Question

The equivalent resistance of the given circuit between the terminals A and B is : 

• A. $0\,\Omega$
• B. $3\,\Omega$
• C. $1\,\Omega$
• D. $\frac{9}{2}\,\Omega$

Hint:

Always look for short circuits first. A wire in parallel with a resistor or a whole section of a circuit effectively deletes it from the calculation, drastically simplifying the problem.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To find the equivalent resistance between two terminals, we identify parallel and series combinations of resistors and simplify the circuit from the farthest end toward the terminals. Additionally, any branch that is short-circuited (parallel to a wire of zero resistance) is effectively removed from the circuit calculations.
Step 2: Key Formula or Approach:
1. For resistors in series: $R_{eq} = R_1 + R_2 + ...$
2. For resistors in parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$
3. A short circuit (zero resistance wire) in parallel with any resistor makes the effective resistance of that combination $0\,\Omega$.
Step 3: Detailed Explanation:
Let's analyze the nodes starting from the left:
1. The first vertical line on the far left is a simple wire. This wire is in parallel with the $5\,\Omega$ vertical resistor. Therefore, the $5\,\Omega$ resistor is shorted and its effective resistance is $0\,\Omega$. Let the top node of this shorted section be at the same potential as ground (terminal B).
2. Moving to the right, there is a horizontal $2\,\Omega$ resistor connected between this grounded node and the top node of the next vertical $2\,\Omega$ resistor. This means the horizontal $2\,\Omega$ and vertical $2\,\Omega$ resistors are both connected between this new node and ground (Node B), making them in parallel. Their equivalent resistance is: \[ R_{p1} = \frac{2 \times 2}{2 + 2} = 1\,\Omega \]
3. Next, we have another horizontal $2\,\Omega$ resistor connecting the previous node to the final part of the circuit. Terminal A is at the top of two $3\,\Omega$ vertical resistors. These two $3\,\Omega$ resistors are connected in parallel to each other. Their equivalent resistance is: \[ R_{p2} = \frac{3 \times 3}{3 + 3} = 1.5\,\Omega \]
4. Now, the circuit simplifies to Terminal A being connected to ground (B) through $R_{p2}$ and another branch consisting of the second horizontal $2\,\Omega$ resistor in series with $R_{p1}$. The resistance of the side branch is $R_{branch} = 2 + 1 = 3\,\Omega$.
5. Finally, Terminal A is connected to ground through two parallel paths: $1.5\,\Omega$ and $3\,\Omega$. \[ R_{AB} = \frac{1.5 \times 3}{1.5 + 3} = \frac{4.5}{4.5} = 1\,\Omega \]
Step 4: Final Answer:
The equivalent resistance between terminals A and B is $1\,\Omega$.