A capacitor consists of a conducting sphere of radius \(a\) surrounded by a concentric conducting shell of radius \(b\). Show that its capacitance is \(C=a b / k(b-a)\).

Electrostatics is a branch of physics that studies stationary or slow-moving electric charges and the forces they exert on each other. When dealing with problems related to electrostatics, we mainly focus on how charges interact through electric fields, even when they happen to be some distance apart.

In the context of capacitors, electrostatic principles help us understand how electric potential and electric fields operate between the plates of a capacitor. A concentric capacitor, which consists of two conductors separated by an insulating material, stores electric energy in the form of an electrostatic field. The charge distribution on the inner and outer conductors gives rise to an electric field in the region between them, directly affecting the device's ability to store charges and consequently, its capacitance.

Vacuum permittivity, denoted as \( \varepsilon_0 \), is a fundamental physical constant that describes the ability of free space (a vacuum) to permit the existence of an electric field. Also referred to as the 'electric constant', it is a measure of the resistance encountered when forming an electric field in a vacuum.

The value of vacuum permittivity is key in calculating capacitance, as it directly influences the strength of the electric field that can be established between the conductors of a capacitor. In the formula \( C = \varepsilon_0 \times (4\pi ab) / (b - a) \), the factor \( \varepsilon_0 \) determines how much energy can be stored for a given electric potential difference across the conductors. It ensures that every calculation we perform aligns with the physical reality of how electric fields behave in a vacuum.

Coulomb's constant, \( k \), is another fundamental physical constant used in electrostatics, defined by the inverse relationship to vacuum permittivity as \( k = 1/ (4\pi \varepsilon_0) \). It is essential in Coulomb's law, which quantifies the electrostatic force between two point charges.

In our concentric capacitor scenario, Coulomb's constant allows us to simplify the original capacitive formula by recognizing that \( k \) is interdependent with vacuum permittivity. By substituting the expression for Coulomb's constant into the capacitance formula, we can derive a form that is often more practical. While \( k \) might seem abstract, it is intimately connected with the electrostatic forces that govern the behavior of the capacitor, ultimately influencing its ability to store charge and energy.