Laplace equation in rectangular geometry

[jaobyccdee]

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Homework Statement

A battery consists of a cube of side L filled with fluid of conductivity s. The electrodes in the battery consist of two plates on
the base at y = 0, one grounded and one at potential V = 12 Volts. The other sides of the battery casing are not
conductive. Find the potential f everywhere inside the battery.
Hint: current density j(x, y) flows according to Ohn’s law, j = σ Ewhere E = -div ∅. For equilibrium current flow, div.j = 0 which
implies σ (div)^2 f = 0. Therefore you must slove Laplace’s equation. Note that on nonconducting surfaces, j.n `
= 0wheren `
is normal
to the surface, so such surfaces have a Neumann boundary condition.

Homework Equations

The Attempt at a Solution

I think the boundary at y=L is a Neumann boundary, and the boundary at y=0 is a Dirichlet boundary, but i don't know what is the boundary condition for y=0 since half of it is 0V and half of it is 12 V.

 

[latentcorpse]

[/itex][/itex]

Homework Statement

A battery consists of a cube of side L filled with fluid of conductivity s. The electrodes in the battery consist of two plates on
the base at y = 0, one grounded and one at potential V = 12 Volts. The other sides of the battery casing are not
conductive. Find the potential f everywhere inside the battery.
Hint: current density j(x, y) flows according to Ohn’s law, j = σ Ewhere E = -div ∅. For equilibrium current flow, div.j = 0 which
implies σ (div)^2 f = 0. Therefore you must slove Laplace’s equation. Note that on nonconducting surfaces, j.n `
= 0wheren `
is normal
to the surface, so such surfaces have a Neumann boundary condition.

Homework Equations

The Attempt at a Solution

I think the boundary at y=L is a Neumann boundary, and the boundary at y=0 is a Dirichlet boundary, but i don't know what is the boundary condition for y=0 since half of it is 0V and half of it is 12 V.

If it's a cube with lower left corner on origin then x=0 would be Dirichlet and x=L would be Neumann wouldn't they?

Also, a cube is 3d but j is just a function of x and y suggesting you don't have to worry about z.

It's been a while since I did this stuff though!