Water bottle fire starting optics problem · CAP 2019 #B2

Physics · optics

Problem

A water bottle with diameter of 6.5 cm, made of transparent material, is left lying sideways on the wooden floor on the terrace, which is fully exposed to the sun during the whole day. 1. Show that it can start a fire on the wood floor. 2. How far away from the line along which the bottle touches floor will the fire start? 3. What will be the angular position of the sun above the horizon? State clearly all the assumptions and approximations you made. The solar constant s (the amount of solar radiation energy hitting the square meter surface) at the Earth surface is about 1 kW/m². Ignition temperature of wood is about 500 °C. The object at the temperature T emits the energy

P = eA\sigma(T^4 - T_0^4)

where

T_0

is the surrounding temperature, σ is the Stefan-Boltzmann constant σ = 5.67×10⁻⁸ W/m²·K and A is the emitting surface. Assume emissivity of wood e = 1. Ignore the heat conductivity of wood. You might find the following lens equations (which also approximately apply to the thick and cylindrical lenses) useful:

\frac{1}{d_0} + \frac{1}{d_i} = \frac{1}{f}

where

d_0

is distance from lens to object,

d_i

is the distance from lens to image, and f is the focal length of the lens.

M = \frac{h_i}{h_0} = \frac{d_i}{d_0}

where

h_i

and

h_0

are the sizes of the image and object. M is the magnification, which is the ratio of image size to object size.

Related theory

optics

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