A parallel-plate capacitor with only air between its plates...

A parallel-plate capacitor with only air between its plates is charged by connecting the capacitor to a battery. The capacitor is then disconnected from the battery, without any of the charge leaving the plates. (a) A voltmeter reads $45.0 \mathrm{V}$ when placed across the capacitor. When a dielectric is inserted between the plates, completely filling the space, the voltmeter reads 11.5 V. What is the dielectric constant of the material? (b) What will the voltmeter read if the dielectric is now pulled away out so it fills only one-third of the space between the plates?

Key Concepts

Composite Capacitor (Partial Filling)

When a dielectric fills only part of the space between the plates of a capacitor, the system can be modeled as a combination of capacitors with different dielectric properties connected in series or parallel, depending on the configuration. Analyzing such composite capacitors involves calculating an effective capacitance, which is essential for understanding how partially filling the capacitor with a dielectric material alters the overall voltage and energy storage characteristics.

Capacitance-Voltage Relationship

The relationship between capacitance (C), voltage (V), and charge (Q) is governed by the equation Q = CV. When the configuration of a capacitor changes, such as by inserting a dielectric, the capacitance changes and so does the voltage across the capacitor if the charge remains constant. This relationship is key to predicting how measurements like voltage will change when modifications are made to the capacitor.

Charge Conservation

When a capacitor is charged and then disconnected from any voltage source, the charge on its plates remains constant. This conservation of charge is crucial for analyzing scenarios where the physical configuration of the capacitor is changed (for example, by inserting or partially inserting a dielectric), as any change in voltage or capacitance must occur under the constraint that the stored charge is fixed.

Dielectric Constant

The dielectric constant (or relative permittivity) is a dimensionless quantity that indicates how much a dielectric material can increase the capacitance of a capacitor compared to vacuum or air. It represents the factor by which the electric field is reduced and the capacitance is increased, making it central to analyses involving changes in capacitor behavior when different materials are used as the insulating medium.

Dielectric

A dielectric is an insulating material placed between the plates of a capacitor. It reduces the electric field within the capacitor for the same amount of charge on the plates, effectively increasing the capacitance. The presence of a dielectric means that the stored energy and voltage characteristics of a capacitor change, which is a key principle in applications that require tuning capacitor properties without altering physical dimensions.

Parallel-Plate Capacitor

A parallel-plate capacitor is an electrical component that consists of two conductive plates separated by an insulating material (or dielectric). Its capacitance, which determines how much charge it can store per unit voltage, is primarily determined by the area of the plates, the separation distance, and the properties of the dielectric between them. This concept is fundamental in understanding how capacitors function in various circuits and how modifications (like inserting a dielectric) affect their behavior.

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