# 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 of $$l$$ between the outer tips. The contact point is infinitely small and the voltage measurement does not affect the electric fields inside the foil.
• The points $$I_+ \text{ and } I_-$$ can be treated as the point source and point drain, respectively, for the planar current density $$\vec j(x,y).$$

Important Relations:

• Current Density (Ohm's law): $\vec j (\vec r) = \sigma \cdot \vec E (\vec r) \ \ \text{with} \ \ I = \int_A \vec j (\vec r) \cdot d\vec A,$ where $$\vec E$$ is the electric field and $$I$$ is the current flow through the plane $$A$$.
• Electric Voltage: $U_{12} = \phi(\vec r_2) - \phi(\vec r_1) = -\int_1^2 \vec E (\vec r) \cdot d\vec r,$ measured between points $$\vec r_1$$ and $$\vec r_2$$ with the electric potential $$\phi(\vec r)$$.
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