = 230◦ , (nearly) = 180◦ + 50◦ Finally considering violet light and taking vio = 1.349, we obtain from the relation (7) for the case of two total internal reflections (i.e. for n = 2), cos i = −0.32012 = cos(1800 − 71.329980 ) or,i = 108.67002◦
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Fig. 2. Diagram for explaining the formation of rainbow when only one total internal reflection takes place inside the raindrop.
Hence from the relation (5) we obtain,
sin r = 0.70228 = sin 1800 − 44.610350
or,r = 135.38965◦ Using these values of ‘i’ and ‘r’ and remembering that in this case, n = 2, we obtain from the relation (3), the value of minimum deviation (ómin ) for a ray of violet colour as ómin = 234.99788◦ = 235◦ , (nearly) = 180◦ + 55◦ 4. Explanation of the formation of rainbow When light rays from the sun fall on water droplets in the atmosphere, each of the incident rays of white light will get dispersed into its constituent colours. We have seen above that due to only one total internal reflection inside the raindrop, the red components suffer a deviation having minimum value equal to (180◦ − 43◦ ). If we consider a cone having semivertical angle 43◦ with the water droplet at its vertex and its axis coinciding with the direction of the sun rays as shown in Fig. 2, then the red rays suffering minimum deviation will all lie on the curved surface of this cone. All other red rays which will suffer deviations greater than the minimum deviation will lie inside the curved surface of the above cone. Now that the rate of change of deviation with angle of incidence is very small in the neighbourhood of minimum deviation, there must be more concentration of the red rays close to the curved surface of the cone as a result of which the intensity of red rays will be maximum in the direction of minimum deviation. Again we have seen earlier by considering one total internal reflection that, for violet components, the minimum deviation is (180◦ − 40◦ ) . Hence, as before it can be said that the intensity of violet rays will be maximum close to the curved surface of a cone of semi-vertical angle equal to 400 . In between the curved surfaces of the two aforesaid cones of semi-vertical angles 430 and 400 , intermediate spectral colours will be found to concentrate. Now, considering two total internal reflections we have found earlier that for red rays, the value of minimum deviation is (180◦ + 50◦ ), and also for violet rays, the minimum deviation has been found to be (180◦ + 55◦ ). Thus as shown in Fig. 3, in this case, concentration of red rays will be found close to the curved surface of the cone of semi-vertical angle 500 and violet rays will be found to be concentrated close to the curved surface of the cone of semi-vertical angle 550 . Red rays suffering deviation greater than the minimum deviation, will lie outside the curved surface of the cone of semi-vertical angle 500 . In a similar manner violet rays in this case will be found to be concentrated close to the curved surface of the cone of semi-vertical angle 550 . Violet and other rays suffering deviation greater than the minimum deviation, will lie outside the curved surface of the cone of semi-vertical angle 550 . All other intermediate colours will lie in the conical space in between the cones of semi-vertical angles 500 and 550 . But there will exist no ray within the cone of semi-vertical angle 500 . 5. Conclusion A novel unambiguous theory of the formation of rainbow has been offered in this paper. The theory presented is unambiguous because of the fact that unlike the traditional theory of formation of rainbow [3–8], which makes use of the traditional ambiguous angles of incidence, reflection, and refraction [1], it makes use of the refined unambiguous angles of incidence,
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Fig. 3. Diagram for explaining the formation of rainbow when there is two total internal reflections inside the raindrop.
reflection, and refraction [2]. The incorporation of the refined unambiguous angles of incidence, reflection, and refraction makes the theory novel as well, there by leading to novel expressions: (i) for the net deviation of an incident ray in coming out of a raindrop after suffering ‘n’ number of total internal reflections inside the raindrop, and (ii) for the primary angle of incidence corresponding to minimum deviation in terms of the refractive index of water and the total number of total internal reflections taking place inside the raindrop. On account of preciseness and sophistication involved in the definitions of the refined unambiguous angles of incidence, reflection, and refraction, the development of such a theory of formation of rainbow with the incorporation of these three refined unambiguous angles, will enhance the relevant optical physics literature there by enriching the same as well. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
P.R. Bhattacharjee, The generalized vectorial laws of reflection and refraction, Eur. J. Phys. 26 (5) (2005) 901–911. P.R. Bhattacharjee, Launching of the new world of geometrical optics, Optik 126 (24) (2015) 5134–5138. R.A. Herman, A Treatise on Geometrical Optics, The University Press, Cambridge, 1900, pp. 291–293. P.N. Tverskoi, Physics of the atmosphere, a course in Meteorology, 1965, 470–472, NASA Technical Document, Identifier: nasa techdoc 19660014173. Mathematics of rainbows – Feature column, retrieved on 03 October 2017 from: www.ams.org/samplings/feature-column/fcarc-rainbows. L.V. Tarasov, A.N. Tarasova, Discussions on Refraction of Light, Mir Publishers, Moscow, 1984, pp. 119–122. R.S. Heath, A Treatise on Geometrical Optics, The University Press, Cambridge, 1887, pp. 340–347. M. Nelcon, P. Parker, Advanced Level Physics, Heinemann Educational Books Ltd., London, 1958, pp. 475–476. P.R. Bhattacharjee, Deviation problems in ray optics in the light of the refined unambiguous definitions of angles of incidence, reflection and refraction, Optik 125 (16) (2014) 4257–4261. [10] P.R. Bhattacharjee, Dealing with the dispersive power of prism on the basis of the refined unambiguous angles of incidence and refraction, Optik 127 (4) (2016) 2177–2182.