Q.A wire of resistance 2 ohms per meter is bent to form a circle of radius 1 m. The equivalent resistance between its two diametrically opposite points, A and B, is

Wheatstone Bridge Symmetry — Why the Formula Holds

The Wheatstone bridge is a circuit used to measure an unknown resistance by balancing two legs of a bridge. The key result is:

When the bridge is balanced, no current flows through the galvanometer, and:

$\frac{R_1}{R_2} = \frac{R_3}{R_4}$

Let's understand why this is true — not just memorize it.

1. The Circuit Setup

A Wheatstone bridge has four resistors arranged in a diamond:

• $R_1$ and $R_2$ in series on the left branch
• $R_3$ and $R_4$ in series on the right branch
• A galvanometer (sensitive current detector) connects the midpoints of the two branches
• A battery connects across the top and bottom

        A
       / \
      /   \
     R1    R3
     |     |
     G-----|   (G = galvanometer)
     |     |
     R2    R4
      \   /
       \ /
        B

2. The Condition for Balance — "No Current Through G"

Balance means the galvanometer shows zero deflection — no current flows through it. This implies points C (between $R_1$ and $R_2$) and D (between $R_3$ and $R_4$) are at the same electric potential. If $V_C = V_D$, no potential difference exists across the galvanometer, so no current flows.

3. Deriving the Ratio — Step by Step

Step 1: Branch currents

With no current through G, the left branch carries a single current $I_1$ and the right branch a single current $I_2$.

Step 2: Equal potentials at the midpoints

From A (common top) to the midpoints, since $V_C = V_D$:

From the midpoints to B (common bottom), since $V_C = V_D$:

Step 3: Divide (1) by (2)

The currents $I_1$ and $I_2$ cancel:

4. Why This Makes Physical Sense

The bridge is essentially two voltage dividers sharing the same input voltage. Balance occurs when both dividers produce the same output voltage at their midpoints. Notice that $I_1$ and $I_2$ need not be equal — the balance condition fixes only the ratio of resistances in each branch, not their individual values.

5. Key Exam Takeaways

Balance = equal potentials → equal voltage ratios → $\frac{R_1}{R_2} = \frac{R_3}{R_4}$