When to use Laplace's or Poisson's equation for calculating potential

[mdwerner]

Homework Statement

The problem that I'm having is simply understanding the difference in when to use the laplace's equation vs the poisson's equation. Here is an example of a question : "A rectangular pipe, running parallel to the z-axis (from -∞ to + ∞), has three grounded metal sides, at y = 0, y = a, and x = 0. The fourth side, at x = b, is maintained at a specified potential V(y)."
I can see in the solution that we are to use Laplace's equation and not Poisson's equation. Why? What determines this? The solution does not state this, it simply states that it IS.
Any help is much appreciated.

Homework Equations

∇^2 V = 0 (Laplace's Equation)
∇^2 V = - ρ / ε (Poisson's Equation)

The Attempt at a Solution

I see that Laplace's equation would be a special case in which the region we are talking about is devoid of any charges - In the example we have a conducting material with a potential in one region that isn't present in another. Does this not mean that there are charges present? If charges are present, then Poisson's Equation must be used. Any help clarifying this would mean a lot - thanks.

 

[vela]

You're presumably trying to find the potential inside the pipe. Since there are no charges inside the pipe, you use Laplace's equation.