A uniform wire of length l and radius r has a resistance of `100 Omega`. It is recast into wire of radius `(R )/(2)`

Options

A. `1600 Omega`
B. `400 Omega`
C. `200 Omega`
D. `100 Omega`

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the initial conditions
We have a uniform wire of length \( L \) and radius \( R \) with a resistance of \( 100 \, \Omega \).

### Step 2: Calculate the volume of the original wire
The volume \( V \) of the original wire can be calculated using the formula for the volume of a cylinder:
\[ V = \text{Cross-sectional Area} \times \text{Length} = \pi R^2 L \]

### Step 3: Determine the new dimensions after recasting
The wire is recast into a new wire with a radius of \( \frac{R}{2} \). Let the new length of the wire be \( L' \). The volume of the new wire is:
\[ V' = \pi \left(\frac{R}{2}\right)^2 L' = \pi \frac{R^2}{4} L' \]

### Step 4: Set the volumes equal
Since the wire is recast, the volume remains constant:
\[ \pi R^2 L = \pi \frac{R^2}{4} L' \]
Cancelling \( \pi R^2 \) from both sides gives:
\[ L = \frac{L'}{4} \]
Thus, we can express \( L' \) in terms of \( L \):
\[ L' = 4L \]

### Step 5: Calculate the new resistance
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, and \( A \) is the cross-sectional area.

For the original wire:
\[ R = \frac{\rho L}{\pi R^2} = 100 \, \Omega \]

For the new wire:
\[ R' = \frac{\rho L'}{A'} \]
where \( A' = \pi \left(\frac{R}{2}\right)^2 = \pi \frac{R^2}{4} \).

Substituting \( L' = 4L \) into the resistance formula for the new wire:
\[ R' = \frac{\rho (4L)}{\pi \frac{R^2}{4}} = \frac{16 \rho L}{\pi R^2} \]

### Step 6: Relate new resistance to original resistance
We know from the original resistance:
\[ R = \frac{\rho L}{\pi R^2} = 100 \, \Omega \]
Thus,
\[ R' = 16 \times R = 16 \times 100 = 1600 \, \Omega \]

### Final Answer
The new resistance \( R' \) of the wire after recasting is:
\[ \boxed{1600 \, \Omega} \]