Question

Assertion (A) : In a Wheatstone bridge circuit, if we interchange the position of the cell and the galvanometer, the balance condition \(\frac{P}{Q} = \frac{R}{S}\) remains unchanged.
Reason (R) : \(\frac{P}{Q} = \frac{R}{S} \Rightarrow \frac{P}{R} = \frac{Q}{S}\) so balance condition remains same.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Both Assertion (A) and Reason (R) are false.

Hint:

In any balanced Wheatstone bridge network, the positions of the source (cell) and the detector (galvanometer) are fully interchangeable without disturbing the fundamental null deflection state because they occupy conjugate branches of the circuit.

Answer 1

The Correct Option is A

Concept: A standard Wheatstone bridge consists of four resistance arms arranged in a closed loop forming a quadrilateral network. Let the four resistances be \(P, Q, R,\) and \(S\). A galvanometer is connected across one pair of opposite vertices, and a battery (cell) is connected across the remaining pair of opposite vertices.
• Balanced Condition: The bridge is balanced when no current flows through the galvanometer circuit. This specific null-deflection state occurs if and only if the potential at the two endpoints of the galvanometer branch becomes exactly equal.
• Conjugate Arms: In network theory, the battery arm and the galvanometer arm are called conjugate arms of the bridge network. Interchanging them leaves the essential cross-ratio criterion unchanged.

Step 1: Evaluation of the Assertion (A).
Let the four vertices of the bridge network be labeled sequentially as \(A, B, C,\) and \(D\).
• Suppose the four resistors are connected as: \(P\) between \(A\) and \(B\), \(Q\) between \(B\) and \(C\), \(R\) between \(A\) and \(D\), and \(S\) between \(D\) and \(C\).
• Initially, let the galvanometer be connected across vertices \(B\) and \(D\), and the cell connected across vertices \(A\) and \(C\). The condition for zero current through the galvanometer is: \[ \frac{P}{Q} = \frac{R}{S} \]
• If we interchange the positions of the cell and the galvanometer, the galvanometer is now connected between vertices \(A\) and \(C\), while the cell is connected between vertices \(B\) and \(D\).
• Looking at the reorganized network with the cell feeding terminals \(B\) and \(D\), the loop balances when the potentials at \(A\) and \(C\) are identical. This structural setup leads directly to the revised balancing fraction: \[ \frac{P}{R} = \frac{Q}{S} \] Since interchanging the conjugate arms maintains a valid balancing mechanism, the Assertion statement is factually true.

Step 2: Evaluation of the Reason (R) and its logical link.
Let us perform algebraic operations on the original balance criteria equation: \[ \frac{P}{Q} = \frac{R}{S} \] By multiplying both sides of the equation by \(Q\) and dividing by \(R\) (cross-multiplying terms), we get: \[ P \cdot S = R \cdot Q \quad \Rightarrow \quad \frac{P}{R} = \frac{Q}{S} \] This mathematically matches the expression given in the Reason. Therefore, the statement in the Reason is independently true. Furthermore, because the algebraic manipulation \(\frac{P}{Q} = \frac{R}{S} \Rightarrow \frac{P}{R} = \frac{Q}{S}\) directly maps the first balanced bridge condition to the second layout configuration, the Reason accurately accounts for why the balance state is preserved after switching the components. Hence, Reason (R) is the correct explanation of Assertion (A).