Problem 66 - Physics

Question

Go Concept Questions Capacitor A and capacitor B both have the same voltage across their plates. However, the energy of capacitor A can melt \(m\) kilograms of ice at \(0^{\circ} \mathrm{C}\), while the energy of capacitor B can boil away the same amount of water at \(100{ }^{\circ} \mathrm{C}\). (a) Which requires more energy, melting the ice or boiling the water? (b) Which capacitor has the greater capacitance? Explain your answers. The capacitance of capacitor \(A\) is \(9.3 \mu F .\) What is the capacitance of capacitor B? Be sure that your answer is consistent with your answers to the Concept Questions.

Step-by-Step Solution

Answer

Boiling water requires more energy. Capacitor B has a greater capacitance, approximately 63.01 µF.

Step 1: Understand Energy Requirement for Phase Changes

To melt ice at 0°C, we need the latent heat of fusion, which is approximately 334 kJ/kg. To boil water at 100°C, we need the latent heat of vaporization, which is approximately 2260 kJ/kg. Thus, boiling water requires more energy than melting ice for the same mass.

Step 2: Determine Capacitor Energy Equations

The stored energy in a capacitor is given by the equation \( E = \frac{1}{2} C V^2 \). Since both capacitors have the same voltage \( V \), their energy is dependent on the capacitance \( C \).

Step 3: Compare Energy Required to Capacitor Energy

Since boiling the water requires more energy than melting the ice, capacitor B, which can boil water, must store more energy than capacitor A. Hence, \( C_B V^2 > C_A V^2 \), implying \( C_B > C_A \) due to equal voltage \( V \).

Step 4: Calculate the Capacitance of Capacitor B

Given that \( C_A = 9.3 \mu F \), we know \( C_B \) must be larger because it has to store more energy. The ratio of energy required for boiling over melting is approximately \( \frac{2260}{334} \approx 6.77 \). Thus, \( C_B \approx 6.77 \times 9.3 \mu F \approx 63.01 \mu F \).

Key Concepts

Capacitance

Capacitance is a property that indicates how much electric charge a capacitor can store per unit of voltage (V). It is measured in farads (F). The formula for capacitance is given by the equation:

• Capacitance, \[ C = \frac{Q}{V} \] where \( Q \) is the charge and \( V \) is the voltage applied across the capacitor.

A higher capacitance means that the capacitor can store more charge at the same voltage level. In this exercise, both capacitors A and B have the same voltage across their plates, but different capacitances. Capacitor B has a greater capacitance since it needs to store more energy to boil water compared to capacitor A, which only melts ice.
Capacitance is crucial in energy storage applications because it defines how effective the capacitor will be when it is used to deliver or absorb energy at given voltages.

Energy Storage

Capacitors are widely used for energy storage because they can release energy quickly when a load requires it. The stored energy (\( E \)) in a capacitor is given by the formula:

• Energy, \( E = \frac{1}{2} C V^2 \)

This equation shows that energy stored in a capacitor depends upon its capacitance \( C \) and the square of the voltage \( V \) across its plates. Thus, even a small increase in voltage or capacitance will significantly amplify the energy stored.
In the given exercise, it's explained that capacitor B stores more energy than capacitor A, primarily due to its larger capacitance. This enhanced energy storage capability allows capacitor B to perform the more energy-intensive task of boiling water, unlike capacitor A, which only melts ice.

Latent Heat

Latent heat refers to the amount of energy required to change the phase of a substance, without altering its temperature. Two common types of latent heat are important here:

• Latent Heat of Fusion: Energy needed to melt ice at 0°C is approximately 334 kJ/kg.
• Latent Heat of Vaporization: Energy needed to boil water at 100°C is much higher, at about 2260 kJ/kg.

In this exercise, melting the ice requires less energy compared to boiling the water. Latent heat is an essential concept for understanding thermal processes and energy transformations during phase changes, particularly in systems involving energy storage, like capacitors.

Phase Changes

Phase changes involve transitions of matter from one state to another, like solid to liquid or liquid to gas. These changes require energy but do not change the temperature during the process. Here, the key phase changes are melting (solid to liquid) and boiling (liquid to gas).

• Melting involves absorbing the latent heat of fusion, turning ice into water without changing its temperature.
• Boiling involves absorbing the latent heat of vaporization, transforming water into vapor, again without a temperature change.

Understanding phase changes is vital in this problem to help deduce the relative energy demands of capacitors A and B. Boiling the same amount of water is more energy-intensive than melting ice, which logically means the capacity for energy storage is greater in the system that performs the boiling, i.e., capacitor B. This reflects the larger capacitance of capacitor B, which can hold and use more energy effectively for phase transitions.