Electric Field Between Concentric Cylinders

This introductory model treats the electrostatics problem of two concentric cylinders of infinite length, which is commonly found in textbooks. Since the problem can be solved analytically, the model can be used to compare theory with numerical simulation results. Two cases are considered:

•

A fixed potential on each cylinder

•

A surface charge density on one cylinder and a fixed potential on the other

Model Definition

First, consider the analytical solution to this problem to use for comparison. Let ri be the radius of the inner cylinder and ro the radius of the outer cylinder. In the first case, where we have a fixed potential V0 on the inner boundary and ground on the outer boundary, the electric potential satisfies

which results in the electric field

where

is a unit vector in the radial direction.

In the case where there is a surface charge q0 on the inner cylinder, the electric potential instead reads

which gives the electric field

where ε0 is the permittivity of vacuum.

To simplify the geometry of the model, take advantage of its twofold symmetry. First, since the cylinders are infinitely long, a perpendicular cross section is sufficient to represent the whole length. Furthermore, such a cross section is rotationally symmetric and it — and by extension, the full geometry — can thus be represented by a 1D component.

In the case with a fixed surface charge density on the inner boundary, an interesting note can be made about why it might be preferable to keep ground on the outer boundary instead of another surface charge q0 with the opposite sign. Since a surface charge is added on the inner cylinder, the latter could be considered to be more intuitive. However, there is one important difference: Having ground on a boundary sets the zero level of the electric potential, and therefore acts as a gauge fix. With only the surface charges specified, there is an infinite number of viable solutions for the electric potential. In order to solve such a problem numerically, one would have to change the default direct solver to an iterative one. Iterative solvers can solve even ungauged problems, finding one solution out of the many possible ones. It is of course important to remember that the electric field will be the same no matter which potential is chosen.

Results

The electric potential in the two cases is plotted in Figure 1 and Figure 2, where the numerical solution is compared with its analytical counterpart.

Figure 1: The electric potential from the numerical and analytical solutions for the case with a specified potential.

Figure 2: The electric potential from the numerical and analytical solutions for the case with a specified surface charge density.

Similarly, the solutions for the radial component of the electric field are plotted and compared in Figure 3 and Figure 4. In these plots, it can be seen how well the numerical solutions for the two different approaches agree with the analytical solutions.

Figure 3: The radial component of the electric field from the numerical and analytical solutions for the case with a specified potential.

Figure 4: The radial component of the electric field from the numerical and analytical solutions for the case with a specified surface charge density.

ACDC_Module/Introductory_Electrostatics/electric_field_concentric_cylinders

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

First, define some parameters that will be used when building the model.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
ri	0.1[m]	0.1 m	Radius of inner cylinder
ro	1[m]	1 m	Radius of outer cylinder
V0	100[V]	100 V	Potential at inner cylinder
q0	1e-7[C/m]	1E-7 C/m	Charge at inner cylinder