A force F is needed to break a copper wire having radius R. the force needed to break a copper wire of radius 2R will be

Options

A. `F/2`
B. 2F
C. 4F
D. `F/4`

Explanation

To solve the problem, we need to understand the relationship between the radius of the wire and the force required to break it. The force required to break a wire is related to its cross-sectional area, which depends on the radius.

### Step-by-Step Solution:

1. **Understand the relationship between force and area**:
The force required to break a wire is proportional to its cross-sectional area. For a circular wire, the cross-sectional area \( A \) can be calculated using the formula:
\[ A = \pi R^2 \]
where \( R \) is the radius of the wire.

2. **Calculate the area for the first wire**:
For the wire with radius \( R \):
\[ A_1 = \pi R^2 \]
Let the force required to break this wire be \( F \).

3. **Calculate the area for the second wire**:
For the wire with radius \( 2R \):
\[ A_2 = \pi (2R)^2 = \pi \cdot 4R^2 = 4\pi R^2 \]

4. **Relate the forces**:
Since the force required to break the wire is proportional to the cross-sectional area, we can set up the following relationship:
\[ \frac{F_2}{F} = \frac{A_2}{A_1} \]
Substituting the areas we calculated:
\[ \frac{F_2}{F} = \frac{4\pi R^2}{\pi R^2} = 4 \]

5. **Calculate the force for the second wire**:
Therefore, we can express \( F_2 \) in terms of \( F \):
\[ F_2 = 4F \]

### Final Answer:
The force needed to break a copper wire of radius \( 2R \) will be \( 4F \).

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