Wheatstone Bridge Principle

The Wheatstone bridge is an arrangement of four resistances $P, Q, R, S$, connected to form the four arms of a quadrilateral (conventionally drawn as a diamond), with a battery connected across one diagonal (through a key) and a sensitive galvanometer connected across the OTHER diagonal (through its own key), as shown in the figure. Current from the battery splits at the top corner and flows through the two parallel paths $P$-then-$R$ and $Q$-then-$S$ to reach the bottom corner, while the galvanometer, bridging the left and right corners, detects whether these two paths are at the same potential at the point it connects to.

Balance condition. The bridge is said to be balanced when the resistances $P, Q, R, S$ are adjusted (usually by varying just one of them, often $R$, using a resistance box) so that the galvanometer shows exactly zero deflection -- meaning no current at all flows through the galvanometer branch, i.e. the left and right corners are at exactly the same potential. Applying Kirchhoff's rules at balance (with zero galvanometer current, the currents through $P$ and $Q$ must be equal to the currents through $R$ and $S$ respectively, since no current is diverted at the galvanometer node), the loop rule applied to the two loops on either side of the galvanometer gives the balance condition:

This single relation is the entire working principle of the bridge: it involves only the RATIO of the resistances in each pair of adjacent arms, and holds regardless of the EMF of the battery used or the exact resistance of the galvanometer itself (both drop out of the algebra entirely once the galvanometer current is set to zero) -- which is precisely why the method is so much more accurate than simply reading a voltmeter or ammeter, whose own imperfections (finite resistance, calibration error) would otherwise directly limit the measurement.

Measuring an unknown resistance. If three of the four arms are known (say $P$, $Q$ from fixed "ratio arms", and $R$ from an adjustable resistance box), and the fourth arm $S$ is the unknown resistance to be measured, then once the bridge is balanced,

$S = \frac{Q}{P}\,R$ …