Question

The equivalent resistance between the points \(A\) and \(B\) in the given circuit is \[ \frac{x}{5}\,\Omega. \] Find the value of \(x\).

Hint:

In symmetric resistor networks, always check if a bridge resistor carries zero current—it often simplifies the circuit drastically.

Answer 1

Correct Answer: 22.5

Concept: To find equivalent resistance:
Identify symmetry in the circuit
Use series and parallel combinations
Use the concept of balanced bridge when applicable
Step 1: Observe symmetry of the circuit The circuit is symmetric about the vertical line through the middle. Hence, the potentials at the midpoints of the top and bottom branches are equal. Therefore, no current flows through the central \(3\,\Omega\) resistor.
Step 2: Remove the central resistor The circuit now reduces to two parallel branches between \(A\) and \(B\):
Top branch: \(6\,\Omega + 3\,\Omega = 9\,\Omega\)
Bottom branch: \(3\,\Omega + 6\,\Omega = 9\,\Omega\)
Step 3: Find equivalent resistance Two \(9\,\Omega\) resistors in parallel: \[ R_{\text{eq}}=\frac{9\times9}{9+9}=\frac{81}{18}=4.5\,\Omega \]
Step 4: Compare with given form \[ \frac{x}{5}=4.5 \Rightarrow x=22.5 \] Final Answer: \[ \boxed{22.5} \]