A wire of radius r has resistance R. If it is stretched to a radius of `(3r)/(4)`, its resistance becomes

Options

A. `(9R)/(16)`
B. `(16R)/(9)`
C. `(81R)/(256)`
D. `(256R)/(81)`

Answer

To solve the problem, we need to determine the new resistance of a wire after it has been stretched. The initial resistance of the wire is given as \( R \) and its initial radius is \( r \). After stretching, the radius becomes \( \frac{3r}{4} \).

### Step-by-Step Solution:

1. **Understand the relationship between resistance, length, and area**:
The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity, \( L \) is the length of the wire, and \( A \) is the cross-sectional area. The area \( A \) of a wire with radius \( r \) is:
\[ A = \pi r^2 \]

2. **Determine the volume of the wire**:
The volume \( V \) of the wire is constant before and after stretching. The volume can be expressed as:
\[ V = A \cdot L = \pi r^2 L \]

3. **Express the new length after stretching**:
When the wire is stretched to a new radius \( \frac{3r}{4} \), let the new length be \( L' \). The new area \( A' \) becomes:
\[ A' = \pi \left(\frac{3r}{4}\right)^2 = \pi \cdot \frac{9r^2}{16} \]
The volume remains the same, so we have:
\[ V = A' \cdot L' = \pi \cdot \frac{9r^2}{16} \cdot L' \]
Setting the two expressions for volume equal gives:
\[ \pi r^2 L = \pi \cdot \frac{9r^2}{16} \cdot L' \]
Simplifying this, we find:
\[ L' = \frac{16L}{9} \]

4. **Calculate the new resistance \( R' \)**:
Now, we can find the new resistance \( R' \) using the new length \( L' \) and new area \( A' \):
\[ R' = \frac{\rho L'}{A'} = \frac{\rho \cdot \frac{16L}{9}}{\pi \cdot \frac{9r^2}{16}} = \frac{16\rho L}{9} \cdot \frac{16}{9\pi r^2} \]
This simplifies to:
\[ R' = \frac{256 \rho L}{81 \pi r^2} \]

5. **Relate the new resistance to the initial resistance**:
Since the initial resistance \( R \) is given by:
\[ R = \frac{\rho L}{\pi r^2} \]
We can express \( R' \) in terms of \( R \):
\[ R' = R \cdot \frac{256}{81} \]

6. **Final Result**:
Thus, the new resistance after stretching the wire is:
\[ R' = \frac{256}{81} R \]

### Conclusion:
The resistance of the wire after being stretched to a radius of \( \frac{3r}{4} \) becomes \( \frac{256}{81} R \).