If the resistance of wire is `1Omega` . If the radius of wire is halved then the new resistance will be:

Options

A. 2 R
B. 16 R
C. 8 R
D. 4 R

Explanation

To solve the problem, we need to understand how the resistance of a wire changes when its radius is altered. The resistance \( R \) of a wire is given by the formula:

\[ R = \frac{\rho L}{A} \]

where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.

### Step 1: Initial Conditions
Let the initial resistance \( R = 1 \, \Omega \). The initial radius of the wire is \( R \) and the length is \( L \). The cross-sectional area \( A \) can be expressed as:

\[ A = \pi R^2 \]

### Step 2: Change in Radius
If the radius of the wire is halved, the new radius \( R' \) becomes:

\[ R' = \frac{R}{2} \]

### Step 3: Calculate New Area
The new cross-sectional area \( A' \) after the radius is halved is:

\[ A' = \pi (R')^2 = \pi \left(\frac{R}{2}\right)^2 = \pi \frac{R^2}{4} = \frac{A}{4} \]

### Step 4: Volume Conservation
Since the volume of the wire remains constant, we can equate the initial volume \( V \) and the final volume \( V' \):

\[ V = A \cdot L = \pi R^2 L \]
\[ V' = A' \cdot L' = \frac{A}{4} \cdot L' \]

Setting these equal gives:

\[ \pi R^2 L = \frac{A}{4} \cdot L' \]

Substituting \( A = \pi R^2 \):

\[ \pi R^2 L = \frac{\pi R^2}{4} \cdot L' \]

### Step 5: Solve for New Length
Cancelling \( \pi R^2 \) from both sides:

\[ L = \frac{L'}{4} \]

Thus, we find:

\[ L' = 4L \]

### Step 6: Calculate New Resistance
Now we can find the new resistance \( R' \):

\[ R' = \frac{\rho L'}{A'} = \frac{\rho (4L)}{\frac{A}{4}} = \frac{4\rho L}{\frac{\pi R^2}{4}} = \frac{16 \rho L}{\pi R^2} \]

### Step 7: Relate to Initial Resistance
Since \( R = \frac{\rho L}{A} \), we can express \( R' \) in terms of \( R \):

\[ R' = 16 \cdot \frac{\rho L}{A} = 16R \]

### Conclusion
Therefore, the new resistance when the radius of the wire is halved is:

\[ R' = 16 \, \Omega \]

### Final Answer
The new resistance will be \( 16R \).

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