In Fig. 25-46, how much charge is stored on the...

In Fig. $25-46$, how much charge is stored on the parallel-plate capacitors by the $12.0 \mathrm{~V}$ battery? One is filled with air. and the other is filled with a dielectric for which $\kappa=3.00 ;$ both capacitors have a plate area of $5.00 \times 10^{-3} \mathrm{~m}^{2}$ and a plate separation of $2.00 \mathrm{~mm}$.

Key Concepts

Capacitance

Capacitance is a measure of a capacitor's ability to store charge per unit voltage across its plates. It is defined by the relation C = Q/V, where Q is the stored charge and V is the voltage. For a parallel-plate capacitor, the capacitance is given by the formula C = ??A/d, where ?? is the permittivity of free space, A is the plate area, and d is the separation between the plates.

Dielectric Constant

A dielectric is an insulating material that, when placed between the plates of a capacitor, increases the overall capacitance. The dielectric constant (?) quantifies this increase by indicating how much more charge a capacitor with the dielectric can hold relative to one with a vacuum. Mathematically, the capacitance in the presence of a dielectric is given by C = ???A/d, reflecting the ability of the material to reduce the effective electric field and enhance charge storage.

Parallel-Plate Capacitor

A parallel-plate capacitor consists of two conductive plates separated by an insulating material or a vacuum. Its geometry simplifies the calculation of the electric field between the plates and the capacitance, making it a fundamental example in electrostatics. The key parameters are the plate area and the distance between the plates, which determine how much electric field can be stored and, consequently, how much charge can be held at a given voltage.

Charge-Voltage Relationship

The charge stored on a capacitor is directly proportional to the voltage applied across its plates, as described by the equation Q = CV. This relationship allows one to determine the amount of charge for a given capacitance and applied voltage, making it central to understanding how capacitors work in electric circuits and energy storage applications.

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