Question

When a wire of resistance r is cut into two equal halves?

Answer 1

When a wire of resistance \( R \) is cut into two equal halves, each half will have half the original length. Since resistance \( R \) is directly proportional to the length of the wire, each half will have half the resistance of the original wire.

Mathematically:

1. If the original wire has resistance \( R \) and is cut into two equal lengths, the resistance of each half is \( \frac{R}{2} \).

So, each of the two halves will have a resistance of \( \frac{R}{2} \).

Answer 2

When a wire of resistance \( R \) is cut into two equal halves, the resistance of each half changes. Let's go through the details to understand what happens:

1. **Original Wire:**
   - The wire initially has a resistance \( R \). This resistance is determined by the material's resistivity, the length of the wire, and its cross-sectional area.

2. **Cutting the Wire:**
   - When you cut the wire into two equal halves, you effectively halve its length.

3. **Resistance of Each Half:**
   - The resistance \( R \) of a wire is given by the formula:
     \[
     R = \rho \frac{L}{A}
     \]
     where:
     - \( \rho \) is the resistivity of the material,
     - \( L \) is the length of the wire,
     - \( A \) is the cross-sectional area of the wire.

   - After cutting the wire into two equal halves, the length of each half is \( \frac{L}{2} \).

   - Since the cross-sectional area \( A \) remains the same, the resistance \( R' \) of each half is:
     \[
     R' = \rho \frac{\frac{L}{2}}{A}
     \]

   - Simplifying this, we get:
     \[
     R' = \frac{R}{2}
     \]

   - Therefore, the resistance of each half of the wire is half the original resistance \( R \).

**Summary:**

- If you cut a wire of resistance \( R \) into two equal halves, the resistance of each half will be \( \frac{R}{2} \). This is because resistance is directly proportional to the length of the wire. Reducing the length by half reduces the resistance by half.