Four Point Probe

Four metallic measuring tips are contacted to a thin foil of thickness \(d\) and conductivity \(\sigma\). A current source is connected to the two outer tips (\(I_+\) and \(I_-\)) which transmit a constant current \(I\) through the sample. A voltage is measured at the two inner tips (\(U_+\) and \(U_-\)). Which formula describes the electrical resistance \[R = \frac{U_+ - U_-}{I} \] in this geometry?

Details and Assumptions:

  • The foil has an almost infinite size (\(L \gg l\)) but a very small thickness (\(d \ll l\)).

  • The tips are equidistant with a distance \(l\) between the outer tips. The contact point are infinite small and the voltage measurement does not affect the electric fields in inside the foil.

  • The points \(I_+\) and \(I_-\) can be treated as point source and point drain for the planar current densitiy \(\vec j(x,y)\).

Important Relations:

  • Current density (Ohm's law) \[ \vec j (\vec r) = \sigma \cdot \vec E (\vec r) \qquad \text{with} \qquad I = \int_A \vec j (\vec r) \cdot d\vec A \] with electric field \(\vec E\) and current flow \(I\) through the plane \(A\).

  • Electric voltage \[ U_{12} = \phi(\vec r_2) - \phi(\vec r_1) = -\int_1^2 \vec E (\vec r) \cdot d\vec r \] between points \(\vec r_1\) and \(\vec r_2\) and the electric potential \(\phi(\vec r)\).

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