In addition to measuring currents and voltages, it is often necessary to measure resistance values. A precise method for this is provided by the so-called Wheatstone bridge circuit, which is usually implemented with a sliding-wire potentiometer. This acts as a variable voltage divider whose resistance can be continuously adjusted by moving a contact point, since the resistance increases proportionally with the length of the wire.
The circuit consists of a fixed resistor $R_1$ (reference resistor), a potentiometer split into $R_2$ and $R_3$, and the unknown resistor $R_x$.
For the voltage $U_1$ across $R_1$ we obtain:
\[
U_1 = U_0\frac{R_x}{R_1 + R_2}
\]
and analogously for $U_2$:
\[
U_2 = U_0\frac{R_2}{R_1 + R_2}
\]
The current through the ammeter vanishes exactly when these two voltages are identical.
By equating and rearranging, we obtain the condition:
\[
R_x = \frac{R_1R_2}{R_3}
\]
The ratio of the two resistances $R_2$ and $R_3$ can be expressed using the total length $L$ of the sliding-wire bridge and the contact position $x$:
\[
\frac{R_2}{R_3} = \frac{L-x}{x}
\]
Thus, the unknown resistance can be calculated as:
\[
\boxed{R_x = R_1\frac{L-x}{x}}
\]
If the total length $L$ is known, it is sufficient to measure the segment $x$ once the potentiometer has been adjusted so that the ammeter shows no current.
Example:
A Wheatstone bridge consists of a sliding-wire bridge of length 1 m and a reference resistor of $27\,\Omega$. The sliding contact is located 30 cm from the zero point.
What is the value of the unknown resistance?
Solution: $63\,\Omega$