A certain parallel-plate capacitor is filled with a...

A certain parallel-plate capacitor is filled with a dielectric for which $\kappa=5.5 .$ The area of each plate is $0.034 \mathrm{m}^{2},$ and the plates are separated by 2.0 $\mathrm{mm}$ . The capacitor will fail (short out and burn up) if the electric field between the plates exceeds 200 $\mathrm{kN} / \mathrm{C}$ . What is the maximum energy that can be stored in the capacitor?

A certain parallel-plate capacitor is filled with a dielectric
for which $\kappa=5.5 .$ The area of each plate is $0.034 \mathrm{m}^{2},$ and the plates
are separated by 2.0 $\mathrm{mm}$ . The capacitor will fail (short out and burn up) if the electric field between the plates exceeds 200 $\mathrm{kN} / \mathrm{C}$ . What
is the maximum energy that can be stored in the capacitor?

Key Concepts

Energy Storage in Capacitors

The energy stored in a capacitor is a function of its capacitance and the voltage applied across its plates. Typically expressed as one-half the product of the capacitance and the square of the voltage (U = ½ CV²), this relationship is fundamental in evaluating the performance and safety limits of capacitors. It connects the capacitor's physical properties and operating voltage conditions to the maximum energy it can safely handle.

Dielectric Material

A dielectric is an insulating material placed between the plates of a capacitor to increase its capacitance. Its effectiveness is measured by the dielectric constant (?), which indicates how much the electric field is reduced inside the material compared to vacuum. This concept is crucial because the presence of a dielectric enhances the capacitor’s ability to store electrical energy without causing conduction between the plates.

Parallel-Plate Capacitor

A parallel-plate capacitor consists of two conductive plates separated by a small distance, and its capacitance is directly proportional to the plate area and inversely proportional to the distance between them. In the case with a dielectric, the capacitance is increased by the factor of the dielectric constant. This foundational concept is key for analyzing how physical dimensions and materials affect energy storage in capacitors.

Dielectric Breakdown

Dielectric breakdown refers to the point at which an insulating material, such as the dielectric in a capacitor, becomes conductive due to an excessive electric field. This is a critical limitation in capacitor design, as exceeding the breakdown electric field can cause the capacitor to short out and potentially burn up. Therefore, understanding the breakdown field strength is essential for determining the maximum operating limit and energy that can be safely stored in the capacitor.

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