Angle of Deviation in a Raindrop

An idealized raindrop forms a sphere. These are the ones that are favoured when drawing diagrams of both raindrops and rainbows because they suggest that when light, air and water droplets interact they produce predictable and replicable outcomes.

• In real-life, full-size raindrops don’t form perfect spheres because they are composed of water which is fluid and held together solely by surface tension.
• In normal atmospheric conditions, the air a raindrop moves through is itself in constant motion, and, even at a cubic metre scale or smaller, is composed of areas at slightly different temperatures and pressure.
• As a result of turbulence, a raindrop is rarely in free-fall because it is buffeted by the air around it, accelerating or slowing as conditions change from moment to moment.
• The more spherical raindrops are, the better defined is the rainbow they produce because each droplet affects incoming sunlight in a consistent way. The result is stronger colours and more defined arcs.

Real-life raindrops

• Raindrops start to form high in the atmosphere around tiny particles called condensation nuclei — these can be composed of particles of dust and smoke or fragments of airborne salt left over when seawater evaporates.
• Raindrops form around condensation nuclei as water vapour cools producing clouds of microscopic droplets each of which is held together by surface tension and starts off roughly spherical.
• Surface tension is the tendency of liquids to shrink to the minimum surface area possible as their molecules cohere to one another.
• At water-air interfaces, the surface tension that holds water molecules together results from the fact that they are attracted to one another rather than to the nitrogen, oxygen, argon or carbon dioxide molecules also present in the atmosphere.
• As clouds of water droplets begin to form, they are between 0.0001 and 0.005 centimetres in diameter.
• As soon as droplets form they start to collide with one another. As larger droplets bump into other smaller droplets they increase in size — this is called coalescence.
• Once droplets are big and heavy enough they begin to fall and continue to grow. Droplets can be thought to be raindrops once they reach 0.5mm in diameter.
• Sometimes, gusts of wind (updraughts) force raindrops back into the clouds and coalescence starts over.
• As full-size raindrops fall they lose some of their roundness, the bottom flattens out because of wind resistance whilst the top remains rounded.
• Large raindrops are the least stable, so once a raindrop is over 4 millimetres it may break apart to form smaller more regularly shaped drops.
• In general terms, raindrops are different sizes for two primary reasons,  initial differences in particle (condensation nuclei) size and different rates of coalescence.
• As raindrops near the ground, the biggest are the ones that bump into and coalesce with the most neighbours.

An idealised raindrop forms a geometrically perfect sphere. Although such a form is one in a million in real-life,  simplified geometrical raindrops help to make sense of rainbows and reveal general rules governing why they appear.

The insights that can be gained from exploring the geometry of raindrops apply to every rainbow, whilst the rainbows we come across in everyday life demonstrate that each individual case is unique.

Don’t forget that the idea of light rays is also a way to simplify the behaviour of light:

• The idea that light is made up of rays is so commonplace when describing and explaining rainbows that it is easily taken for granted.
• The idea of light rays is useful when trying to model how light and raindrops produce the rainbow effects seen by an observer.
• Light rays don’t exist in the sense that the term accurately describes a physical property of light. More accurate descriptions use terms like photons or waves.

Basics of raindrop geometry

• A line drawing of a spherical raindrop is the starting point for exploring how raindrops produce rainbows.
• The easiest way to represent a raindrop is as a cross-section that cuts it in half through the middle.
• A dot or small circle can be used to mark the centre whilst the larger circle marks the circumference.
• Marking the centre makes it easy to add lines that show the radius and diameter.
• Marking the centre also makes it easy to add lines that are normal to the circumference.
• A normal (or the normal) refers to a line drawn perpendicular to and intersecting another line, plane or surface.
• A normal is used in a diagram to connect the centre with a point where a ray strikes the circumference.
• The diameter of a circle is a line that passes through its centre and is drawn from the circumference on one side to the other.
• The radius of a circle is a line from the centre to any point on the circumference.
• The horizontal axis of a raindrop is a line drawn through its centre and parallel to incident light. The vertical axis intersects the horizontal at 90⁰ and also passes through the centre point.
• The angle at which incident light strikes the surface of a raindrop can be calculated by drawing a line that shows where an incident ray strikes a droplet and then drawing the normal. The angle of incidence is measured between them.
• The path of light as it strikes the surface and changes direction as it is refracted at the boundary between air and water can be calculated using the Law of Refraction (Snell’s law).
• When light is refracted as it enters a droplet it bends towards the normal.
• The law of reflection can be used to calculate the change of direction each time light reflects off the inside surface of the raindrop.
• When light exits a raindrop the angle of refraction is the same as when it entered but this time bends away from the normal.