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For electrolysis and computation of resistances of grounding
plates, we have a conductive medium with conductivity σ and
a steady current. The current density **J** is
related to the electric field **E** through **J** = *σ***E**. Combining the continuity equation ∇ · **J** = *Q*, where *Q* is
a current source, with the definition of the electric potential *V* yields
the elliptic Poisson's equation

–∇ · (*σ*∇*V*)
= *Q*.

The only two PDE parameters are the conductivity *σ* and
the current source *Q*.

The Dirichlet boundary condition assigns values of the electric
potential *V* to the boundaries, usually metallic
conductors. The Neumann boundary condition requires the value of the
normal component of the current density (**n** · (*σ*∇*V*))
to be known. It is also possible to specify a generalized Neumann
condition defined by **n** · (*σ*∇*V*) + *qV* = *g*, where *q* can be
interpreted as a *film conductance* for thin plates.

The electric potential *V*, the electric field **E**, and the current density **J** are
all available for plotting. Interesting quantities to visualize are
the current lines (the vector field of **J**)
and the equipotential lines of *V*. The equipotential
lines are orthogonal to the current lines when *σ* is
isotropic.

Two circular metallic conductors are placed on a plane, thin conductor like a blotting paper wetted by brine. The equipotentials can be traced by a voltmeter with a simple probe, and the current lines can be traced by strongly colored ions, such as permanganate ions.

The physical model for this problem consists of the Laplace equation

–∇ · (*σ*∇*V*)
= 0

for the electric potential *V* and the 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 *σ* = 1 (constant).

Open the PDE app by typing

at the MATLABpdetool

^{®}command prompt.Click

**Options**>**Application**>**Conductive Media DC**.Click

**Options**>**Grid Spacing...**, deselect the**Auto**check boxes for**X-axis linear spacing**and**Y-axis linear spacing**, and choose a spacing of 0.3, as pictured. Ensure the Y-axis goes from –0.9 to 0.9. Click**Apply**, and then**Done**.Click

**Options**>**Snap**Click and draw the blotting paper as a rectangle with corners in (-1.2,-0.6), (1.2,-0.6), (1.2,0.6), and (-1.2,0.6).

Click and add two circles with radius 0.3 that represent the circular conductors. Place them symmetrically with centers in (-0.6,0) and (0.6,0).

To express the 2-D domain of the problem, enter

for theR1-(C1+C2)

**Set formula**parameter.To decompose the geometry and enter the boundary mode, click .

Hold down

**Shift**and click to select the outer boundaries. Double-click the last boundary to open the Boundary Condition dialog box.Select

**Neumann**and click**OK**.Hold down

**Shift**and click to select the left circular conductor boundaries. Double-click the last boundary to open the Boundary Condition dialog box.Set the parameters as follows and then click

**OK**:**Condition type**=**Dirichlet****h**=`1`

**r**=`1`

Hold down

**Shift**and click to select the right circular conductor boundaries. Double-click the last boundary to open the Boundary Condition dialog box.Set the parameters as follows and then click

**OK**:**Condition type**=**Dirichlet****h**=`1`

**r**=`-1`

Open the PDE Specification dialog box by clicking

**PDE**>**PDE Specification**.Set the current source,

**q**, parameter to`0`

.Initialize the mesh by clicking

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

**Mesh**>**Refine Mesh**twice.Improve the triangle quality by clicking

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

The resulting potential is zero along the

*y*-axis, which is a vertical line of anti-symmetry for this problem.Visualize the current density $$J$$ by clicking

**Plot**>**Parameters**, selecting**Contour**and**Arrows**check box, and clicking**Plot**.The current flows, as expected, from the conductor with a positive potential to the conductor with a negative potential.

**The Current Density Between Two Metallic Conductors**

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