A wire of length 'I' is bent into a circular loop of radius R and carries a current I. The magnetic field at the centre of the loop is 'B '. The same wire is now bent into a double loop. If both loops carry the same current I and it is in the same direction, the magnetic field at the centre of the double loop will be

Options

A. zero
B. 2B
C. 4B
D. 8B

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Magnetic Field due to a Single Loop
The magnetic field \( B \) at the center of a circular loop of radius \( R \) carrying a current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2R} \]
where \( \mu_0 \) is the permeability of free space.

### Step 2: Relate the Length of the Wire to the Radius of the Loop
The length of the wire \( L \) is equal to the circumference of the loop:
\[ L = 2\pi R \]
From this, we can express the radius \( R \) in terms of the length of the wire:
\[ R = \frac{L}{2\pi} \]

### Step 3: Substitute \( R \) into the Magnetic Field Formula
Substituting \( R \) into the magnetic field formula gives:
\[ B = \frac{\mu_0 I}{2 \left(\frac{L}{2\pi}\right)} = \frac{\mu_0 I \cdot 2\pi}{2L} = \frac{\mu_0 \pi I}{L} \]

### Step 4: Analyze the Double Loop Configuration
When the wire is bent into a double loop, the total length of the wire remains the same, \( L \), but now it forms two loops. Therefore, the length of wire for each loop is:
\[ L' = \frac{L}{2} \]
The radius for each loop in the double loop configuration can be calculated as:
\[ 2\pi R' = \frac{L}{2} \implies R' = \frac{L}{4\pi} \]

### Step 5: Calculate the Magnetic Field for the Double Loop
The magnetic field at the center of the double loop is given by:
\[ B' = 2 \cdot \frac{\mu_0 I}{2R'} = \frac{\mu_0 I}{R'} \]
Substituting \( R' \):
\[ B' = \frac{\mu_0 I}{\frac{L}{4\pi}} = \frac{4\pi \mu_0 I}{L} \]

### Step 6: Relate \( B' \) to \( B \)
Now we can relate \( B' \) to \( B \):
\[ B' = \frac{4\pi \mu_0 I}{L} = 4 \left(\frac{\mu_0 \pi I}{L}\right) = 4B \]

### Conclusion
The magnetic field at the center of the double loop is:
\[ B' = 4B \]