# 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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