Three resistors each of 2 ohm are connected together in a triangular shape. The resistance between any two vertices will be

Options

A. `4//3Omega`
B. `3//4Omega`
C. `3Omega`
D. `6Omega`

Answer

To find the equivalent resistance between any two vertices of a triangular arrangement of three resistors, each of 2 ohms, we can follow these steps:

### Step 1: Understand the Circuit Configuration
We have three resistors (R1, R2, and R3), each of 2 ohms, connected in a triangular shape. We need to find the equivalent resistance between two vertices, say A and B.

### Step 2: Identify the Series and Parallel Connections
From vertex A to vertex B, there are two paths:
1. The path through resistor R1 (2 ohms) and R2 (2 ohms) in series.
2. The path through resistor R3 (2 ohms) directly.

### Step 3: Calculate the Resistance in Series
The resistors R1 and R2 are in series. The equivalent resistance (R_eq1) of resistors in series is given by:
\[ R_{eq1} = R_1 + R_2 = 2 \, \text{ohm} + 2 \, \text{ohm} = 4 \, \text{ohm} \]

### Step 4: Calculate the Equivalent Resistance in Parallel
Now we have two resistances: R_eq1 (4 ohms from the series path) and R3 (2 ohms from the direct path). These two resistances are in parallel. The equivalent resistance (R_eq_total) for resistors in parallel is given by:
\[ \frac{1}{R_{eq\_total}} = \frac{1}{R_{eq1}} + \frac{1}{R_3} \]

Substituting the values:
\[ \frac{1}{R_{eq\_total}} = \frac{1}{4} + \frac{1}{2} \]

### Step 5: Simplify the Parallel Resistance
To add these fractions, we find a common denominator (which is 4):
\[ \frac{1}{R_{eq\_total}} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \]

Now, taking the reciprocal to find R_eq_total:
\[ R_{eq\_total} = \frac{4}{3} \, \text{ohm} \]

### Step 6: Conclusion
Thus, the equivalent resistance between any two vertices (A and B) of the triangular arrangement of resistors is:
\[ R_{eq\_total} = \frac{4}{3} \, \text{ohm} \]