A spherical capacitor has the space between the plates filled with a material of conductivity \(\sigma\) and a relative permittivity \(\varepsilon_{\mathrm{v}}\) and has its outer plate earthed. Show that the resistance \(R\) between the plates and the capacitance \(C\) are related by \(C R=\varepsilon_{t} \varepsilon_{0} / \sigma\).

Capacitance is a measure of a capacitor's ability to store charge per unit voltage. It can be imagined as a tiny battery with the capability to hold an electric charge. Capacitance, represented by the symbol \(C\), is dependent on the geometry of the capacitor and the material between the plates, known as the dielectric.

For a spherical capacitor, which consists of two concentric spherical conducting shells, the capacitance is given by the formula: \(C = \frac{4 \pi \epsilon_a}{b - a}\), where \(\epsilon_a\) is the absolute permittivity, and \(a\) and \(b\) are the radii of the inner and outer spheres, respectively. The capacitance of a capacitor increases when the permittivity of the dielectric material increases or when the separation between the conductive plates decreases. This concept is critical for understanding how capacitors function and their role in electric circuits, such as filtering signals or storing energy.

Electrical resistance is a measure of the opposition to the flow of electric current through a conductor. The greater the resistance, the harder it is for current to flow. Resistance, denoted by \(R\), can be calculated using the formula: \(R = \frac{\rho l}{A}\), where \(\rho\) is the material's resistivity, \(l\) is the length through which the current flows, and \(A\) is the cross-sectional area.

In a spherical capacitor, the resistance between the plates can be determined by considering the unique geometry of a sphere. Since resistance is directly proportional to the length of the conductor and inversely proportional to its area, the configuration of the capacitor's plates affects the overall resistance. The formula, adjusted for a spherical geometry, results in: \(R = \frac{\frac{1}{\sigma}(b - a)}{4 \pi a^2}\), which reflects the resistance of the dielectric material between the inner and outer spherical shells.

Relative permittivity, often denoted as \(\epsilon_v\) or \(\epsilon_r\), is a dimensionless measure comparing the permittivity of a material to the vacuum permittivity \(\epsilon_0\). Relative permittivity indicates how much the electric field within the material is reduced compared to a vacuum. This property greatly influences the behavior of capacitors, as it determines the capacitance when a dielectric material is used.

The relationship between absolute permittivity \(\epsilon_a\) and relative permittivity is given by the equation: \(\epsilon_a = \epsilon_v \epsilon_0\). In the context of our spherical capacitor exercise, the relative permittivity shows how the material between the plates modifies the electric field, and consequently, the capacitance.

Conductivity, represented by the Greek letter \(\sigma\), quantifies a material's ability to conduct electric current. It is the reciprocal of resistivity \(\rho\), and its unit is siemens per meter (S/m). Conductivity in materials can vary dramatically, from very high in metals to very low in insulators.

In a spherical capacitor, the conductivity of the material between the plates is crucial. It's important to note that while capacitors are not designed to allow direct current flow between their plates, for the purposes of this exercise, we assume there's a leakage current through the dielectric. The relationship between conductivity, resistance, and capacitance in the spherical capacitor is captured in the formula \(CR = \frac{\epsilon_v \epsilon_0}{\sigma}\), illustrating how these properties are interconnected.