Question

In the figure shown, three resistors of \(4\,\Omega\) are connected. If point \(D\) divides the resistor between \(B\) and \(C\) into two equal parts, then the equivalent resistance between points \(A\) and \(D\) is:

• A. \(12\,\Omega\)
• B. \(6\,\Omega\)
• C. \(3\,\Omega\)
• D. \(\dfrac{1}{3}\,\Omega\)

Hint:

Whenever a point divides a resistor into equal parts, first split the resistor accordingly. Here: \[ 4\,\Omega \rightarrow 2\,\Omega + 2\,\Omega \] Then identify all possible paths between the required terminals and reduce the circuit using series and parallel combinations.

Answer 1

The Correct Option is C

Step 1: Identify the resistance values. Each side of the triangle has resistance \[ 4\,\Omega \] Since \(D\) is the midpoint of the resistor \(BC\), \[ BD=DC=2\,\Omega \]

Step 2: Find the two paths between \(A\) and \(D\). Path 1: \[ A \rightarrow B \rightarrow D \] Resistance along this path: \[ R_1=4+2=6\,\Omega \] Path 2: \[ A \rightarrow C \rightarrow D \] Resistance along this path: \[ R_2=4+2=6\,\Omega \]

Step 3: Combine the two paths. The two \(6\,\Omega\) paths are connected in parallel between \(A\) and \(D\). Therefore, \[ R_{AD} = \frac{6\times6}{6+6} \] \[ = \frac{36}{12} \] \[ = 3\,\Omega \] Therefore, the equivalent resistance between \(A\) and \(D\) is \[ \boxed{3\,\Omega} \] Hence, the correct answer is: \[ \boxed{\text{(C)}} \]