A cylindrical rod is reformed to half of its original length keeping volume constant. If its resistance before this change were R, then the resistance after reformation of rod will be

Options

A. R
B. R/4
C. 3R/4
D. R/2

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the initial conditions
We have a cylindrical rod with:
- Original length \( L \)
- Original radius \( R_1 \)
- Original resistance \( R \)

### Step 2: Write the formula for resistance
The resistance \( R \) of a cylindrical conductor is given by the formula:
\[ R = \frac{\rho L}{A} \]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the rod,
- \( A \) is the cross-sectional area.

For a cylinder, the cross-sectional area \( A \) is:
\[ A = \pi R_1^2 \]
Thus, the initial resistance can be expressed as:
\[ R = \frac{\rho L}{\pi R_1^2} \]

### Step 3: Understand the changes after reformation
After reformation, the rod's length is halved:
- New length \( L_2 = \frac{L}{2} \)

Since the volume is kept constant, we can express the volume before and after reformation:
\[ \text{Volume} = A_1 \cdot L_1 = A_2 \cdot L_2 \]
Substituting the areas:
\[ \pi R_1^2 \cdot L = \pi R_2^2 \cdot \left(\frac{L}{2}\right) \]
Cancelling \( \pi \) and rearranging gives:
\[ R_1^2 \cdot L = R_2^2 \cdot \frac{L}{2} \]
This simplifies to:
\[ R_1^2 = \frac{R_2^2}{2} \]

### Step 4: Write the new resistance formula
Using the new length and radius, the new resistance \( R_2 \) can be expressed as:
\[ R_2 = \frac{\rho L_2}{A_2} = \frac{\rho \left(\frac{L}{2}\right)}{\pi R_2^2} \]

### Step 5: Relate the resistances
From the relationship derived in Step 3, we have:
\[ \frac{R_1^2}{R_2^2} = 2 \]
This implies:
\[ R_2^2 = \frac{R_1^2}{2} \]
Taking the square root gives:
\[ R_2 = \frac{R_1}{\sqrt{2}} \]

### Step 6: Substitute \( R_1 \) with \( R \)
Since \( R_1 = R \), we have:
\[ R_2 = \frac{R}{\sqrt{2}} \]

### Step 7: Final expression for resistance
The final expression for the resistance after reformation is:
\[ R_2 = \frac{R}{4} \]

### Conclusion
Thus, the resistance after reformation of the rod will be:
\[ \boxed{\frac{R}{4}} \]