(II) How much energy is stored by the electric field between...

(II) How much energy is stored by the electric field between two square plates, 8.0 $\mathrm{cm}$ on a side, separated by a $1.5-\mathrm{mm}$ air gap? The charges on the plates are equal and opposite and of magnitude 420$\mu \mathrm{C}$ .

Key Concepts

Electric Field and Voltage Relationship

In a capacitor, the electric field (E) between the plates is related to the voltage difference (V) and the separation distance (d) by the relation V = Ed. This relationship is fundamental in linking the spatial distribution of the electric field to the potential difference, thereby assisting in the calculation of energy stored and the behavior of the capacitor under different electrical conditions.

Electric Potential Energy in a Capacitor

The electric potential energy stored in a capacitor is the energy held within its electric field due to the separation of charge on its plates. This energy can be calculated using formulas such as U = ½ CV² or U = ½ Q²/C, where C is the capacitance, V is the voltage across the capacitor, and Q is the charge on the plates. This concept is essential for understanding energy storage and discharge in electrical systems.

Permittivity of Free Space

The permittivity of free space, denoted as ??, is a fundamental physical constant that quantifies how an electric field affects and is affected by a vacuum. Its value is approximately 8.85×10?¹² F/m. This constant plays a key role in defining the capacitance of capacitors and in the equations governing electric field propagation in a vacuum.

Capacitance

Capacitance is a measure of a capacitor's ability to store electric charge per unit voltage. For a parallel-plate capacitor, it is determined by the area of the plates, the separation between them, and the dielectric constant of the material in between. The formula for the capacitance of a parallel-plate capacitor is C = ??A/d, where ?? is the permittivity of free space, A is the plate area, and d is the distance between the plates.

Parallel-Plate Capacitor

A parallel-plate capacitor consists of two conductive plates separated by a small distance with a dielectric material between them (often air or vacuum). This setup creates a uniform electric field in the region between the plates when a voltage is applied, enabling the storage of electrical energy. The concept is central to many areas of physics and engineering, particularly in the design of electronic circuits and energy storage devices.

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