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

A certain parallel-plate capacitor is filled with a dielectric for which $\kappa=2.4$. The area of each plate is $0.017 \mathrm{~m}^{2}$, 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

Parallel-Plate Capacitor

A parallel-plate capacitor is a fundamental electrical component that stores energy by maintaining a separation of charge between two conductive plates. Its capacitance depends on the plate area, their separation distance, and the permittivity of the intervening medium. The standard formula, C = ?A/d, becomes modified when a dielectric is inserted between the plates.

Dielectric Constant

The dielectric constant is a dimensionless quantity that indicates how much a given material increases the capacitance of a capacitor relative to a vacuum. By multiplying the vacuum permittivity by the dielectric constant, the effective permittivity becomes higher, which in turn enhances the capacitor’s ability to store charge for a given geometry.

Dielectric Breakdown

Dielectric breakdown is the process where an insulating material becomes conductive due to an excessively high electric field. Each dielectric has a maximum electric field threshold, above which it can no longer prevent the flow of current, leading to a short circuit or failure of the capacitor. Understanding the breakdown field is crucial to ensuring reliable operation within safe limits.

Energy Storage in a Capacitor

The energy stored in a capacitor is determined by the formula U = (1/2)CV^2, where C is the capacitance and V is the voltage across the capacitor. When a capacitor is limited by a dielectric breakdown voltage, the maximum energy that can be safely stored is dictated by that voltage limit. Alternatively, by expressing the energy density using the permittivity and electric field (U = (1/2)?E^2 times the volume), one can connect the field strength limit to the stored energy.

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