Chapter 24: Problem 40
Polystyrene has dielectric constant 2.6 and dielectric strength 2.0 \(\times\) 10\(^7\) V/m. A piece of polystyrene is used as a dielectric in a parallel-plate capacitor, filling the volume between the plates. (a) When the electric field between the plates is 80% of the dielectric strength, what is the energy density of the stored energy? (b) When the capacitor is connected to a battery with voltage 500.0 V, the electric field between the plates is 80% of the dielectric strength. What is the area of each plate if the capacitor stores 0.200 mJ of energy under these conditions?
Short Answer
Step by step solution
Compute Electric Field
Energy Density Formula
Calculate Energy Density
Relate Energy to Capacitor Specifications
Capacitance Calculation
Relate Capacitance to Physical Dimensions
Calculate Plate Area
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Dielectric Constant
Dielectric materials are insulators, and they do not conduct electricity but can support electrostatic fields. This ability enhances the capacitor's potential to store charge, an essential aspect of various electrical applications.
- The dielectric constant is a dimensionless number and has no units.
- It varies depending on the material type and influences the energy-storing capacity of a capacitor.
Dielectric Strength
When a dielectric material reaches or exceeds its dielectric strength, it can no longer function as an insulator and may allow the enclosed charges to move freely across the material, causing a short circuit or damage to the capacitor.
- Dielectric strength ensures that under normal operating conditions, the material prevents breakdown and maintains its insulating properties.
- It is vital to select a material with a sufficient dielectric strength for applications with high voltages to prevent failure.
Energy Density
For polystyrene with a dielectric constant of 2.6 and an electric field of 1.6 × 10⁷ V/m, we substitute these values to calculate the energy density. This expresses how the internal structure of a capacitor is capable of storing energy given the specific properties of the material used.
- Higher permittivity and dielectric constants allow for greater energy density.
- Energy density is a critical factor in applications where size and efficiency are constraints.
Parallel-Plate Capacitor
\[ C = \frac{\varepsilon_0 \varepsilon_r A}{d} \]
where \( \varepsilon_0 \) is the permittivity of free space.
The parallel-plate capacitor serves a wide array of applications due to its straightforward design and reliable functionality. Typically, it is used whenever a precise and stable capacitance value is needed, as the physical parameters influencing the capacitance can be accurately controlled.
- The area of the plates and the nature of the dielectric material directly affect the overall capacitance.
- The simplicity of the design makes it easy to manufacture and integrate into various circuits.