A force F is needed to break a copper wire having radius R. The force needed to break a copper wire of same length and radius 2R will be

Options

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

Explanation

To solve the problem, we need to understand how the breaking strength of a wire is related to its radius. The breaking strength of a material is defined as the force required to break the wire divided by its cross-sectional area.

### Step-by-Step Solution:

1. **Identify the Breaking Strength Formula**:
The breaking strength (σ) of the wire can be expressed as:
\[ \sigma = \frac{F}{A} \]
where \( F \) is the force required to break the wire and \( A \) is the cross-sectional area of the wire.

2. **Calculate the Cross-Sectional Area**:
For a wire with radius \( R \), the cross-sectional area \( A \) is given by:
\[ A = \pi R^2 \]
Therefore, the breaking strength for the first wire can be expressed as:
\[ \sigma = \frac{F}{\pi R^2} \]

3. **Consider the New Wire with Radius \( 2R \)**:
Now, we need to find the force \( F_2 \) required to break a wire of the same length but with a radius \( 2R \). The cross-sectional area for this wire is:
\[ A_2 = \pi (2R)^2 = \pi (4R^2) = 4\pi R^2 \]

4. **Express the Breaking Strength for the New Wire**:
The breaking strength for the new wire can be expressed as:
\[ \sigma = \frac{F_2}{A_2} = \frac{F_2}{4\pi R^2} \]

5. **Set the Breaking Strengths Equal**:
Since the breaking strength of the material remains the same, we can set the two expressions for breaking strength equal to each other:
\[ \frac{F}{\pi R^2} = \frac{F_2}{4\pi R^2} \]

6. **Solve for \( F_2 \)**:
By simplifying the equation, we can eliminate \( \pi R^2 \) from both sides:
\[ F = \frac{F_2}{4} \]
Rearranging gives:
\[ F_2 = 4F \]

### Conclusion:
The force needed to break a copper wire of radius \( 2R \) is \( 4F \).