Boundary Conditions for an infinite rectangular pipe

[guyvsdcsniper]

Homework Statement

An infinite rectangular pipe with sides a, has two opposite sides at voltage V
(front and back) and at voltage V=0 (top and bottom).

Find the potential inside the pipe.

Relevant Equations

Fourier Sine Trick

Does setting up the problem symmetrically on this axis and the boundary conditions applied make sense? I don't believe I will have a problem solving for the potential inside, but i just want to make sure I have my B.C and axis correct before proceeding.

EDIT:

Or should this be a 2-D lapace equation since the pipe is infinitely long, making this independent of the z axis?

 

[haruspex]

I'm a bit confused about how you are assigning the axes.
I'll call the left right axis X, the vertical axis Z and the lower left/ upper right axis in the picture Y. From that and the text description, I would say the pipe is infinite in the X axis and width a in the other two. The voltage is V on the faces normal to the Y axis and 0 on those normal to the Z axis.

 

[guyvsdcsniper]

I'm a bit confused about how you are assigning the axes.
I'll call the left right axis X, the vertical axis Z and the lower left/ upper right axis in the picture Y. From that and the text description, I would say the pipe is infinite in the X axis and width a in the other two. The voltage is V on the faces normal to the Y axis and 0 on those normal to the Z axis.

Sorry I missed that. y is the vertical axis.z is the axis coming out of the page. X is horizontal.

But I think it should be independent of Z since it is infinitely long. It mirrors an example straight out my textbook.

 

[haruspex]

z is the axis coming out of the page.

independent of Z since it is infinitely long.

Looking at the diagram, the infinitely long direction is horizontal.

 

[rude man]

https://tutorial.math.lamar.edu/classes/de/laplaceseqn.aspx
Bit of a challenge as you can see.