Rainbow — Revision Notes

⚡ 30-Second Revision

Primary Rainbow: — 1 TIR, Red outside ($approx 42^circ$), Violet inside ($approx 40^circ$), Brighter.
Secondary Rainbow: — 2 TIRs, Violet outside ($approx 54^circ$), Red inside ($approx 51^circ$), Fainter.
Phenomena: — Refraction, Dispersion, Total Internal Reflection.
Observation: — Sun behind observer, rain in front. Center at anti-solar point.
Critical Angle: — $sin \theta_c = \frac{n_{\text{air}}}{n_{\text{water}}} approx \frac{1}{1.33} approx 48.75^circ$.
Alexander's Dark Band: — Dark region between primary and secondary rainbows ($approx 42^circ - 51^circ$) due to lack of scattered light.

2-Minute Revision

Rainbows are natural spectacles formed by sunlight interacting with water droplets, involving refraction, dispersion, and total internal reflection (TIR). White sunlight first enters a raindrop, refracts, and disperses into its constituent colors (VIBGYOR).

These dispersed rays then undergo one or more TIRs inside the droplet before refracting out towards the observer. The primary rainbow is brighter, formed by one TIR. Red light, deviating least, appears on the outer edge at about $42^circ$ from the anti-solar point, while violet is on the inner edge at $40^circ$.

The secondary rainbow is fainter, formed by two TIRs. Its colors are inverted, with violet on the outer edge at about $54^circ$ and red on the inner edge at $51^circ$. The region between these two rainbows, known as Alexander's Dark Band, appears darker because no significant light is scattered by raindrops into that angular range.

For observation, the sun must be behind the observer, and rain in front.

5-Minute Revision

Rainbows are stunning optical phenomena resulting from the interplay of sunlight and water droplets. The process begins with dispersion, where white sunlight splits into its spectrum (VIBGYOR) upon entering a raindrop.

This happens because water's refractive index varies with wavelength, causing different colors to bend at slightly different angles during refraction. Violet light bends most, red light least. After entering, the dispersed light travels to the back of the droplet.

Here, it undergoes total internal reflection (TIR), bouncing back into the water if the angle of incidence exceeds the critical angle (approx. $48.75^circ$ for water-air). Finally, the light refracts a second time as it exits the raindrop, further separating the colors and directing them towards the observer.

There are two main types: the primary rainbow and the secondary rainbow. The primary rainbow is formed by one total internal reflection. It is brighter, with red on the outer (top) edge and violet on the inner (bottom) edge, spanning an angular radius of approximately $40^circ$ (violet) to $42^circ$ (red) from the anti-solar point (opposite the sun).

The secondary rainbow is formed by two total internal reflections. This extra reflection makes it fainter and inverts the color order, so violet is on the outer (top) edge and red on the inner (bottom) edge, with a larger angular radius of approximately $51^circ$ (red) to $54^circ$ (violet).

The dark region between these two rainbows is called Alexander's Dark Band, occurring because no significant light is scattered into that angular range ($42^circ - 51^circ$). To see a rainbow, the sun must always be behind the observer, and water droplets must be in front.

Each observer sees a unique rainbow, as it's an optical effect, not a fixed object.

Worked Example: If the critical angle for water is $48.75^circ$, and a light ray hits the back of a raindrop at $50^circ$, will it contribute to a rainbow? Yes, because $50^circ > 48.75^circ$, so it will undergo TIR, a necessary step for rainbow formation. If it hit at $40^circ$, it would refract out and not contribute to the main rainbow arcs.

Prelims Revision Notes

Rainbows are optical phenomena caused by sunlight interacting with water droplets. Key processes involved are refraction, dispersion, and total internal reflection (TIR).

1. Dispersion: White light splits into VIBGYOR because water's refractive index ($n$) varies with wavelength ($lambda$). $n_{\text{violet}} > n_{\text{red}}$, so violet light bends more than red light.

2. Refraction: Light bends upon entering (air to water) and exiting (water to air) the raindrop. Governed by Snell's Law: $n_1 sin \theta_1 = n_2 sin \theta_2$.

3. Total Internal Reflection (TIR): Occurs when light travels from a denser medium (water) to a rarer medium (air) and the angle of incidence ($i$) is greater than the critical angle ($heta_c$). For water-air, $heta_c approx 48.75^circ$.

Primary Rainbow:

Formation: — Two refractions + One total internal reflection.
Brightness: — Brighter and more intense.
Color Order (Top to Bottom): — Red, Orange, Yellow, Green, Blue, Indigo, Violet (ROYGBIV).
Angular Radius (from anti-solar point): — Red $approx 42^circ$, Violet $approx 40^circ$. (Red is on the outer edge).

Secondary Rainbow:

Formation: — Two refractions + Two total internal reflections.
Brightness: — Fainter due to two reflections.
Color Order (Top to Bottom): — Violet, Indigo, Blue, Green, Yellow, Orange, Red (VIBGYOR - inverted).
Angular Radius (from anti-solar point): — Violet $approx 54^circ$, Red $approx 51^circ$. (Violet is on the outer edge).

Conditions for Observation:

Sun must be behind the observer.
Water droplets (rain/mist) must be in front of the observer.
The center of the rainbow always lies on the anti-solar point (directly opposite the sun).

Alexander's Dark Band:

A darker region observed between the primary ($42^circ$) and secondary ($51^circ$) rainbows.
Caused by the lack of significant light scattering by raindrops into this specific angular range.

Supernumerary Rainbows:

Faint, narrow bands of color seen inside the primary or outside the secondary rainbow.
Explained by diffraction and interference, not just geometric optics.

Key takeaway: Focus on the number of reflections, color order, relative brightness, and angular positions for primary vs. secondary rainbows.

Vyyuha Quick Recall

For Primary vs. Secondary Rainbows: Primary: Pretty (Brighter), Proper (ROYGBIV), Precise (1 TIR), Positive angle (Red $42^circ$ outer). Secondary: Subdued (Fainter), Switched (VIBGYOR), Second (2 TIRs), Superior angle (Violet $54^circ$ outer).