A wire of length l carries a steady current. It is bent first to form a circular plane loop of one turn. The magnetic field at the centre of the loop is B. The same length is now bent more sharply to give a double loop of smaller radius. The magnetic field at the centre caused by the same is

Options

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

Explanation

To solve the problem step by step, we will analyze the magnetic field produced by the wire when it is bent into a circular loop and then into a double loop.

### Step 1: Magnetic Field of a Single Loop
When the wire of length \( L \) is bent into a single circular loop, the circumference of the loop is equal to the length of the wire:
\[ L = 2\pi R \]
where \( R \) is the radius of the loop.

From this, we can express the radius \( R \):
\[ R = \frac{L}{2\pi} \]

The magnetic field \( B \) at the center of a circular loop carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2R} \]
Substituting \( R \) from the previous equation:
\[ B = \frac{\mu_0 I}{2 \left(\frac{L}{2\pi}\right)} = \frac{\mu_0 I \cdot \pi}{L} \]

### Step 2: Magnetic Field of a Double Loop
Now, the same wire is bent to form a double loop (two turns) of smaller radius. The total length of the wire remains the same, \( L \), but now it forms two loops. The circumference of each loop will be:
\[ L = 2\pi r \cdot 2 = 4\pi r \]
where \( r \) is the radius of each smaller loop.

From this, we can express the radius \( r \):
\[ r = \frac{L}{4\pi} \]

The magnetic field \( B' \) at the center of the double loop (two turns) is given by:
\[ B' = 2 \cdot \frac{\mu_0 I}{2r} = \frac{\mu_0 I}{r} \]
Substituting \( r \):
\[ B' = \frac{\mu_0 I}{\left(\frac{L}{4\pi}\right)} = \frac{4\pi \mu_0 I}{L} \]

### Step 3: Relating \( B' \) to \( B \)
Now we can relate \( B' \) to \( B \):
\[ B = \frac{\mu_0 I \cdot \pi}{L} \]
\[ B' = \frac{4\pi \mu_0 I}{L} \]

To find the ratio \( \frac{B'}{B} \):
\[ \frac{B'}{B} = \frac{\frac{4\pi \mu_0 I}{L}}{\frac{\mu_0 I \cdot \pi}{L}} = \frac{4\pi \mu_0 I}{\mu_0 I \cdot \pi} = 4 \]

Thus, we find:
\[ B' = 4B \]

### Final Answer
The magnetic field at the center caused by the double loop is:
\[ B' = 4B \]