Problem 48 - Fundamentals of Physics

Question

Exploding shoes. The rain-soaked shoes of a person may explode if ground current from nearby lightning vaporizes the water. The sudden conversion of water to water vapor causes a dramatic expansion that can rip apart shoes. Water has density \(1000 \mathrm{~kg} / \mathrm{m}^{3}\) and requires \(2256 \mathrm{~kJ} / \mathrm{kg}\) to be vaporized. If horizontal current lasts \(2.00 \mathrm{~ms}\) and encounters water with resistivity \(150 \Omega \cdot \mathrm{m}\), length \(12.0 \mathrm{~cm}\), and vertical cross-sectional area \(15 \times 10^{-5} \mathrm{~m}^{2}\), what average current is required to vaporize the water?

Step-by-Step Solution

Answer

The average current required is approximately 412.31 A.

Step 1: Calculate the Volume of Water

First, we calculate the volume \( V \) of the water using the formula \( V = A \times L \), where \( A \) is the cross-sectional area and \( L \) is the length of the water column. Given \( A = 15 \times 10^{-5} \text{ m}^2 \) and \( L = 12.0 \text{ cm} = 0.12 \text{ m} \), the volume is: \( V = 15 \times 10^{-5} \cdot 0.12 = 1.8 \times 10^{-5} \text{ m}^3 \).

Step 2: Calculate the Mass of Water

Using the density \( \rho = 1000 \text{ kg/m}^3 \), the mass \( m \) is given by \( m = \rho \times V \). Thus, \( m = 1000 \cdot 1.8 \times 10^{-5} = 0.018 \text{ kg} \).

Step 3: Calculate the Energy Required to Vaporize Water

The energy \( Q \) required to vaporize the water is given by \( Q = m \cdot L_v \), where \( L_v = 2256 \text{ kJ/kg} = 2.256 \times 10^6 \text{ J/kg} \). Therefore, \( Q = 0.018 \cdot 2.256 \times 10^6 = 40608 \text{ J} \).

Step 4: Calculate the Resistance of Water

The resistance \( R \) is calculated using \( R = \frac{\rho L}{A} \), where \( \rho = 150 \Omega \cdot \text{ m} \). Substituting the given values, \( R = \frac{150 \times 0.12}{15 \times 10^{-5}} = 120 \Omega \).

Step 5: Calculate the Power

Power \( P \) is given by \( P = \frac{Q}{t} \), where \( t = 2 \text{ ms} = 2 \times 10^{-3} \text{ s} \). Thus, \( P = \frac{40608}{2 \times 10^{-3}} = 20304000 \text{ W} \).

Step 6: Calculate the Average Current

The power can also be expressed in terms of current and resistance as \( P = I^2 \cdot R \). Rearranging the formula gives \( I = \sqrt{\frac{P}{R}} \). Substituting the calculated values, \( I = \sqrt{\frac{20304000}{120}} = 412.31 \text{ A} \).

Key Concepts

Thermodynamics

Thermodynamics is the study of heat, energy, and the movement of particles. In this scenario, thermodynamics helps explain how energy transitions from one form to another. When lightning strikes, it releases a significant amount of electrical energy. This energy gets transferred into the water trapped in the shoes, transforming it from liquid to vapor.
The energy required for this phase change is known as the latent heat of vaporization. For water, it requires significant energy to break the hydrogen bonds holding the water molecules in liquid form. This is quantified by the latent heat of 2256 kJ/kg.
Understanding how energy transfers and transforms helps explain how a seemingly harmless puddle in a shoe can become an explosive event.

Electric Current

Electric current is essentially the flow of electric charge. In electronics and physics, it's often measured in amperes (A), which tell us how much charge flows through a point in a circuit over a period of time. In the problem, we need to calculate the current necessary to generate enough energy to vaporize the water using Ohm's Law.
The current’s role is vital, as it's the main culprit that provides the required energy to turn the water in the shoe from a liquid into gas. The faster and more current there is, the more energy is transferred, leading to the shoe bursting open. It's crucial to understand how current, resistance, and energy interplay to predict such occurrences effectively.

Resistivity

Resistivity is a material-specific property that indicates how much the material opposes the flow of electric current. It's denoted by the Greek letter rho (\( \rho \)) and is crucial for understanding how charged particles move through different substances.
In the exercise, water's resistivity is given as 150 Ω·m. Higher resistivity means the material will resist the flow more, needing more electrical power (or energy over time, in watts) to move the current through it. Understanding resistivity helps determine the resistance presented to electric current, which directly affects the amount of current needed to achieve the energy conversion necessary for vaporization.

Energy Conversion

Energy conversion is the process of transforming one form of energy into another. This exercise exemplifies how electrical energy, produced by lightning, can be converted into thermal energy in wet shoes.
This conversion of electric energy into thermal energy generates heat, which is required to transform water into vapor. The specific energy required for this phase change is determined by the mass of water and the latent heat of vaporization. Since this is a rapid process, the quick and vast amount of energy converted results in the sudden and dramatic expansion, explaining the explosion of the shoes.
Understanding energy conversion is essential for deciphering the instantaneous events around us, like how simple physical principles can lead to complex outcomes such as in this scenario.