Question

Derive the condition for a balanced Wheatstone Bridge.

Hint:

In a balanced Wheatstone bridge: - No current flows through the galvanometer. - Ratio of resistances in opposite arms are equal. - Used in meter bridge and strain gauge measurements.

Answer 1

Wheatstone Bridge
A Wheatstone bridge is an electrical circuit used to measure an unknown resistance by comparing it with known resistances. It consists of four resistors arranged in the form of a bridge. A battery is connected across one diagonal of the bridge, and a galvanometer is connected across the other diagonal to detect the current flowing through the circuit.

Let the four resistances of the bridge be \(P\), \(Q\), \(R\), and \(S\). The galvanometer is connected between the junctions \(B\) and \(D\), while the battery is connected between \(A\) and \(C\).

Condition for Balance
When the bridge is balanced, no current flows through the galvanometer. This means the potential at point \(B\) becomes equal to the potential at point \(D\).

Let the current \(I_1\) flow through the branch containing resistances \(P\) and \(Q\), and the current \(I_2\) flow through the branch containing resistances \(R\) and \(S\).

The potential drop across resistance \(P\) is
\[ V_P = I_1 P \] The potential drop across resistance \(R\) is
\[ V_R = I_2 R \] Since the potentials at points \(B\) and \(D\) are equal in the balanced condition, the potential drops must satisfy
\[ I_1 P = I_2 R \] Similarly, the potential drop across resistance \(Q\) and \(S\) must satisfy
\[ I_1 Q = I_2 S \] Dividing the two equations:
\[ \frac{I_1 P}{I_1 Q} = \frac{I_2 R}{I_2 S} \] \[ \frac{P}{Q} = \frac{R}{S} \] Balanced Wheatstone Bridge Condition
\[ \frac{P}{Q} = \frac{R}{S} \] This relation represents the condition for a balanced Wheatstone bridge.

Conclusion
A Wheatstone bridge is said to be balanced when no current flows through the galvanometer. The condition for balance is that the ratio of resistances in one arm of the bridge must be equal to the ratio of resistances in the other arm, that is
\[ \frac{P}{Q} = \frac{R}{S} \] Under this condition, the unknown resistance can be determined accurately using known resistances.