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What is the First Color of the Primary Rainbow? What Angle is Secondary Rainbow Formed?
An interesting natural phenomenon of dispersion of light is the rainbow. It is formed by the splitting of sunlight into different colours [Fig.]. The formation of rainbow is due to refraction and reflection of sunlight by water droplets. After rain, huge droplets of water remain suspended in the sky. An observer standing with his back towards the sun, can see the rainbow.
Sometimes, a little above the primary rainbow, another rainbow called the secondary rainbow of lesser brightness can be seen. The rainbow can be seen in the form of a circular arc spread across the sky. The primary rainbow has the red colour at the outside of the arc and violet colour at the inside. In the secondary rainbow, the colour spectrum is arranged in reverse order.
Formations of the Primary (First Order) Rainbow
Let a sun ray AB be incident on a suspended water droplet at the point B. (Fig.). The refracted ray, inside the water droplet is BC. From point C, the ray gets reflected inside the droplet to D and then refracted to air through DE to reach the eyes of observer.
Generally at the point C, inside the water droplet, a partial reflection of the ray takes place and a part of it gets refracted into the air. So, the brightness of emerging ray DE becomes less than that of the incident ray AB.
Fig. shows only one reflection of the refracted ray, inside the water droplet, at point C. For different angles of incidence more than one reflection inside the curved surface of the water droplet is possible.
More the reflections less is the brightness of the rays reaching the eyes.
The deviation of the light ray: According to Fig. angle of incidence at B = i; angle of refraction = r; deviation of the ray = (i – r), angle of Incidence at C = angle of reflection = r, deviation of the ray = 180° – 2r; angle of incidence at D = r, angle of refraction = i, the deviation of the ray = i – r.
So, total deviation of incident ray with respect to the observer,
δ = (i – r) + (180° – 2r) + (i – r)
∴ δ = 2i – 4r + 180° ….. (1)
Condition of deviation angle to be minimum:
Differentiating equation (1) with respect to i,
\(\frac{d \delta}{d i}\) = 2 – 4\(\frac{d r}{d i}\)
For, deviation angle δ to be minimum
\(\frac{d \delta}{d i}\) = 0
So, 2 – 4\(\frac{d r}{d i}\) = 0 or, \(\frac{d r}{d i}\) = \(\frac{1}{2}\) ….. (2)
Again, refractive index of water,
μ = \(\frac{\sin i}{\sin r}\) or, sinr = \(\frac{1}{\mu}\)sini
In this case, differentiating w.r.t. i, we get,
cosr\(\frac{d r}{d i}\) = \(\frac{1}{\mu}\)cosi or, \(\frac{d r}{d i}\) = \(\frac{\cos i}{\mu \cos r}\) ….. (3)
Comparing equation (2) and (3), we get
\(\frac{1}{2}\) = \(\frac{\cos i}{\mu \cos r}\) or, 2 cosi = μcos r
∴ 4cos2i = µ2cos2r = µ2(1 – sin2r)
= µ2 – µ2sin2r = µ2 – sin2i
∴ µ2 = 4cos2i + sin2i = 3 cos2i + 1
or, cos i = \(\sqrt{\frac{\mu^2-1}{3}}\) ….. (4)
So, if the value of µ is known, the value of i can be calculated from equation (4).
Then, putting the values of i and in the equation µ = \(\frac{\sin i}{\sin r}\), the value of r can be calculated.
From equation (1), with the help of values µ, i and r, the value of minimum angle of deviation can be obtained.
Now, refractive index of colour red in water is 1.331. Minimum angle of deviation of colour red, due to one-time reflection inside the water droplet = 138°. When parallel red rays coming from sun reach the eyes of the observer with minimum deviation, the Inclination angle of these rays with respect to sun rays, become (180° – 138°) = 42° [Fig.(a)].
An important point to note is that only in minimum deviation, many light rays remain in overlapping condition which increases the brightness considerably. As the angle of minimum deviation of other colours is not 138°, hence with inclination angle 42°, the red colour appears to be most prominent, whereas other colours remain almost obscure. One can find rainbow’s bright spectrum of red-colour along an arc subtending angle 42° at the eye, making eye as centre.
Similarly, the minimum angle of deviation for violet ray is 140°. So, the colour seen by observer with an inclination angle (180° – 140°) or 40° is deep violet. Clearly the other colours are seen as bright spectrum in between red and violet. In this way, by one reflection inside the water droplet, a first order rainbow is formed with red colour at the top (outside) and violet colour at the bottom (inside) [Fig.(b)].
Formation of Second Order Rainbow
Sunrays can also reach the eyes of observer after having two reflections on the inside surface of the water droplet [Fig.]. Calculations show that the inclination angle of the red ray at the eye of the observer, due to reflection at minimum deviation angle is 52°. For violet ray, this inclination angle is 55°. Hence the other rainbow, which forms at an angular interval of 3° (from 52° to 55°) is called the second order rainbow.
The inclination of rainbow can easily be shown with the help of a figure like Fig. At an inclination angle of 40° to 42°, the rainbow seen in the direction of R1v1, is the first order rainbow. A little above it, the second order rainbow is seen at an inclination angle of 52°-55°.
Points to note:
i) At each reflection, inside the planes of the water droplet, a portion of the light ray is absorbed or refracted. Hence, its brightness also gets reduced. For this reason, the second order rainbow is of much lesser brightness than the first one. For more reflections, the higher order rainbows can be formed, but their brightness would be so low that they are almost invisible.
ii) Rainbow is an example of an almost pure spectrum. Due to dispersion of sunrays, its seven colours can be seen almost distinctly. But some impurity still creeps in. Due to the finite size of the sun, the rays do not remain perfectly parallel. As a result the inclination angles for minimum deviation also vary. So there is bound to be some overlapping of colours in the rainbow, and consequently the spectrum must incur some impurity, however small.
iii) Two observers, standing near each other, do not see the same rainbow because the circular arcs drawn, taking their eyes as centre, are always different. This is why the two of them see two different rainbows at the same time. More finely, it can be said, one single observer sees two different rainbows in his two eyes.
iv) Sometimes, an observer [Fig.], looking straight sees a rainbow in the sky and at the same time a reflection of the rainbow in a pond or in a reflector on the earth surface. It is very clear from the Fig., that, this image is not the image of the one seen by the observer. Looking straight, the observer sees the rainbow at P1 position in the sky, on the other hand the image P’2 seen through the reflector is the image of the rainbow which is at a position P2 in the sky.
A comparatively darker zone lying in between the primary and secondary rainbow is called Alexander’s dark band. In 200AD, Greek philosopher Alexander explained the phenomenon. The dark band is formed due to difference in the angles of deviation of the primary and secondary rainbows. The appearance of this dark band to a person standing at a partic-ular place means that no light is being dispersed to his eyes from the region of the dark band. Just like the rainbow, the position of the dark band is not fixed. Light dispersed from this region may reach the eyes of another person standing at a different position. So, he may see a rainbow at the same region and a dark band at some other region.