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. `6Omega`
B. `(8)/(3)Omega`
C. `9Omega`
D. `12Omega`

Answer

To solve the problem of finding 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 total resistance of the wire
The total resistance of the wire is given as \( R_0 = 12 \, \Omega \).

### Step 2: Calculate the length of each side of the triangle
Since the wire is bent into the shape of an equilateral triangle, the total length of the wire is divided into three equal parts. Therefore, the length of each side of the triangle is:
\[ L = \frac{L_{\text{total}}}{3} \]
The resistance of each side of the triangle can be calculated using the formula for resistance:
\[ R_{\text{side}} = \frac{R_0}{3} = \frac{12 \, \Omega}{3} = 4 \, \Omega \]

### Step 3: Draw the circuit
Label the corners of the triangle as A, B, and C. The resistances between the corners are:
- \( R_{AB} = 4 \, \Omega \) (between A and B)
- \( R_{AC} = 4 \, \Omega \) (between A and C)
- \( R_{BC} = 4 \, \Omega \) (between B and C)

### Step 4: Analyze the circuit between two corners
To find the effective resistance between points A and B, we can see that:
- \( R_{AC} \) and \( R_{BC} \) are in series with each other.
- Therefore, the total resistance of the path from A to C to B is:
\[ R_{AC} + R_{BC} = 4 \, \Omega + 4 \, \Omega = 8 \, \Omega \]

### Step 5: Combine the resistances in parallel
Now, we have two paths between A and B:
1. Directly through \( R_{AB} = 4 \, \Omega \)
2. Through \( R_{AC} \) and \( R_{BC} \) which totals to \( 8 \, \Omega \)

These two resistances are in parallel. The formula for equivalent resistance \( R_{AB} \) for two resistors in parallel \( R_1 \) and \( R_2 \) is given by:
\[ \frac{1}{R_{AB}} = \frac{1}{R_1} + \frac{1}{R_2} \]
Substituting the values:
\[ \frac{1}{R_{AB}} = \frac{1}{4 \, \Omega} + \frac{1}{8 \, \Omega} \]

### Step 6: Calculate the equivalent resistance
Calculating the right-hand side:
\[ \frac{1}{R_{AB}} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8} \]
Thus,
\[ R_{AB} = \frac{8}{3} \, \Omega \]

### Conclusion
The effective resistance between any two corners of the triangle is:
\[ R_{AB} = \frac{8}{3} \, \Omega \]