A piece of wire of resistance 4 ohm s is bent through `180^(@)` at its mid point and the two halves are twisted together, then the resistance is

Options

A. 8 ohms
B. 1 ohm
C. 2 ohms
D. 5 ohms

Answer

To solve the problem, we need to find the equivalent resistance of a piece of wire that has been bent and twisted. Here’s the step-by-step solution:

### Step 1: Understand the Initial Resistance
We start with a wire of total resistance \( R = 4 \, \Omega \).

### Step 2: Determine the Length of Each Half
When the wire is bent at its midpoint, it divides into two equal halves. Therefore, the resistance of each half of the wire can be calculated as follows:
- The total length of the wire is \( L \).
- The length of each half is \( \frac{L}{2} \).

### Step 3: Calculate the Resistance of Each Half
The resistance of a wire is given by the formula:
\[ R = \rho \frac{L}{A} \]
where \( \rho \) is the resistivity, \( L \) is the length, and \( A \) is the cross-sectional area. Since the resistivity and area remain constant, the resistance of each half will be:
\[ R_{half} = \frac{R}{2} = \frac{4 \, \Omega}{2} = 2 \, \Omega \]

### Step 4: Analyze the Configuration After Twisting
After bending the wire, the two halves are twisted together. This means that the two halves are connected in parallel.

### Step 5: Calculate the Equivalent Resistance
For two resistors \( R_1 \) and \( R_2 \) in parallel, the equivalent resistance \( R_{eq} \) is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]
Substituting \( R_1 = 2 \, \Omega \) and \( R_2 = 2 \, \Omega \):
\[ \frac{1}{R_{eq}} = \frac{1}{2} + \frac{1}{2} = \frac{2}{2} = 1 \]
Thus,
\[ R_{eq} = 1 \, \Omega \]

### Conclusion
The equivalent resistance of the twisted wire is \( 1 \, \Omega \).

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