A vertical parallel-plate capacitor is half filled with a...

A vertical parallel-plate capacitor is half filled with a dielectric for which the dielectric constant is $2.00$ (Fig. P26.71a). When this capacitor is positioned horizontally, what fraction of it should be filled with the same diclectric (Fig. P26.71b) so that the two capacitors have equal capacitance?

Key Concepts

Series and Parallel Combinations of Capacitors

The methods of combining capacitors in series or parallel are used to calculate an effective capacitance when multiple capacitive elements are connected together. For spatially distributed dielectrics in a capacitor, the configuration of the dielectric—whether it occupies contiguous regions along the direction of the electric field (series) or spans the entire voltage difference (parallel)—determines how the individual capacitance contributions add together to yield the overall capacitance. This concept is key when determining the proper fraction of a capacitor that should be filled with a dielectric to match a given effective capacitance.

Partial Dielectric Insertion in Capacitors

When a capacitor is only partially filled with a dielectric, its overall behavior can be modeled by dividing the capacitor into regions with different permittivities. Depending on how the dielectric is arranged (for instance, covering part of the plate area or part of the gap), these regions can be treated as capacitors in parallel or in series. This concept is important for determining the effective capacitance when only a fraction of the space between plates contains a dielectric material.

Dielectric Constant

The dielectric constant, or relative permittivity, is a measure of a material’s ability to increase the capacitance of a capacitor compared to a vacuum. It quantifies how much the electric field is reduced within the dielectric, which in turn enhances the capacitor’s charge storage capacity. This concept is crucial when analyzing changes in capacitance due to the introduction of different materials between the capacitor plates.

Parallel?Plate Capacitor

A parallel?plate capacitor consists of two conducting plates separated by a distance, with a dielectric medium in between. Its capacitance is given by the formula C = ?A/d, where ? is the permittivity (the product of the dielectric constant and the permittivity of free space), A is the area of the plates, and d is the separation distance. Understanding this formula is essential when modifying the dielectric filling as it directly affects the capacitance.

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