A thin membrane with conductivity and thickness separates two regions with conductivity . Assume uniform current flow in the z-direction in the figure

A thin membrane with conductivity σ and thickness δ separates two regions with conductivity σ.

Assume uniform current flow in the z-direction in the figure above. When δ is small, it makes sense to seek “across-the-membrane” matching conditions for the electrostatic potential ϕ(z) defined entirely in terms of quantities defined outside the membrane. Find the potential in all three regions of the figure and prove that suitable matching conditions are

A voltage difference V₀ causes a steady current to flow from the top conductor to the bottom conductor (in the sketch below) through an ohmic medium with conductivity σ. Find an approximate expression for the current I that flows into the hemispherical bump (radius R) portion of the lower conductor. Assume that d >> R.

0=2- 9=2 2 a 0