Consider a long piece of conducting material ($\sigma$) with a length $L$ and arbitrarily-shaped cross-sectional of area $A$. The ends of this wire are connected to a battery with potential differences $\Delta V$, so that current flows.

Using Laplace’s equation ($\nabla^2 V = 0$), show that electric field in the wire is constant. Make sure you can argue how your solution to Laplace’s equation satisfies the relevant boundary conditions. What do your results mean for how we define steady state?