A long cylindrical capacitor has radii \(a\) and \(b(b>a)\) and the space between the cylinders is filled with a dielectric of relative permittivity \(\varepsilon_{t}\) and dielectric strength \(K\). What is the maximum energy which can be stored per unit leagth of the capacitor?

Capacitance is a measure of a capacitor's ability to store electric charge per unit voltage. When we talk about a cylindrical capacitor, we refer to a system composed of two coaxial cylinders, each having a different radius. The capacitance of such a system can be determined using the formula:

\[C = 2\frac{pi}{ln{\frac{b}{a}}})\]
where \(\varepsilon_{0}\) represents the vacuum permittivity, a constant value which quantifies the electric permeability in a vacuum, while \(\varepsilon_{t}\) is the relative permittivity, a factor that describes how much the dielectric material can increase the capacitor's capacitance compared to a vacuum. In the context of a capacitor, a dielectric is simply an insulator between the conductors.

Knowing these variables—\(a\) and \(b\) being the inner and outer radii, respectively, and \(L\) the length of the capacitor—allows us to accurately calculate the capacitance. It's important to use consistent units when substituting values to ensure the correct calculation of capacitance.

Dielectric strength is a critical property of insulating material—often called a dielectric—placed between the conductors of a capacitor. It's defined as the maximum electric field the dielectric can endure without breakdown. In a cylindrical capacitor, breakdown implies that the insulating material becomes conductive, leading to a potential capacitor failure.

The dielectric strength, represented by \(K\), is measured in volts per meter (V/m). To determine the maximum voltage \(V_{max}\) that can be applied to the capacitor without causing a dielectric breakdown, one can use the relation:

\[V_{max} = K(b - a)\]
Here, \((b - a)\) is the difference between the radii of the cylinders, which effectively is the distance between them and determines the physical space the dielectric occupies. Therefore, knowing the dielectric strength is crucial to ensure we operate the capacitor within safe limits, hence preserving its integrity and functionality over time.

Capacitors are not only useful for their ability to store charge but also for their ability to store energy. The energy stored within a capacitor is contingent upon the capacitance and the voltage applied across it. The general formula for calculating this energy is:

\[U = \frac{1}{2}CV^2\]
where \(U\) is the energy stored in joules, \(C\) is the capacitance in farads, and \(V\) is the voltage in volts. For a cylindrical capacitor with a dielectric, when considering the maximum energy that can be stored per unit length, we use the derived equation:

\[U_{max} = \frac{1}{2}(2\frac{pi}{ln{\frac{b}{a}}))(K(b - a))^2\]
It’s interesting to note that not all energy input into the capacitor is stored; some energy is always dissipated as heat due to dielectric losses. Understanding energy storage and dissipation in capacitors is essential, especially in applications involving power electronics, energy systems, and signal processing where capacitor performance and reliability are critical.