Poisson's equation in cylindrical coordinates is $$ \frac{1}{r} \partial V / \partial r+\partial^{2} V / \partial r^{2}+\frac{1}{r^{2}} \partial^{2} V / \partial \theta^{2}+\partial^{2} V / \partial z^{2}=-\rho / \varepsilon_{\theta} $$ For constant \(\rho\), show that if cylindrical symmetry is complete (no variation with \(\theta\) ) and all cross-sections perpendicular to the \(z\) axis are identical (no variation with \(z), V\) must be of the form \(k_{1} \log _{2} r+k_{2} r^{2}+k_{3}\), where \(k_{1}, k_{2}\) and \(k_{3}\) are constants. Apply this to the solution of Problem \(4.10\).

Cylindrical symmetry refers to a situation where the properties of a system or an equation do not change when rotated around a central axis, which in this case is the axis of a cylinder. In the context of Poisson's equation, when cylindrical symmetry is assumed, it implies that the electrostatic potential V does not depend on the angle \( \theta \) and the position along the axis z. Such assumptions greatly simplify the mathematical complexity of the problem.

Poisson's equation, which is a second-order partial differential equation, can have its terms involving \( \partial^{2} V / \partial \theta^{2} \) and \( \partial^{2} V / \partial z^{2} \) removed, as the potential is constant with respect to these variables due to cylindrical symmetry. Consequently, the problem is reduced to a function of the radial distance r alone. This simplification is pivotal in solving many problems in physics and engineering that exhibit cylindrical symmetry, such as the electrostatic potential within a coaxial cable.

Electrostatic potential, commonly denoted as V, is a scalar quantity that represents the electric potential energy per unit charge at a point in space due to an electric field. It is a central concept in electrostatics and is related to the way in which electric charges interact with each other and with electric fields.

In Poisson's equation, the electrostatic potential is what we aim to solve for, given a charge distribution \( \rho \). The equation describes how the potential varies in space due to the presence of the charge and is governed by the permittivity \( \varepsilon_{\theta} \) of the medium in which the charges reside. In a system with cylindrical symmetry, the electrostatic potential is independent of the angular and vertical coordinates, and it allows us to express V as a function of the radial coordinate alone, which leads to a form that combines logarithmic and polynomial terms in r.

Differential equations are mathematical equations that relate some function with its derivatives. They play an important role in engineering, physics, economics, and other sciences, as they describe numerous phenomena in the real world. In the context of Poisson's equation in cylindrical coordinates, we are dealing with a partial differential equation, which involves partial derivatives of a function that depends on several variables.

When we talk about solving a differential equation like Poisson's, it means finding the function V(r) that satisfies the equation given certain conditions. This process typically involves making an educated guess, known as an ansatz, that can simplify and allow integration of the equation. For constant charge density \( \rho \), the associated differential equation, after applying the symmetry constraints, becomes an ordinary differential equation in terms of r. By solving it, we find the electrostatic potential as a function of the radial distance that takes into account the nature of the charge distribution and the geometry of the space.