## Current Density Between Two Metallic Conductors: PDE Modeler App

Two circular metallic conductors are placed on a brine-soaked blotting paper which serves as a plane, thin conductor. The physical model for this problem consists of the Laplace equation

–∇ · (*σ*∇*V*) = 0

for the electric potential *V* and these boundary conditions:

*V*= 1 on the left circular conductor*V*= –1 on the right circular conductorthe natural Neumann boundary condition on the outer boundaries

$$\frac{\partial V}{\partial n}=0$$

The conductivity is *σ* = 1.

To solve this equation in the PDE Modeler app, follow these steps:

Model the geometry: draw the rectangle with corners at (-1.2,-0.6), (1.2,-0.6), (1.2,0.6), and (-1.2,0.6), and two circles with a radius of 0.3 and centers at (-0.6,0) and (0.6,0). The rectangle represents the blotting paper, and the circles represent the conductors.

pderect([-1.2 1.2 -0.6 0.6]) pdecirc(-0.6,0,0.3) pdecirc(0.6,0,0.3)

Model the geometry by entering

`R1-(C1+C2)`

in the**Set formula**field.Set the application mode to

**Conductive Media DC**.Specify the boundary conditions. To do this, switch to the boundary mode by selecting

**Boundary**>**Boundary Mode**. Use**Shift**+click to select several boundaries. Then select**Boundary**>**Specify Boundary Conditions**.For the rectangle, use the Neumann boundary condition with

`g = 0`

and`q = 0`

.For the left circle, use the Dirichlet boundary condition with

`h = 1`

and`r = 1`

.For the right circle, use the Dirichlet boundary condition with

`h = 1`

and`r = -1`

.

Specify the coefficients by selecting

**PDE**>**PDE Specification**or clicking the button on the toolbar. Specify`sigma = 1`

and`q = 0`

.Initialize the mesh by selecting

**Mesh**>**Initialize Mesh**.Refine the mesh by selecting

**Mesh**>**Refine Mesh**.Improve the triangle quality by selecting

**Mesh**>**Jiggle Mesh**.Solve the PDE by selecting

**Solve**>**Solve PDE**or clicking the button on the toolbar. The resulting potential is zero along the*y*-axis, which, for this problem, is a vertical line of antisymmetry.Plot the current density

**J**. To do this:Select

**Plot**>**Parameters**.In the resulting dialog box, select the

**Color**,**Contour**, and**Arrows**options.Set the

**Arrows**value to`current density`

.

The current flows, as expected, from the conductor with a positive potential to the conductor with a negative potential. The conductivity

*σ*is isotropic, and the equipotential lines are orthogonal to the current lines.