The Theory of The Rainbows

Physic 464ProjectBy: Thao Dang

What is a rainbow?

• Rainbow is an arc of colors appearing opposite the sun as a result of the refraction of sunlight in rain.

Presentation points

• How does a rainbow form?• What make colors in the rainbows?• Mathematical of Rainbows• Types of rainbows• Experiment

How does a rainbow form?

• Rainbows are created by sunlight shine and water droplets.

• Rainbows are created by refraction and reflection of sun’s rays in falling rain or mist.

• Sunlight strikes the front of raindrop and is refracted and separated into it’s colors. Then the light hit the back of the raindrop and is reflected, and is refracted again as it leaves the raindrop.

Light travel through the raindrop to create rainbow

What makes colors in the rainbow?

• The sunlight consists of all wavelengths of visible light, which we simply see as white because they are all combined together.

• When light is refracted, different wavelengths refract at different angles, therefore the various colors are separated.

• Rainbows have 7 colors; however, they actually consist of every color. The color is constantly changing from red to violet.

What makes the colors of rainbow?

• Each of the colors presents in white light are bent a different amount in passing through the drop.

• 7 colors in a rainbow is formed follow the order: Red, Orange, Yellow, Green, Blue, Indigo, and Violet.

• The red bends the least and violet bends the most. • The rest of the color lie in between the red and the

violet.

What makes colors in the rainbow?

What makes colors in the rainbow?

Mathematics of Rainbows

The mathematics of rainbow relates toLaw of Refraction (Snell’s Law)Law of ReflectionLaw of SinesGeometryDerivative

Rainbow Angle

Calculation Rainbow Angle

• Using Snell’s law• n*sin(a) = n*sin(b)• The radius of the circle is r• We have: sin(a) = h/r = M

Rainbow angle

• The angle of incident equal the angle of reflection

• The angle we want to find is the one between the incoming ray and outgoing ray which is named e

• The sum of the interior angles of a triangle is 180 degree

Rainbow Angle

• Sum of all angle at the center of circles is 360 degree• 360 = a + (180 – 2b) + (180 – 2b) + f• f = 4b – a• f = e + a• e = f – a• e = 4b – 2 a• sin (a) = h/r = M• a = (1/sin)M

Rainbow angle

• n*sin(b) = sin(a)• b= (1/sin)(M/n)

Finally we have final formula• e= 4 * (1/sin) (M) – 2*(1/sin)(M)

Take derivative of e with respect to (M) and find where it has an extreme by setting the derivative

Equal 0 and solve for (M) we get the final result for M = ((4-n^2)/3)^(1/2) (a)

Rainbow Angle

• e has a maximum value given by• e = (4*(1/sin)(((4-n^2)/3)^(1/2))/n- 2*(1/sin)(((4-

n^2)/3)^(1/2)). (b)• n water= 1.333• Substitute n = 1.33 into (a) and (b) we have• M = 0.8608• e = 42.0 degree

Nature Rainbow

• Nature rainbow is created by nature. We can see the rainbow when it rains or at the waterfall under the sunshine.

Nature Rainbow

Nature Rainbow

Manmade Rainbows

• Manmade Rainbows are the rainbows that are created by human.

• We can make our own rainbows in many ways• We can use a garden hose, a sprinkler, and a water

spout out, and the sunlight to make our own rainbows.

• We can use a prism and a flashlight to make a rainbow

• We can also make a rainbow by using a flashlight, a round container, and a large piece of cardboard.

Stern, an artist and visual arts professor, uses fire trucks, boats, pumps, and hoses to create rainbows at Germany.

I used a spherical bottle, a flashlight, water, and a piece of cardboard to create a simple rainbow

References

• http://brantancan.co.uk/rainbow.htm• http://www.albemarle.cc.nc.us/~aaldridg/mat172/proje

ct2.html• http://www.psupress.org/samplechapters/justataste_lee.

html• http://www.unidata.ucar.edu/staff/blynds/rnbw.html• Overheim, R.Daniel and Wagner, David L. Light and

Color.1982.• Reese, Ronald Lane. University of Physics.1998.

Thanks for your listening