Boundary Value and Separation of Variables.

[zhuang382]

Homework Statement

Consider a rectangular box with sides, $0\leq x \leq a$, $0\leq y \leq b$, $0\leq z\leq c$
The potential is 0 on four sides of the box: $x=0$, $x=a$, $y=0$, $y=b$, except at $z= 0$ and $z = c$, (top and bottom) with electric field $E_0 = E_z = constant$

Relevant Equations

$\nabla^2 V = 0$

If the boundary condition is not provided in the form of electric potential, how do we solve such problem?
In this case, I want to use $V = - \int \vec{E} \cdot{d\vec{l}}$, but I don't know how to choose an appropriate reference point.

 

[BvU]

You have a second order differential equation with first order ($\vec E = \vec\nabla V\$ given) boundary conditions.
Make a guess for a simple $V$ that satisfies the equation and fill in the boundary conditions.