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.