Solution

Problem Statement

$13.4.17.$ A vessel containing 100 g of water is placed at the focus of a spherical mirror. What should be the diameter of the mirror so that the water boils in 1 minute, assuming the mirror is directed straight at the Sun? The energy flux density from the Sun is 0.14 W/cm². Neglect any losses.

Solution

To boil the water, the required amount of energy is the heat of vaporization:

$$Q = L \cdot m \tag{1}$$

where $L$ is the specific heat of vaporization of water. The power needed to deliver this energy in time $t$ is:

$$P = \frac{Q}{t} = \frac{L m}{t} \tag{2}$$

This power is provided by the Sun’s radiation via the mirror:

$$P = \varepsilon \cdot S \tag{3}$$

where $\varepsilon$ is the energy flux density given in the problem, and $S$ is the cross-sectional area of the spherical mirror. If the diameter of the mirror is denoted by $D$, then $S = \frac{\pi D^2}{4}$. Equating equations (2) and (3):

$$\frac{L m}{t} = \varepsilon \cdot \frac{\pi D^2}{4}$$

Solving for $D$:

$$D = \sqrt{\frac{4 L m}{\pi \varepsilon t}}$$

Substituting the numerical values gives:
$D = 1.85 \text{ m}$