Question

The resultant resistance between $ A $ and $ B $ in the given figure is:

• A. \( 1 \, \Omega \)
• B. \( 2 \, \Omega \)
• C. \( 3 \, \Omega \)
• D. \( 6 \, \Omega \)

Hint:

In a balanced Wheatstone bridge, no current flows through the middle resistor, simplifying the circuit to series and parallel combinations.

Answer 1

The Correct Option is B

We need to find the equivalent resistance between points \( A \) and \( B \) in the given circuit.
Step 1: Identify the circuit configuration.
The circuit is a Wheatstone bridge with resistors: \( AB = 3 \, \Omega \), \( AD = 3 \, \Omega \), \( BC = 3 \, \Omega \), \( CD = 3 \, \Omega \), and \( BD = 6 \, \Omega \).
Step 2: Check if the bridge is balanced.
\[ \frac{R_{AB}}{R_{AD}} = \frac{3}{3} = 1, \quad \frac{R_{BC}}{R_{CD}} = \frac{3}{3} = 1 \] The bridge is balanced, so no current flows through the 6 \( \Omega \) resistor.
Step 3: Simplify the circuit.
\( AB \) and \( BC \) in series: \( 3 + 3 = 6 \, \Omega \).
\( AD \) and \( DC \) in series: \( 3 + 3 = 6 \, \Omega \).
These two 6 \( \Omega \) branches are in parallel between \( A \) and \( C \):
\[ R_{AC} = \frac{6 \times 6}{6 + 6} = 3 \, \Omega \]
Step 4: Find resistance between \( A \) and \( B \).
In the balanced bridge, the resistance between \( A \) and \( B \) is:
Path \( AB \): 3 \( \Omega \).
Effective resistance considering the parallel paths: \[ R_{AB} = \frac{(R_{AB} + R_{BC}) \times (R_{AD} + R_{DC})}{(R_{AB} + R_{BC}) + (R_{AD} + R_{DC})} \times \frac{1}{2} = \frac{6 \times 6}{6 + 6} \times \frac{1}{2} = 2 \, \Omega \] Final Answer: \[ \boxed{2} \]