Question

For the given mixed combination of resistors, calculate the total resistance between points \(A\) and \(B\).

• A. \(9\,\Omega\)
• B. \(18\,\Omega\)
• C. \(4\,\Omega\)
• D. \(14\,\Omega\)

Hint:

Whenever a triangular resistor network contains: \[ R \text{ in parallel with } (R_1+R_2), \] reduce that section first. Here: \[ 8\Omega \parallel (4\Omega+4\Omega) = 8\Omega \parallel 8\Omega = 4\Omega \] After simplification, the remaining network becomes a simple series-parallel combination.

Answer 1

The Correct Option is B

Step 1: Analyze the upper triangular section. Between the two upper junctions, there are two possible paths: Direct path: \[ 8\,\Omega \] Upper path through the apex: \[ 4+4=8\,\Omega \] Therefore these two \(8\,\Omega\) resistances are in parallel. \[ R_{\text{upper}} = 8 \parallel 8 = \frac{8\times8}{8+8} = 4\,\Omega \]

Step 2: Replace the upper section. The circuit now becomes a rectangle with: \[ 4\,\Omega \] between the upper junctions and \[ 12\,\Omega \] between the lower junctions. The side resistances are: \[ 4\,\Omega \] on the left and \[ 4\,\Omega \] on the right.

Step 3: Find equivalent resistance between the lower junctions. One path is directly: \[ 12\,\Omega \] The other path goes through the upper branch: \[ 4+4+4 = 12\,\Omega \] Hence, \[ R = 12 \parallel 12 = 6\,\Omega \]

Step 4: Include the terminal resistors. The \(6\,\Omega\) resistor from \(A\) to the left junction and the \(6\,\Omega\) resistor from the right junction to \(B\) are in series with the equivalent found above. Thus, \[ R_{AB} = 6+6+6 \] \[ R_{AB} = 18\,\Omega \] Therefore, \[ \boxed{R_{AB}=18\,\Omega} \] Hence, the correct answer is: \[ \boxed{\text{(B)}} \]