A wire stretches by a certain amount under a load. If the load and radius are increased to four times, find the stretch caused in the wire.

Options

A. l
B. `(l)/(2)`
C. `(l)/(3)`
D. `(l)/(4)`

Explanation

To solve the problem, we need to understand the relationship between stress, strain, and the resulting stretch in a wire.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- Let the initial load on the wire be \( F \).
- Let the initial radius of the wire be \( r \).
- The initial area of cross-section \( A \) of the wire is given by:
\[ A = \pi r^2 \]

2. **Calculate Initial Stress**:
- Stress (\( \sigma \)) is defined as force per unit area:
\[ \sigma = \frac{F}{A} = \frac{F}{\pi r^2} \]

3. **Determine the New Conditions**:
- The load is increased to \( 4F \).
- The radius is increased to \( 4r \).
- The new area of cross-section \( A' \) becomes:
\[ A' = \pi (4r)^2 = \pi (16r^2) = 16\pi r^2 \]

4. **Calculate New Stress**:
- The new stress (\( \sigma' \)) is given by:
\[ \sigma' = \frac{4F}{A'} = \frac{4F}{16\pi r^2} = \frac{F}{4\pi r^2} \]

5. **Relate New Stress to Original Stress**:
- We can express the new stress in terms of the original stress:
\[ \sigma' = \frac{1}{4} \sigma \]
- This means the new stress is one-fourth of the original stress.

6. **Determine the Stretch**:
- The stretch in the wire is proportional to the stress applied. Therefore, if the new stress is one-fourth of the original stress, the stretch caused in the wire will also be one-fourth of the original stretch.
- If the original stretch is \( \Delta L \), the new stretch \( \Delta L' \) will be:
\[ \Delta L' = \frac{1}{4} \Delta L \]

### Final Answer:
The stretch caused in the wire when the load and radius are increased to four times is one-fourth of the original stretch.