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# DC Conduction

Solve PDEs that model direct current electrical conduction

The direct current conduction problems, such as electrolysis and computation of resistances of grounding plates, involve a steady current passing through a conductive medium. The current density J is related to the electric field E as J = σ E, where σ is the conductivity of the medium. The electric field E is the gradient of the electric potential V, E = –∇V. Thus, the continuity equation ∇ · J = Q, where Q is the current source, yields the elliptic Poisson's equation:

–∇ · (σV) = Q.

The toolbox supports the following boundary conditions for DC conduction problems:

• Dirichlet boundary condition assigning values of V at the boundaries, which are typically metallic conductors.

• Neumann boundary condition assigning the value of the normal component of the current density (n · (σV)).

• Generalized Neumann condition n · (σV) + qV = g, where q is film conductance for thin plates.

## Apps

 PDE Modeler Solve partial differential equations in 2-D regions

## Topics

### Programmatic Workflow

Poisson's Equation on Unit Disk

Use command-line functions to solve a simple elliptic PDE in the form of Poisson's equation on a unit disk.

Minimal Surface Problem

Use command-line functions to solve a nonlinear elliptic problem.

Poisson's Equation with Point Source and Adaptive Mesh Refinement

Solve a Poisson's equation with a delta-function point source on the unit disk using the `adaptmesh` function.

### PDE Modeler App Workflow

Poisson's Equation on Unit Disk: PDE Modeler App

Use the PDE Modeler app to solve a simple elliptic PDE in the form of Poisson's equation on a unit disk.

Minimal Surface Problem: PDE Modeler App

Use the PDE Modeler app to solve a nonlinear elliptic problem.

Current Density Between Two Metallic Conductors

Solve the Laplace equation for a geometry consisting of two circular metallic conductors placed on a plane.

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