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Wheatstone Bridge: Theory and Applications

Wheatstone Bridge

Wheatstone Bridge was designed by British physicist Sir Charles F. Wheatstone in 1833.
It is an arrangement of four resistors used to determine the resistance of one resistor in terms of the other three resistors.
Consider the figure given below, which is an arrangement of resistors and is known as the Wheatstone Bridge:

Wheatstone Bridge consists of four resistances $P$, $Q$, $R$, and $S$ with a battery of EMF $E$. Two keys $K_1$ and $K_2$ are connected across terminals $A$ and $C$, and $B$ and $D$, respectively.
When $K_1$ is pressed first and then $K_2$ is pressed, if the galvanometer shows no deflection, the Wheatstone Bridge is said to be balanced.
If the galvanometer shows no deflection, it means no current is flowing through the galvanometer, and terminals $B$ and $D$ are at the same potential. Thus, for a balanced bridge:

To find the condition for a balanced Wheatstone Bridge, we apply Kirchhoff's loop rule:

From equations (1) and (2), we get:
Or:
Equation (3) gives the condition for a balanced Wheatstone Bridge.
Thus, if the ratio of resistances $P$ and $Q$ and the value of resistance $R$ is known, the unknown resistance $S$ can easily be calculated.
One important thing to note is that when a Wheatstone bridge is balanced, the positions of the cell (or power source) and the galvanometer can be swapped without affecting the balance of the bridge. This is because, in the balanced condition, the potential difference across the galvanometer is zero, meaning no current flows through it.
In a balanced Wheatstone bridge, the ratio of the resistances in one pair of opposite arms is equal to the ratio of the resistances in the other pair. As long as this condition is maintained, the balance of the bridge will remain the same, regardless of the arrangement of the power supply and the galvanometer.
The sensitivity of the bridge depends on the relative magnitudes of the resistances in the four arms of the bridge and is maximum when the resistances are of the same order.