Hello all! I just finished the following problem: Consider a thin semi-infinite plate of negligible thickness made of an isotropic conductive material. A voltage V0=1V is applied at x=0 on the plate (across the short dimension). At a distance x=d=1cm from the end (x=0) V is measured to be .1V. Find the voltage V(x) at an arbitrary distance x from the end. In my first attempt I got V(x)=-90*x+1, which is a solution to the Laplace equation in 1D, but does not match the boundary condition at infinity. I tried the problem again and got V(x)=V0*10^(-x/d), which matches all boundary conditions and is the correct answer. My question is why doesn't this solution satisfy the Laplace equation? Does it have to? Why/why not?