When current is passed through a circular wire prepared from a conducting material, the magnetic field produced at its centre is B. Now a loop having two turns is prepared from the same wire and the same current is passed through it. The magnetic field at its centre will be :

Options

A. `4B`
B. `B//4`
C. `B//2`
D. `16B`

Explanation

To solve the problem, we need to analyze the magnetic field produced by a circular wire and how it changes when we create a loop with two turns from the same wire.

### Step-by-Step Solution:

1. **Understanding the Magnetic Field from a Single Loop**:
The magnetic field \( B \) at the center of a single circular loop of radius \( R \) carrying a current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I n}{2R} \]
where \( n \) is the number of turns (which is 1 for a single loop).

2. **Magnetic Field for the First Case**:
In the first case, we have a single loop (1 turn) with radius \( R_1 \) and the magnetic field at its center is given as \( B \):
\[ B = \frac{\mu_0 I \cdot 1}{2R_1} \]
This can be marked as Equation (1).

3. **Creating a Loop with Two Turns**:
Now, we create a new loop with 2 turns from the same wire. When we double the number of turns, the radius of the new loop \( R_2 \) will be half of the original radius \( R_1 \) (since the same wire is used, and the circumference must remain the same):
\[ R_2 = \frac{R_1}{2} \]
Thus, the number of turns \( n_2 = 2 \).

4. **Magnetic Field for the Second Case**:
For the second case, the magnetic field at the center of the new loop with 2 turns is:
\[ B' = \frac{\mu_0 I n_2}{2R_2} \]
Substituting \( n_2 = 2 \) and \( R_2 = \frac{R_1}{2} \):
\[ B' = \frac{\mu_0 I \cdot 2}{2 \cdot \frac{R_1}{2}} = \frac{\mu_0 I \cdot 2}{\frac{R_1}{1}} = \frac{2\mu_0 I}{R_1} \]

5. **Relating the Two Magnetic Fields**:
Now, we can relate \( B' \) to \( B \):
From Equation (1), we have:
\[ B = \frac{\mu_0 I}{2R_1} \]
Therefore:
\[ B' = 4B \]

6. **Conclusion**:
The magnetic field at the center of the loop with two turns is \( 4B \).

### Final Answer:
The magnetic field at the center of the loop with two turns is \( 4B \).