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

Step 1: Understanding the Concept:
The circuit is a bridge network. Since the ratio of the resistances on the left side (\(6/3\)) is not equal to the ratio on the right side (\(3/6\)), it is an unbalanced Wheatstone bridge. Such circuits are solved using nodal analysis or Delta-Wye transformation.
Step 2: Key Formula or Approach:
Nodal Analysis: Apply a potential difference across A and B, determine the currents, and use \(R_{eq} = V/I\).
Step 3: Detailed Explanation:
Let the potential at \(A\) be \(V_A = 1\) V and at \(B\) be \(V_B = 0\) V.
Let the potential at the top-middle junction be \(V_1\) and at the bottom-middle junction be \(V_2\).
Applying KCL at node \(V_1\): \[ \frac{V_1 - 1}{6} + \frac{V_1 - 0}{3} + \frac{V_1 - V_2}{3} = 0 \] Multiply by 6: \[ (V_1 - 1) + 2V_1 + 2(V_1 - V_2) = 0 \implies 5V_1 - 2V_2 = 1 \quad \dots(i) \] Applying KCL at node \(V_2\): \[ \frac{V_2 - 1}{3} + \frac{V_2 - 0}{6} + \frac{V_2 - V_1}{3} = 0 \] Multiply by 6: \[ 2(V_2 - 1) + V_2 + 2(V_2 - V_1) = 0 \implies 5V_2 - 2V_1 = 2 \quad \dots(ii) \] Solving (i) and (ii): From (i), \(2V_2 = 5V_1 - 1\). Substitute into (ii): \[ 5\left(\frac{5V_1 - 1}{2}\right) - 2V_1 = 2 \implies 25V_1 - 5 - 4V_1 = 4 \implies 21V_1 = 9 \implies V_1 = \frac{3}{7} \text{ V} \] Then \(V_2 = \frac{5(3/7) - 1}{2} = \frac{8/7}{2} = \frac{4}{7} \text{ V}\).
Total current \(I\) from source: \[ I = I_{top\_left} + I_{bottom\_left} = \frac{1 - V_1}{6} + \frac{1 - V_2}{3} \] \[ I = \frac{1 - 3/7}{6} + \frac{1 - 4/7}{3} = \frac{4/7}{6} + \frac{3/7}{3} = \frac{4}{42} + \frac{1}{7} = \frac{2}{21} + \frac{3}{21} = \frac{5}{21} \text{ A} \] \[ R_{eq} = \frac{V}{I} = \frac{1}{5/21} = \frac{21}{5} \, \Omega \] Given \(R_{eq} = \frac{x}{5}\), therefore \(x = 21\).
Step 4: Final Answer:
The value of \(x\) is 21.