Laplace Equation on semi-infinite plate

[ZombieCat]

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?

 

[gabbagabbahey]

The electrostatic potential only satisfies Laplace's equation in regions where the charge density is zero. The rest of the time it satisfies Poisson's equation.

 

[ZombieCat]

Oh dangit! Cause we're dealing with a conductor gotcha! So I guess the epsilon_0*(ln10)^2*exp(-x/d) would be the charge density as a function of x.

Thanks!

*Maybe I should rename myself "the phorgetful physicist"...