- Email: [email protected]
The generalized vectorial laws of reflection and refraction applied to the rotation problems in ray optics
Optik 121 (2010) 2128–2132
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
The generalized vectorial laws of reflection and refraction applied to the rotation problems in ray optics Pramode Ranjan Bhattacharjee Department of Physics, M.B.B. College, Agartala, Tripura 799 004, India
a r t i c l e in f o
a b s t r a c t
Article history: Received 7 April 2009 Accepted 20 July 2009
This paper deals with the mirror rotation problem and the problem of rotation of refracting surface in ray optics. These two problems of rotation in ray optics have been dealt with on the basis of the generalized vectorial laws of reflection and refraction discovered by the author in 2005. In addition to the development of many interesting physical insights to the aforesaid rotation problems in ray optics, the most remarkable fact that has been discovered in the present study is that the proposition ‘Velocity of light is unattainable’ is not correct. Rather, it is possible to have velocity exceeding the velocity of light – a result not in agreement with the special theory of relativity. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Reflection Refraction Vector algebra Vector calculus
1. Introduction This paper concentrates on the generalized vectorial laws of reflection and refraction [1]. The discovery [1] proved that the traditional laws of reflection and refraction of light are ambiguous and to get rid of the ambiguity, the generalized vectorial laws of reflection and refraction have been offered in [1]. In [2], the theoretical proof of the discovered generalized vectorial laws of reflection and refraction has been accomplished. In order to verify how the generalized vectorial laws of reflection and refraction will advance the field of optics, the rotation problems in ray optics have been considered in this paper. The generalized vectorial laws of reflection and refraction have been applied to the mirror rotation problem as well as to the problem of rotation of refracting surface. The overall study reveals that a lot of interesting physical insights to the aforesaid rotation problems in ray optics could be achieved on the basis of the generalized vectorial laws of reflection and refraction unlike the ambiguous traditional laws of reflection and refraction[3-10]. The paper consists of five more sections. In Section 2, the generalized vectorial law of reflection as well as that of refraction have been presented. Section 3 deals with some useful theorems. In Section 4, the treatment of the mirror rotation problem on the basis of the generalized vectorial law of reflection has been offered. The generalized vectorial law of refraction has been subsequently employed in Section 5 to deal with the problem of rotation of refracting surface. Section 6 deals with the conclusion.
2. The discovered laws In this section the laws reported in [1] are being presented. 2.1. The generalized vectorial law of reflection If ˆı and ˆr represent unit vectors along the directions of incident ˆ represents unit vector ray and reflected ray respectively, and if n along the direction of the positive unit normal to the reflector at the point of incidence, then n^ i^ ¼ n^ r^
2.2. The generalized vectorial law of refraction ˆ represent unit vectors along the directions of the If ˆı and R incident ray and refracted ray of particular colour respectively, ˆ represents unit vector along the direction of the positive and if n unit normal to the surface of separation at the point of incidence, then ^ n^ i^ ¼ mðn^ RÞ where m is the refractive index of the second medium with respect to the first medium for the particular colour of light under consideration.
3. Some useful theorems Correspondence to: 5 Mantri Bari Road, P.O. Agartala, Tripura (West) 799 001, India. E-mail address: [email protected]
0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.07.011
This section deals with some useful theorems. They are being ˆ and n ˆ having their considered with the unit vectors ˆı, ˆr, R meanings same as that given in Section 2.
P.R. Bhattacharjee / Optik 121 (2010) 2128–2132
2129
ˆ is parallel to the resultant of the vectors ˆr Theorem 1. The vector n and ˆı.
A is a non-null vector of constant magnitude and if Theorem 4. If ~ jd~ A=dtj a 0, then ~ A is perpendicular to d~ A=dt.
Proof. We have
Proof. We have
^ ¼ n^ r^ n^ i^ n^ ðr^ iÞ ¼ 0 ½since by the generalized vectorial law of reflection;
~ A~ A ¼ j~ Aj2
n^ i^ ¼ n^ r^
ˆ and (rˆ ˆı) are parallel to each other. ‘ The vectors n
&
ˆ and the vector (rˆ ˆı) are coincident vectors. Theorem 2. The vector n Proof. By the law of parallelogram of vectors, it follows from Fig. 1 that the vector (rˆ ˆı), which is the resultant of the vectors ˆr and ˆı acting at O must be represented in magnitude and direction by the diagonal of the parallelogram AOBC and will pass ˆ and (rˆ ˆı) will both through O. Thus it is seen that the vectors n pass through O. ˆ is parallel to (rˆ ˆı) and also since Now since by Theorem 1, n ˆ and (rˆ ˆı) pass through O, if follows that n ˆ and both the vectors n (rˆ ˆı) are coincident vectors. & Theorem 3. The vectors ˆr and ˆı are equally inclined to their resultant (rˆ ˆı). Proof. We have considering Fig. 2, ^ ¼1 OA ¼ BC ¼ j ij Also; OB ¼ AC ¼ jr^ j ¼ 1 ‘OA ¼ OB ¼ BC ¼ AC ‘OACB is a rhombus ‘+AOC ¼ +OCB; being alternate angles ¼ +BOC ½since OB ¼ BC
‘ The vectors ˆr and ˆı are equally inclined to their resultant (rˆ ˆı) &
¼ constant ½‘~ A has constant magnitude d ~ ~ ‘ ðA AÞ ¼ 0 dt
or d~ A d~ A ~ ~ þ A¼0 A dt dt or d~ A 2~ A ¼0 dt or d~ A ~ A ¼0 dt
This implies that ~ A is perpendicular to d~ A=dt provided ~ A is a non-null vector of constant magnitude and jd~ A=dtj a0. & 4. The problem of rotation of plane mirror In this section, the generalized vectorial law of reflection of light has been employed to find interesting physical insights to the mirror rotation problem in ray optics. The following theorem is being offered in the said context. Theorem 5. In the mirror rotation problem, if for a particular direction of ˆı, the unit vectors corresponding to the positive unit normal to the reflector and the reflected ray at any instant t be ˆ (t)= n ˆ and ˆr (t)= ˆr respectively, then denoted by n dr^ dn^ ¼ 2 dt dt Proof. In the mirror rotation problem, let us represent the fixed direction of the incident ray by the unit vector ˆı and as shown in Fig. 3, let the position of the rotating plane mirror at any instant t be MM0 , the corresponding directions of the reflected ray and the unit positive normal to the reflector MM0 at the instant t being ˆ respectively, so that ˆr (t)= ˆr and n ˆ (t) =n ˆ. represented by ˆr and n Then from the generalized vectorial law of reflection of light at the instant t we have
ˆ and (rˆ ˆı). Fig. 1. Figure showing ˆı, ˆr, ˆı, n
Fig. 2. Figure showing ˆr, ˆı and (rˆ ˆı).
n^ i^ ¼ n^ r^
ˆ and ˆr at any instant t of rotation of Fig. 3. Figure representing the directions of ˆı, n the plane mirror.
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Differentiating both sides of this relation with respect to the time variable t we get n^
di^ ^ dn^ dr^ dn^ i ¼ n^ r^ dt dt dt dt
or i^
dn^ dr^ dn^ ¼ n^ r^ dt dt dt
[since here ˆı is a constant vector being independent of the time variable t] or ðr^ ^iÞ
dn^ dr^ ¼ n^ dt dt
ð1Þ
ˆ are like parallel Now since by Theorems 1 and 2, (rˆ ˆı) and n vectors, we have, r^ i^ ¼ jr^ ^ijn^
Further since by Theorem 3, ˆr and ˆı are equally inclined to (rˆ ˆı), we have considering Fig. 2, ^ ¼ jr^ ij
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ cos 2yÞ ðjr^ j2 þj ^ij2 þ 2jr^ jj ij
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1þ 1 þ 2:1:1 cos 2y ¼ 2 cos y ^ ‘r ^i ¼ ð2 cos yÞn^
Thus Eq. (1) reduces to ð2 cos yÞn^
dn^ dr^ ¼ n^ dt dt
or ^ ^ ð2cos yÞn^ dn ¼ n^ dr dt dt
ˆ. Fig. 5. Figure representing ˆr, drˆ, ˆr + drˆ and n
The aforesaid rigorous treatment of the problem of rotation of a plane mirror leads to the following interesting results: ˆ /dt| shows that, if a plane mirror (i) The relation |drˆ/dt| =2|dn turns through an angle, then the reflected ray will turn through twice that angle regardless of whether the plane mirror rotates with uniform speed or not. ˆ /dt| clearly reveals that the linear (ii) The relation |drˆ/dt| = 2|dn speed of the reflected ray, i.e. |drˆ/dt| increases with the ˆ/ increase of the speed of rotation of the plane mirror, i.e. |dn dt| ˆ /dt| that the (iii) It is also clear from the relation |drˆ/dt| = 2|dn linear speed of the reflected ray at any instant depends only ˆ (i.e. on the linear on the linear speed of the normal vector n speed of the rotating mirror) at the said instant but is totally independent of the values of the angle of incidence or angle of reflection at the said instant. (iv) If the plane mirror undergoes rotation on its own plane, then ˆı ˆ will both be constant vectors being independent of t. and n Under this condition we have from the generalized vectorial law of reflection at the instant t,
or dn^ dr^ 2cos y 1 sin 903 ¼ 1 sin ð903 yÞ dt dt [making use of Theorem 4 and considering Figs. 4 and 5] or dn^ dr^ 2 cos y ¼ cos y dt dt or dr^ ^ ¼ 2dn dt dt
&
n^ ^i ¼ n^ r^
or d d ðn^ ^iÞ ¼ ðn^ r^ Þ dt dt
or n^
di^ ^ dn^ dr^ dn^ i ¼ n^ r^ dt dt dt dt
or 0 ¼ n^
dr^ 0 dt
ˆ are constant vectors, each being independent of [since ˆı and n t] or n^ ˆ , dn ˆ and n ˆ + dn ˆ. Fig. 4. Figure representing n
dr^ ¼0 dt
P.R. Bhattacharjee / Optik 121 (2010) 2128–2132
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or ^ n^ dr ¼ 0 dt or dr^ ^ sinð903 yÞ ¼ 0 ½considering Fig: 5 jnj dt
or dr^ cos y ¼ 0 dt
This implies that |drˆ/dt| =0, i.e. ˆr would then be a constant vector independent of time. Thus if the plane mirror undergoes rotation on its own plane, the reflected ray will not undergo any kind of rotation at all. ˆ /dt|, it also follows that, if |dn ˆ/ (v) From the relation |drˆ/dt|= 2|dn dt| ZC/2 where C is the magnitude of the velocity of light in free space, then |drˆ/dt|ZC. This implies that the reflected ray can be made to rotate with a linear speed greater than or equal to the value of the velocity of light in free space if the plane mirror undergoes rotation with a linear speed greater than or equal to half the magnitude of velocity of light in free space. This is a very important result violating the fact ‘Space travel grater than the speed of light is not possible’ according to the special theory of relativity.
Fig. 6. Diagram for refraction showing directions of various kinds of vectors.
ˆ at the instant t will be as shown in Fig. 6 because in that case the n two vectors represented, respectively, by the two cross-products on the two sides of Eq. (2) will both be directed perpendicularly into the plane of paper. Now, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jmR^ ^ij ¼ m2 þ 12 þ 2m 1 cos a ¼ m2 þ1 þ 2m cos f1803 i þ Rg qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ m2 þ 12m cos ði RÞ Further, from Fig. 6, it is seen that the angle between the ^ ˆ ˆı) and dn vectors (mR dt ¼ +AOB ¼ f +BOC ¼ f ð903 iÞ ¼ f903 ðf þ iÞg
5. The problem of rotation of refracting surface This section deals with the case of rotation of refracting surface on the basis of the generalized vectorial law of refraction of light. As in case of the problem of rotation of plane mirror, here also we are interested to know what will happen to the refracted ray (the direction of the incident ray being fixed) when the refracting surface undergoes rotation. At any instant t of rotation of the ˆ (t)= R ˆ and n ˆ (t) =n ˆ . Clearly in this refracting surface, let ˆı (t)=ˆı, R problem, ˆı (t)=ˆı is a constant vector independent of the time variable t. Then from the generalized vectorial law of refraction of light we have
ˆ /dt as seen from ˆ and dR Also, the angle between the vectors mn Fig. 6 is (901 R). Then from Eq. (2) we have ^ ^ ðmR^ ^iÞ dn ¼ mn^ dR dt dt or dR^ dn^ ^ sinð903 RÞ jmR^ ^ij sin½ 903 ðf þiÞ ¼ jmnj dt dt or qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ dR^ dn m2 þ 1 2m cos ði RÞ cosðf þ iÞ ¼ m cosR dt dt
^ n^ i^ ¼ mðn^ RÞ
Now, it can be readily seen from Fig. 6 that,
Differentiating both sides with respect to the time variable t we get d ^ ¼ d ½mðn^ RÞ ^ ðn^ iÞ dt dt
a ¼ +COD ¼ +BOC þ+BOD ¼ ð903 iÞ þ ð903 þ RÞ ¼ 1803 ði RÞ m sin f1803 ði RÞg ‘tan f ¼ 1þ m cosf1803 ði RÞg or
or "
n^
ð3Þ
di^ ^ dn^ dR^ dn^ i ¼ m n^ R^ dt dt dt dt
#
m sin ði RÞ 1 m cosði RÞ
Hence Eq. (3) takes the form
or i^
f ¼ tan1
" # dn^ dR^ dn^ ¼ m n^ R^ dt dt dt
[since ˆı is a constant vector] or dn^ dR^ ¼ mn^ ð2Þ ðmR^ ^iÞ dt dt ˆ /dt, which is This relation demands that directions of dR ˆ and that of dn ˆ /dt, which is perpendicular to perpendicular to R
^ jdn=dtj ^ jdR=dtj m cos R ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ 1 2m cos ði RÞ cosftan1 ððm sinði RÞÞ=ð1 m cosði RÞÞþ ig
ð4Þ
Eq. (4) is novel and it connects the linear speed of the normal vector at any instant with the linear speed of the refracted ray vector at the said instant. It may be noted that unlike the case of reflection, the linear speed of the refracted ray at any instant is not only dependent on the linear speed of the normal vector
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(i.e. that of the refracting surface) at that instant but also on the actual value of i at the said instant and the value of m, because value of R depends on the value of i for a particular value of m. In particular, if at any instant t of rotation, i= R=01, then Eq. (4) becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ dR^ dn 2 m þ 1 2m ¼ m dt dt
ˆ /dt| =0 and hence the linear speed of the This implies that |dR refracted ray vector being zero, the refracted ray will not undergo any kind of rotation in such a case.
or dR^ 1 dn^ ¼ 1 dt m dt
In this paper the author has attempted to apply his discovered laws, viz. The generalized vectorial laws of reflection and refraction reported in 2005 to the cases of the rotation problems in ray optics. The mirror rotation problem as well as the problem of rotation of refracting surface has been dealt with. The quantitative physics incorporated in both the aforesaid cases is theoretically worthwhile and it will enrich the branch of optics. The overall study provides the following interesting physical insights to the said rotation problems in ray optics:
Thus it implies that in this particular case, at the instant t we must have, dR^ dn^ o dt dt i.e. the refracted ray will move with a linear speed less than that of the corresponding refracting surface. Again if m = 1.5 for the refracting medium under consideration and if at any instant of rotation of the said refracting surface, i ¼ 303 then sin R ¼
sin i
m
¼
sin 303 sin 19:53 1:5
so that R= 19.51. Thus from Eq. (4) it can be easily seen that ^ jdR=dtj ¼ 0:3874 ^ jdn=dtj ˆ /dt| o|dn ˆ /dt|. This shows that in this particular case also, |dR In this way the correlation between the linear speed of the refracted ray and that of the unit normal vector can be studied by considering typical cases as above although the actual appearance of Eq. (4) is not so simple. Finally, if the refracting surface undergoes rotation about an axis ˆ will be perpendicular to the surface of separation, then both ˆı and n constant vectors, each being independent of the time variable t. Then from the generalized vectorial law of refraction of light we have ^ n^ i^ ¼ mðn^ RÞ or n o d ^ ¼ d mðn^ RÞ ^ ðn^ iÞ dt dt or n^ or,
( ) di^ ^ dn^ dR^ dn^ i ¼ m n^ R^ dt dt dt dt (
dR^ 0 0 ¼ m n^ dt
)
ˆ are constant vectors in this case] [since ˆı and n or dR^ n^ ¼0 dt or ^ dR^ n ¼0 dt or dR^ ^ sinð903 RÞ ¼ 0 ½considering Fig: 6 jnj dt
6. Conclusion
(i) While the linear speed of the reflected ray at any instant (in the mirror rotation problem) depends exclusively on the linear ˆ (i.e. on the linear speed of the speed of the normal vector n rotating mirror), the linear speed of the refracted ray at any instant (in the problem of rotation of refracting surface) is ˆ (i.e. on the dependent on the linear speed of the normal vector n linear speed of the refracting surface) as well as on the values of any two of the three parameters i, R (at the said instant) and m. (ii) The fact that velocity of light is unbeatable has been found to be incorrect from the present study. In the mirror rotation problem, if the plane mirror is rotated with a linear speed ZC/2, then the reflected ray will be found to move with a linear speed ZC, where C is the magnitude of the velocity of light in free space, a most remarkable feature not in agreement with the result obtained from the special theory of relativity. (iii) It has been found by considering two typical examples in the problem of rotation of refracting surface that the refracted ray moves with a velocity less than that of the refracting surface. (iv) The cases of no movement of the reflected ray (in the mirror rotation problem) as well as that of the refracted ray (in the problem of rotation of refracting surface) have been investigated. The result shows that in both cases the plane mirror and the refracting surface will have to undergo rotation about ˆ. the normal vector n
References [1] P.R. Bhattacharjee, The generalized vectorial laws of reflection and refraction, Eur. J. Phys. 26 (2005) 901–911. [2] P.R. Bhattacharjee, Momentum conservation leading to the generalized vectorial laws of reflection and refraction, Opt. Int. J. Light Electron Optics 120 (2009) 642–646. [3] E. Hecht, Optics, fourth ed., Pearson Education, Inc., 2004. [4] A. Cox, Optics, The Focal Press, London, 1961. [5] P. Drude, The Theory of Optics, Longmans, Green, London, 1939. [6] M.H. Freeman, Optics, Butterworths, London, 1990. [7] G.A. Fry, Geometrical Optics, Chilton, Philadelphia, 1969. [8] R.S. Longhurst, Geometrical and Physical Optics, Wiley, New York, 1967. [9] J. Morgan, Introduction to Geometrical and Physical Optics, McGraw-Hill, New York, 1953. [10] I. Newton, Optiks, Dover Publications, Addison-Wesley, Reading, MA, 1992.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
The generalized vectorial laws of reflection and refraction applied to the rotation problems in ray optics Pramode Ranjan Bhattacharjee Department of Physics, M.B.B. College, Agartala, Tripura 799 004, India
a r t i c l e in f o
a b s t r a c t
Article history: Received 7 April 2009 Accepted 20 July 2009
This paper deals with the mirror rotation problem and the problem of rotation of refracting surface in ray optics. These two problems of rotation in ray optics have been dealt with on the basis of the generalized vectorial laws of reflection and refraction discovered by the author in 2005. In addition to the development of many interesting physical insights to the aforesaid rotation problems in ray optics, the most remarkable fact that has been discovered in the present study is that the proposition ‘Velocity of light is unattainable’ is not correct. Rather, it is possible to have velocity exceeding the velocity of light – a result not in agreement with the special theory of relativity. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Reflection Refraction Vector algebra Vector calculus
1. Introduction This paper concentrates on the generalized vectorial laws of reflection and refraction [1]. The discovery [1] proved that the traditional laws of reflection and refraction of light are ambiguous and to get rid of the ambiguity, the generalized vectorial laws of reflection and refraction have been offered in [1]. In [2], the theoretical proof of the discovered generalized vectorial laws of reflection and refraction has been accomplished. In order to verify how the generalized vectorial laws of reflection and refraction will advance the field of optics, the rotation problems in ray optics have been considered in this paper. The generalized vectorial laws of reflection and refraction have been applied to the mirror rotation problem as well as to the problem of rotation of refracting surface. The overall study reveals that a lot of interesting physical insights to the aforesaid rotation problems in ray optics could be achieved on the basis of the generalized vectorial laws of reflection and refraction unlike the ambiguous traditional laws of reflection and refraction[3-10]. The paper consists of five more sections. In Section 2, the generalized vectorial law of reflection as well as that of refraction have been presented. Section 3 deals with some useful theorems. In Section 4, the treatment of the mirror rotation problem on the basis of the generalized vectorial law of reflection has been offered. The generalized vectorial law of refraction has been subsequently employed in Section 5 to deal with the problem of rotation of refracting surface. Section 6 deals with the conclusion.
2. The discovered laws In this section the laws reported in [1] are being presented. 2.1. The generalized vectorial law of reflection If ˆı and ˆr represent unit vectors along the directions of incident ˆ represents unit vector ray and reflected ray respectively, and if n along the direction of the positive unit normal to the reflector at the point of incidence, then n^ i^ ¼ n^ r^
2.2. The generalized vectorial law of refraction ˆ represent unit vectors along the directions of the If ˆı and R incident ray and refracted ray of particular colour respectively, ˆ represents unit vector along the direction of the positive and if n unit normal to the surface of separation at the point of incidence, then ^ n^ i^ ¼ mðn^ RÞ where m is the refractive index of the second medium with respect to the first medium for the particular colour of light under consideration.
3. Some useful theorems Correspondence to: 5 Mantri Bari Road, P.O. Agartala, Tripura (West) 799 001, India. E-mail address: [email protected]
0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.07.011
This section deals with some useful theorems. They are being ˆ and n ˆ having their considered with the unit vectors ˆı, ˆr, R meanings same as that given in Section 2.
P.R. Bhattacharjee / Optik 121 (2010) 2128–2132
2129
ˆ is parallel to the resultant of the vectors ˆr Theorem 1. The vector n and ˆı.
A is a non-null vector of constant magnitude and if Theorem 4. If ~ jd~ A=dtj a 0, then ~ A is perpendicular to d~ A=dt.
Proof. We have
Proof. We have
^ ¼ n^ r^ n^ i^ n^ ðr^ iÞ ¼ 0 ½since by the generalized vectorial law of reflection;
~ A~ A ¼ j~ Aj2
n^ i^ ¼ n^ r^
ˆ and (rˆ ˆı) are parallel to each other. ‘ The vectors n
&
ˆ and the vector (rˆ ˆı) are coincident vectors. Theorem 2. The vector n Proof. By the law of parallelogram of vectors, it follows from Fig. 1 that the vector (rˆ ˆı), which is the resultant of the vectors ˆr and ˆı acting at O must be represented in magnitude and direction by the diagonal of the parallelogram AOBC and will pass ˆ and (rˆ ˆı) will both through O. Thus it is seen that the vectors n pass through O. ˆ is parallel to (rˆ ˆı) and also since Now since by Theorem 1, n ˆ and (rˆ ˆı) pass through O, if follows that n ˆ and both the vectors n (rˆ ˆı) are coincident vectors. & Theorem 3. The vectors ˆr and ˆı are equally inclined to their resultant (rˆ ˆı). Proof. We have considering Fig. 2, ^ ¼1 OA ¼ BC ¼ j ij Also; OB ¼ AC ¼ jr^ j ¼ 1 ‘OA ¼ OB ¼ BC ¼ AC ‘OACB is a rhombus ‘+AOC ¼ +OCB; being alternate angles ¼ +BOC ½since OB ¼ BC
‘ The vectors ˆr and ˆı are equally inclined to their resultant (rˆ ˆı) &
¼ constant ½‘~ A has constant magnitude d ~ ~ ‘ ðA AÞ ¼ 0 dt
or d~ A d~ A ~ ~ þ A¼0 A dt dt or d~ A 2~ A ¼0 dt or d~ A ~ A ¼0 dt
This implies that ~ A is perpendicular to d~ A=dt provided ~ A is a non-null vector of constant magnitude and jd~ A=dtj a0. & 4. The problem of rotation of plane mirror In this section, the generalized vectorial law of reflection of light has been employed to find interesting physical insights to the mirror rotation problem in ray optics. The following theorem is being offered in the said context. Theorem 5. In the mirror rotation problem, if for a particular direction of ˆı, the unit vectors corresponding to the positive unit normal to the reflector and the reflected ray at any instant t be ˆ (t)= n ˆ and ˆr (t)= ˆr respectively, then denoted by n dr^ dn^ ¼ 2 dt dt Proof. In the mirror rotation problem, let us represent the fixed direction of the incident ray by the unit vector ˆı and as shown in Fig. 3, let the position of the rotating plane mirror at any instant t be MM0 , the corresponding directions of the reflected ray and the unit positive normal to the reflector MM0 at the instant t being ˆ respectively, so that ˆr (t)= ˆr and n ˆ (t) =n ˆ. represented by ˆr and n Then from the generalized vectorial law of reflection of light at the instant t we have
ˆ and (rˆ ˆı). Fig. 1. Figure showing ˆı, ˆr, ˆı, n
Fig. 2. Figure showing ˆr, ˆı and (rˆ ˆı).
n^ i^ ¼ n^ r^
ˆ and ˆr at any instant t of rotation of Fig. 3. Figure representing the directions of ˆı, n the plane mirror.
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Differentiating both sides of this relation with respect to the time variable t we get n^
di^ ^ dn^ dr^ dn^ i ¼ n^ r^ dt dt dt dt
or i^
dn^ dr^ dn^ ¼ n^ r^ dt dt dt
[since here ˆı is a constant vector being independent of the time variable t] or ðr^ ^iÞ
dn^ dr^ ¼ n^ dt dt
ð1Þ
ˆ are like parallel Now since by Theorems 1 and 2, (rˆ ˆı) and n vectors, we have, r^ i^ ¼ jr^ ^ijn^
Further since by Theorem 3, ˆr and ˆı are equally inclined to (rˆ ˆı), we have considering Fig. 2, ^ ¼ jr^ ij
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ cos 2yÞ ðjr^ j2 þj ^ij2 þ 2jr^ jj ij
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1þ 1 þ 2:1:1 cos 2y ¼ 2 cos y ^ ‘r ^i ¼ ð2 cos yÞn^
Thus Eq. (1) reduces to ð2 cos yÞn^
dn^ dr^ ¼ n^ dt dt
or ^ ^ ð2cos yÞn^ dn ¼ n^ dr dt dt
ˆ. Fig. 5. Figure representing ˆr, drˆ, ˆr + drˆ and n
The aforesaid rigorous treatment of the problem of rotation of a plane mirror leads to the following interesting results: ˆ /dt| shows that, if a plane mirror (i) The relation |drˆ/dt| =2|dn turns through an angle, then the reflected ray will turn through twice that angle regardless of whether the plane mirror rotates with uniform speed or not. ˆ /dt| clearly reveals that the linear (ii) The relation |drˆ/dt| = 2|dn speed of the reflected ray, i.e. |drˆ/dt| increases with the ˆ/ increase of the speed of rotation of the plane mirror, i.e. |dn dt| ˆ /dt| that the (iii) It is also clear from the relation |drˆ/dt| = 2|dn linear speed of the reflected ray at any instant depends only ˆ (i.e. on the linear on the linear speed of the normal vector n speed of the rotating mirror) at the said instant but is totally independent of the values of the angle of incidence or angle of reflection at the said instant. (iv) If the plane mirror undergoes rotation on its own plane, then ˆı ˆ will both be constant vectors being independent of t. and n Under this condition we have from the generalized vectorial law of reflection at the instant t,
or dn^ dr^ 2cos y 1 sin 903 ¼ 1 sin ð903 yÞ dt dt [making use of Theorem 4 and considering Figs. 4 and 5] or dn^ dr^ 2 cos y ¼ cos y dt dt or dr^ ^ ¼ 2dn dt dt
&
n^ ^i ¼ n^ r^
or d d ðn^ ^iÞ ¼ ðn^ r^ Þ dt dt
or n^
di^ ^ dn^ dr^ dn^ i ¼ n^ r^ dt dt dt dt
or 0 ¼ n^
dr^ 0 dt
ˆ are constant vectors, each being independent of [since ˆı and n t] or n^ ˆ , dn ˆ and n ˆ + dn ˆ. Fig. 4. Figure representing n
dr^ ¼0 dt
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or ^ n^ dr ¼ 0 dt or dr^ ^ sinð903 yÞ ¼ 0 ½considering Fig: 5 jnj dt
or dr^ cos y ¼ 0 dt
This implies that |drˆ/dt| =0, i.e. ˆr would then be a constant vector independent of time. Thus if the plane mirror undergoes rotation on its own plane, the reflected ray will not undergo any kind of rotation at all. ˆ /dt|, it also follows that, if |dn ˆ/ (v) From the relation |drˆ/dt|= 2|dn dt| ZC/2 where C is the magnitude of the velocity of light in free space, then |drˆ/dt|ZC. This implies that the reflected ray can be made to rotate with a linear speed greater than or equal to the value of the velocity of light in free space if the plane mirror undergoes rotation with a linear speed greater than or equal to half the magnitude of velocity of light in free space. This is a very important result violating the fact ‘Space travel grater than the speed of light is not possible’ according to the special theory of relativity.
Fig. 6. Diagram for refraction showing directions of various kinds of vectors.
ˆ at the instant t will be as shown in Fig. 6 because in that case the n two vectors represented, respectively, by the two cross-products on the two sides of Eq. (2) will both be directed perpendicularly into the plane of paper. Now, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jmR^ ^ij ¼ m2 þ 12 þ 2m 1 cos a ¼ m2 þ1 þ 2m cos f1803 i þ Rg qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ m2 þ 12m cos ði RÞ Further, from Fig. 6, it is seen that the angle between the ^ ˆ ˆı) and dn vectors (mR dt ¼ +AOB ¼ f +BOC ¼ f ð903 iÞ ¼ f903 ðf þ iÞg
5. The problem of rotation of refracting surface This section deals with the case of rotation of refracting surface on the basis of the generalized vectorial law of refraction of light. As in case of the problem of rotation of plane mirror, here also we are interested to know what will happen to the refracted ray (the direction of the incident ray being fixed) when the refracting surface undergoes rotation. At any instant t of rotation of the ˆ (t)= R ˆ and n ˆ (t) =n ˆ . Clearly in this refracting surface, let ˆı (t)=ˆı, R problem, ˆı (t)=ˆı is a constant vector independent of the time variable t. Then from the generalized vectorial law of refraction of light we have
ˆ /dt as seen from ˆ and dR Also, the angle between the vectors mn Fig. 6 is (901 R). Then from Eq. (2) we have ^ ^ ðmR^ ^iÞ dn ¼ mn^ dR dt dt or dR^ dn^ ^ sinð903 RÞ jmR^ ^ij sin½ 903 ðf þiÞ ¼ jmnj dt dt or qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ dR^ dn m2 þ 1 2m cos ði RÞ cosðf þ iÞ ¼ m cosR dt dt
^ n^ i^ ¼ mðn^ RÞ
Now, it can be readily seen from Fig. 6 that,
Differentiating both sides with respect to the time variable t we get d ^ ¼ d ½mðn^ RÞ ^ ðn^ iÞ dt dt
a ¼ +COD ¼ +BOC þ+BOD ¼ ð903 iÞ þ ð903 þ RÞ ¼ 1803 ði RÞ m sin f1803 ði RÞg ‘tan f ¼ 1þ m cosf1803 ði RÞg or
or "
n^
ð3Þ
di^ ^ dn^ dR^ dn^ i ¼ m n^ R^ dt dt dt dt
#
m sin ði RÞ 1 m cosði RÞ
Hence Eq. (3) takes the form
or i^
f ¼ tan1
" # dn^ dR^ dn^ ¼ m n^ R^ dt dt dt
[since ˆı is a constant vector] or dn^ dR^ ¼ mn^ ð2Þ ðmR^ ^iÞ dt dt ˆ /dt, which is This relation demands that directions of dR ˆ and that of dn ˆ /dt, which is perpendicular to perpendicular to R
^ jdn=dtj ^ jdR=dtj m cos R ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ 1 2m cos ði RÞ cosftan1 ððm sinði RÞÞ=ð1 m cosði RÞÞþ ig
ð4Þ
Eq. (4) is novel and it connects the linear speed of the normal vector at any instant with the linear speed of the refracted ray vector at the said instant. It may be noted that unlike the case of reflection, the linear speed of the refracted ray at any instant is not only dependent on the linear speed of the normal vector
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(i.e. that of the refracting surface) at that instant but also on the actual value of i at the said instant and the value of m, because value of R depends on the value of i for a particular value of m. In particular, if at any instant t of rotation, i= R=01, then Eq. (4) becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ dR^ dn 2 m þ 1 2m ¼ m dt dt
ˆ /dt| =0 and hence the linear speed of the This implies that |dR refracted ray vector being zero, the refracted ray will not undergo any kind of rotation in such a case.
or dR^ 1 dn^ ¼ 1 dt m dt
In this paper the author has attempted to apply his discovered laws, viz. The generalized vectorial laws of reflection and refraction reported in 2005 to the cases of the rotation problems in ray optics. The mirror rotation problem as well as the problem of rotation of refracting surface has been dealt with. The quantitative physics incorporated in both the aforesaid cases is theoretically worthwhile and it will enrich the branch of optics. The overall study provides the following interesting physical insights to the said rotation problems in ray optics:
Thus it implies that in this particular case, at the instant t we must have, dR^ dn^ o dt dt i.e. the refracted ray will move with a linear speed less than that of the corresponding refracting surface. Again if m = 1.5 for the refracting medium under consideration and if at any instant of rotation of the said refracting surface, i ¼ 303 then sin R ¼
sin i
m
¼
sin 303 sin 19:53 1:5
so that R= 19.51. Thus from Eq. (4) it can be easily seen that ^ jdR=dtj ¼ 0:3874 ^ jdn=dtj ˆ /dt| o|dn ˆ /dt|. This shows that in this particular case also, |dR In this way the correlation between the linear speed of the refracted ray and that of the unit normal vector can be studied by considering typical cases as above although the actual appearance of Eq. (4) is not so simple. Finally, if the refracting surface undergoes rotation about an axis ˆ will be perpendicular to the surface of separation, then both ˆı and n constant vectors, each being independent of the time variable t. Then from the generalized vectorial law of refraction of light we have ^ n^ i^ ¼ mðn^ RÞ or n o d ^ ¼ d mðn^ RÞ ^ ðn^ iÞ dt dt or n^ or,
( ) di^ ^ dn^ dR^ dn^ i ¼ m n^ R^ dt dt dt dt (
dR^ 0 0 ¼ m n^ dt
)
ˆ are constant vectors in this case] [since ˆı and n or dR^ n^ ¼0 dt or ^ dR^ n ¼0 dt or dR^ ^ sinð903 RÞ ¼ 0 ½considering Fig: 6 jnj dt
6. Conclusion
(i) While the linear speed of the reflected ray at any instant (in the mirror rotation problem) depends exclusively on the linear ˆ (i.e. on the linear speed of the speed of the normal vector n rotating mirror), the linear speed of the refracted ray at any instant (in the problem of rotation of refracting surface) is ˆ (i.e. on the dependent on the linear speed of the normal vector n linear speed of the refracting surface) as well as on the values of any two of the three parameters i, R (at the said instant) and m. (ii) The fact that velocity of light is unbeatable has been found to be incorrect from the present study. In the mirror rotation problem, if the plane mirror is rotated with a linear speed ZC/2, then the reflected ray will be found to move with a linear speed ZC, where C is the magnitude of the velocity of light in free space, a most remarkable feature not in agreement with the result obtained from the special theory of relativity. (iii) It has been found by considering two typical examples in the problem of rotation of refracting surface that the refracted ray moves with a velocity less than that of the refracting surface. (iv) The cases of no movement of the reflected ray (in the mirror rotation problem) as well as that of the refracted ray (in the problem of rotation of refracting surface) have been investigated. The result shows that in both cases the plane mirror and the refracting surface will have to undergo rotation about ˆ. the normal vector n
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