A wafer of titanium dioxide (κ=173) of area 1.00 cm^2 has a thickness of 0.100 mm. Aluminum is e

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A wafer of titanium dioxide (κ=173) of area 1.00 cm^2 has a...

A wafer of titanium dioxide $(\kappa=173)$ of area $1.00 \mathrm{cm}^{2}$ has a thickness of $0.100 \mathrm{mm}$. Aluminum is evaporated on the parallel faces to form a parallel-plate capacitor. (a) Calculate the capacitance. (b) When the capacitor is charged with a $12.0-\mathrm{V}$ battery, what is the magnitude of charge delivered to each plate? (c) For the situation in part (b), what are the free and induced surface charge densities? (d) What is the magnitude of the electric field?

Key Concepts

Surface Charge Density

Surface charge density is defined as the charge per unit area on the plates of a capacitor. It gives insight into how charge is distributed on the surfaces and is significant in analyzing the local electric field and the impact of induced charges in the presence of dielectrics.

Electric Field in Capacitors

The electric field in a parallel-plate capacitor is related to the voltage and separation distance between the plates, typically given by E = V/d in dielectric-filled capacitors. This concept is crucial for exploring how voltages lead to field formation, which in turn influences the force on charges and the energy stored within the system.

Charge-Voltage Relationship in Capacitors

The charge stored on a capacitor is directly proportional to the applied voltage, expressed by Q = CV. This relationship is fundamental in determining the amount of charge delivered to the capacitor and is pivotal in understanding the energy storage and discharge behavior of capacitors in circuits.

Capacitance Computation with Dielectrics

The capacitance of a parallel-plate capacitor with a dielectric is calculated using the formula C = (???A)/d, where ? is the dielectric constant, ?? is the permittivity of free space, A is the area of one plate, and d is the separation between the plates. This formula demonstrates how geometrical factors and material properties combine to determine the capacitor's charge-storing ability.

Dielectric Materials and Dielectric Constant

Dielectric materials are insulating substances placed between the plates of a capacitor. They enhance the capacitor's ability to store charge by reducing the effective electric field, characterized by the dielectric constant (or relative permittivity), which quantifies how much the material increases the capacitance compared to a vacuum.

Parallel-Plate Capacitors

A parallel-plate capacitor is a system consisting of two conductive plates separated by a distance, where an electric field is established when a voltage is applied. It serves as a basic model in electrostatics for understanding charge storage, energy storage, and the distribution of the electric field between the plates.

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