The Poisson equation two examples of heat conduction

We all have an intuitive idea about heat and temperature, so the easiest way to illustrate the Poisson equation is probably through the heat conduction equation.

One-dimensional heat conduction

Consider a gold rod of length L suspended between two wires both having some temperature T0 that we will specify later. The rod is covered by a heater that transfers heat into the rod at some rate Q given in units of Watts per volume (W m-3). See the figure below:

This means that we have heat flowing into the rod at every point along its axial direction, the x direction, and heat flowing out at the ends. One obvious question is: what is the temperature inside the gold rod? Before we answer this question, we must specify that we are only interested in the final temperature, not what happens when we switch the heater on and off. This is called the steady state temperature profile.

Note

The Poisson equation is used to describe steady state situations...