A wire has a resistance of 12 ohm . It is bent in the form of equilateral triangle. The effective resistance between any two corners of the triangle is

Options

A. 9 ohms
B. 12 ohms
C. 6 ohms
D. `8//3` ohms

Answer

To find the effective resistance between any two corners of a wire bent in the form of an equilateral triangle, we can follow these steps:

### Step 1: Determine the Resistance of Each Side
The total resistance of the wire is given as 12 ohms. Since the wire is bent into the shape of an equilateral triangle, it is divided into three equal sides.

\[ \text{Resistance of each side} = \frac{\text{Total Resistance}}{3} = \frac{12 \, \text{ohms}}{3} = 4 \, \text{ohms} \]

### Step 2: Identify the Configuration of the Resistances
When considering the effective resistance between any two corners of the triangle, we can visualize the triangle as having three sides, each with a resistance of 4 ohms. Let's label the corners of the triangle as A, B, and C. We want to find the effective resistance between points A and B.

### Step 3: Analyze the Circuit
Between points A and B, there are two paths:
1. From A to B directly (4 ohms).
2. From A to C to B (4 ohms + 4 ohms = 8 ohms).

### Step 4: Combine the Resistances
The resistance from A to B directly (4 ohms) is in parallel with the resistance from A to C to B (8 ohms). The formula for combining two resistances \( R_1 \) and \( R_2 \) in parallel is:

\[ \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} \]

Substituting the values:

\[ \frac{1}{R_{\text{eq}}} = \frac{1}{4} + \frac{1}{8} \]

### Step 5: Calculate the Effective Resistance
To combine the fractions, we find a common denominator:

\[ \frac{1}{R_{\text{eq}}} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8} \]

Now, taking the reciprocal to find \( R_{\text{eq}} \):

\[ R_{\text{eq}} = \frac{8}{3} \, \text{ohms} \]

### Conclusion
The effective resistance between any two corners of the triangle is:

\[ \boxed{\frac{8}{3} \, \text{ohms}} \]

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