Thank you @kuruman !kuruman said:
The basic formula for the capacitance of a cylindrical capacitor is given by \( C = \frac{2 \pi \epsilon_0 \epsilon_r L}{\ln(b/a)} \), where \( \epsilon_0 \) is the permittivity of free space, \( \epsilon_r \) is the relative permittivity of the dielectric material, \( L \) is the length of the cylinder, \( a \) is the radius of the inner cylinder, and \( b \) is the radius of the outer cylinder.
To derive the capacitance formula for a cylindrical capacitor, start by considering two coaxial cylinders with radii \( a \) and \( b \). Calculate the electric field \( E \) in the region between the cylinders using Gauss's law. Integrate the electric field to find the potential difference \( V \) between the cylinders. Finally, use the relationship \( C = \frac{Q}{V} \) to find the capacitance, where \( Q \) is the charge on the inner cylinder.
The derivation assumes that the length \( L \) of the cylinders is much greater than their radii, ensuring a uniform electric field along the length. It also assumes that the dielectric material between the cylinders is homogeneous and isotropic, and that edge effects at the ends of the cylinders are negligible.
The natural logarithm arises from the integration of the electric field over the radial distance between the inner and outer cylinders. Specifically, the potential difference \( V \) is found by integrating \( E \) from radius \( a \) to \( b \), resulting in a logarithmic function due to the inverse relationship between the electric field and the radial distance.
The capacitance of a cylindrical capacitor is directly proportional to its length \( L \). This is because a longer cylinder provides more surface area for charge storage, which increases the overall capacitance. The relationship is linear, as seen in the formula \( C = \frac{2 \pi \epsilon_0 \epsilon_r L}{\ln(b/a)} \).