Van Der Pauw completing the missing part B

Van Der Pauw -completing the missing part. B 93901007 許恭銓

The Problem • Van Der Pauw, in his paper, derived the equation for the special case where the four probes are in a line. He later claimed that this equation is suitable for almost all smooth lamellae, and it could be proved by using complex analysis. Unfortunately, in the paper he didn’t prove this, and so a question arises.

Field Representations on complex plane • The electric field in the complex plane • The complex theorem guarantees that

A useful derivation • Now, on the complex plane we observe the current traversing through the paths P 2 between arbitrary points P 1 ->P 2 P 1

Applying the Derivation 1/2 • Back to our main problem, on the half complex plane, z-plane, we A B C D traverse along the edge, the x-axis, and observe • Except near the points A and B, there are no currents flowing perpendicular to the traversal. So by our previous derivation, the value of won’t change.

Applying the Derivation 2/2 • When we get near points A/B, we encounter the currents flowing in/out of the points. • When jumping over point A, by our previous derivation, • Similarly, for point B, • So we have

Applying to arbitrary shape 1/3 • Now, we consider a lamella of arbitrary shape lying on the complex plane. We should call it the t-plane. • Suppose there exists a conformal mapping from z-plane to t-plane, say • The corresponding conformal transform maps A, B, C, D , , ,

Applying to arbitrary shape 2/3 • Now, let the function • Traversing along • If we make , we still have , then we can assure that for the corresponding points. And by the characteristics of the conformal mapping, is guaranteed the potential function for t-plane.

Applying to arbitrary shape 3/3 • Therefore,