In the case of a cylindrical Gaussian surface, the electric field ($\vec{E}$) is constant in the direction of the versor normal to the section ($\hat{n}$). Therefore, using the variables the charge ($Q$), the electric field constant ($\epsilon_0$), and the dielectric constant ($\epsilon$), the integral over the surface where the electric field is constant ($dS$) can be calculated through the following equation:

For a cylinder characterized by the axle distance ($r$) and the conductor length ($L$), the following applies:

Furthermore, the linear charge density ($\lambda$) is calculated using the charge ($Q$) according to the equation:

Thus, it is established that the electric field, infinite conducting cylinder ($E_c$) is:

The electric potential ($\varphi$), in the context of cylindrical geometry, is equal to the base electrical potential ($\varphi_0$) plus the integral along the path of the electric field ($\vec{E}$), taking the dot product with the infinitesimal distance ($ds$):

the electric field, infinite conducting cylinder ($E_c$) is inversely proportional to the radius ($r$):

$E_c\propto\displaystyle\frac{1}{r}$

Therefore, the simplest path is radial. However, the base electrical potential ($\varphi_0$) cannot be at the origin or at infinity, as the integral diverges at both points. Therefore, the base electrical potential ($\varphi_0$) must be referenced to a radius where the potential is zero, which in this case is the cylinder radius ($r_0$):

The reference electrical, infinite conducting cylinder ($\varphi_c$) is derived from the radial integration of the electric field, infinite conducting cylinder ($E_c$) from the cylinder radius ($r_0$) to the axle distance ($r$), resulting in the following equation:

Furthermore, for the variables the charge ($Q$), the dielectric constant ($\epsilon$), and the electric field constant ($\epsilon_0$), the value of the electric field, infinite conducting cylinder ($E_c$) is given as:

This implies that by performing the integration

$\varphi_c = -\displaystyle\int_{r_0}^r du \displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon u }= -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon } \ln\left(\displaystyle\frac{ r }{ r_0 }\right)$

the following equation is obtained:

The reference electrical, infinite conducting cylinder ($\varphi_c$) is derived from the radial integration of the electric field, infinite conducting cylinder ($E_c$) from the cylinder radius ($r_0$) to the axle distance ($r$), resulting in the following equation:

Furthermore, for the variables the charge ($Q$), the dielectric constant ($\epsilon$), and the electric field constant ($\epsilon_0$), the value of the electric field, infinite conducting cylinder ($E_c$) is given as:

This implies that by performing the integration

$\varphi_c = -\displaystyle\int_{r_0}^r du \displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon u }= -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon } \ln\left(\displaystyle\frac{ r }{ r_0 }\right)$

the following equation is obtained:

The reference electrical, infinite conducting cylinder ($\varphi_c$) is derived from the radial integration of the electric field, infinite conducting cylinder ($E_c$) from the cylinder radius ($r_0$) to the axle distance ($r$), resulting in the following equation:

Furthermore, for the variables the charge ($Q$), the dielectric constant ($\epsilon$), and the electric field constant ($\epsilon_0$), the value of the electric field, infinite conducting cylinder ($E_c$) is given as:

This implies that by performing the integration

$\varphi_c = -\displaystyle\int_{r_0}^r du \displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon u }= -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon } \ln\left(\displaystyle\frac{ r }{ r_0 }\right)$

the following equation is obtained: