A parallel-plate capacitor using a dielectric material having an \(\epsilon_{r}\) of \(2.5\) has a plate spacing of \(1 \mathrm{~mm}(0.04 \mathrm{in} .)\). If another material having a dielectric constant of \(4.0\) is used and the capacitance is to be unchanged, what must be the new spacing between the plates?

Understanding the capacitance formula is essential for comprehending how energy is stored in an electrical field. In the context of a parallel-plate capacitor, the formula
\(C = \frac{\epsilon_{0} \epsilon_{r} A}{d}\)
serves as a mathematical relationship among capacitance (\(C\)), vacuum permittivity (\(\epsilon_{0}\)), relative permittivity (\(\epsilon_{r}\)), the plate area (\(A\)) and the distance between the plates (\(d\)). This formula illustrates that the capacitance increases with the plate area and the dielectric constant while decreasing with greater plate spacing. Often, in exercises, you'll find scenarios where you need to adjust one parameter, like plate spacing, to maintain capacitance when another factor, such as the dielectric constant, changes.

A parallel-plate capacitor is a fundamental component in electronics, composed of two conductive plates separated by a small distance. This configuration stores electric charge and energy in the electric field between the plates. In practice, a dielectric material is often inserted between the plates to increase the capacitor's ability to store charge, as indicated by its higher capacitance. The expressions for calculating the characteristics of parallel-plate capacitors derive from the fundamental concept of electric fields in conductors and the behavior of dielectrics, which enhance the stored energy without requiring larger conductive plates or closer spacing.

Relative permittivity, also known as the dielectric constant (\(\epsilon_{r}\)), is a measure of how much a dielectric material can increase the electrical capacitance of a capacitor compared to a vacuum. This property is a dimensionless number, indicating how much the electric field decreases within the material as compared to a vacuum. In other words, a high relative permittivity means the material is very effective as a dielectric, allowing the capacitor to store more charge at a particular voltage. It's crucial to note that when the dielectric constant changes in a capacitor, if we want to keep the capacitance unchanged, we must adjust the geometry of the capacitor appropriately, as demonstrated in textbook exercises.

Vacuum permittivity (\(\epsilon_{0}\)) is a fundamental constant that represents the electric permittivity of a classical vacuum. It is used as a reference point to compare how much more capable materials are at permitting an electric field through them, as compared to the vacuum state. Vacuum permittivity has a fixed value of approximately
\(8.854 \times 10^{-12} \frac{F}{m}\) (farads per meter), and it provides the baseline in the capacitance formula. An understanding of vacuum permittivity is important for solving problems that involve electric fields and capacitances, as it dictates how much energy can be stored in a given configuration in a vacuum.