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

A parallel-plate capacitor with only air between the plates is charged by connecting it 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 $\mathrm{V}$ . What is the dielectric constant of this material? (b) What will the voltmeter read if the dielectric is now pulled partway out so it fills only one-third of the space between the plates?

A parallel-plate capacitor with only air between the plates is charged by connecting it 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 $\mathrm{V}$ . What is the
dielectric constant of this material? (b) What will the voltmeter read if the dielectric is now pulled partway out so it fills only one-third of the space between the plates?

Key Concepts

Equivalent Capacitance in Non-uniform Dielectric Configurations

In scenarios where a dielectric partially fills a capacitor, the system behaves as if it is composed of multiple capacitors with different dielectric properties. Determining the overall or equivalent capacitance involves combining these capacitors, often modeled in series or parallel, to account for the non-uniform distribution of the dielectric material. This concept is crucial when analyzing systems with partial dielectric insertion and its effect on voltage and overall performance.

Charge Conservation in Isolated Capacitors

When a capacitor is disconnected from a battery, it becomes isolated and the charge on its plates remains constant even if its capacitance changes. This principle of charge conservation is essential for understanding how voltage across the capacitor varies when external factors, such as the insertion of a dielectric, alter its capacitance.

Dielectric Materials and Dielectric Constant

Dielectric materials are insulating substances that, when placed between the plates of a capacitor, increase the capacitance by reducing the effective electric field. The dielectric constant (?) is a measure that quantifies how much a dielectric material can increase the capacitance compared to the vacuum. This concept is fundamental when analyzing changes to capacitor behavior upon inserting different substances between the plates.

Parallel-plate Capacitors

A parallel-plate capacitor is a basic electrical component composed of two conductive plates separated by an insulating region. Its capacitance is directly related to the area of the plates and inversely related to the distance between them. This relationship governs how the capacitor stores electrical energy in the electric field established between the plates.

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