Chapter 24: Problem 46
A parallel-plate air capacitor is made by using two plates 12 cm square, spaced 3.7 mm apart. It is connected to a 12-V battery. (a) What is the capacitance? (b) What is the charge on each plate? (c) What is the electric field between the plates? (d) What is the energy stored in the capacitor? (e) If the battery is disconnected and then the plates are pulled apart to a separation of 7.4 mm, what are the answers to parts (a)-(d)?
Short Answer
Step by step solution
Calculating Capacitance (Initial Setup)
Calculating Initial Charge on Plates
Calculating Electric Field Between Plates (Initial Setup)
Calculating Initial Energy Stored
Capacitance after Battery Disconnection and Plate Separation
Charge After Changing Plate Separation
Electric Field After Changing Plate Separation
Energy Stored After Changing Plate Separation
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parallel-plate capacitor
The idea is that when a voltage is applied, one plate accumulates a positive charge and the other a negative charge, creating an electric field between them. With a parallel-plate capacitor, the larger the surface area of the plates and the smaller the distance between them, the more energy it can store. This makes the design quite efficient for several applications, including energy storage, filtering, and in various electronic circuits.
Capacitance calculation
For the given problem, the area \( A \) is \((0.12)^2\) square meters and the initial separation \( d \) is 0.0037 meters. By substituting these values into the formula, you can calculate the initial capacitance.
This capacity reflects how much charge the plates can hold for every volt of electric potential difference applied, establishing a core function of capacitors in storing and releasing charge as needed in circuits.
Electric field between plates
For the initial setup with a 12-V battery and a separation of 0.0037 meters, this formula helps compute the strength of the electric field. The electric field represents the force per unit charge that would be experienced by a charge placed within it.
When the separation is increased while keeping the charge constant (as when the battery is disconnected), the voltage changes and consequently, the electric field can be recalibrated. This variability is essential in understanding capacitor behavior under different physical configurations.
Energy stored in capacitor
Upon disconnecting the battery and altering the plate separation, the voltage across the capacitor changes, although the charge remains the same. The energy stored is then recalculated with the modified capacitance using the alternative expression: \[ U = \frac{Q^2}{2C} \] This highlights how energy is influenced by changes in configuration and the potential energy shift due to altered geometric or material properties. Understanding these concepts emphasizes the dynamic nature of energy storage in capacitors, vital for managing power in various applications.