A circular coil carrying a certain current produces a megnetic field `B_(o)` at its centre. The coil is now rewound so as to have 3 turns and the same current is passed through it. The new magnetic field at the centre is

Options

A. `B_(o)//2`
B. `9B_(o)`
C. `B_(o)//3`
D. `3B_(o)`

Explanation

To solve the problem, we need to understand how the magnetic field produced by a circular coil changes when the number of turns is altered while keeping the current constant.

### Step-by-Step Solution:

1. **Understand the Magnetic Field Formula**:
The magnetic field \( B \) at the center of a circular coil is given by the formula:
\[ B = \frac{\mu_0 n I}{2R} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( n \) is the number of turns,
- \( I \) is the current flowing through the coil,
- \( R \) is the radius of the coil.

2. **Initial Magnetic Field**:
For the initial coil with 1 turn, the magnetic field at the center is:
\[ B_0 = \frac{\mu_0 I}{2R} \]

3. **New Configuration of the Coil**:
The coil is rewound to have 3 turns. Therefore, the new number of turns \( n \) is 3, and the current \( I \) remains the same.

4. **Determine the New Radius**:
When the coil is rewound to have 3 turns, the radius of the coil will change. The length of the wire remains constant. The length of the wire for the original coil is:
\[ L = 2\pi R \]
For 3 turns, the new length of the wire is:
\[ L = 3 \times 2\pi R' \quad \text{(where \( R' \) is the new radius)} \]
Setting these equal gives:
\[ 2\pi R = 3 \times 2\pi R' \implies R' = \frac{R}{3} \]

5. **Calculate the New Magnetic Field**:
Now, substituting \( n = 3 \) and \( R' = \frac{R}{3} \) into the magnetic field formula:
\[ B = \frac{\mu_0 n I}{2R'} = \frac{\mu_0 \cdot 3I}{2 \cdot \frac{R}{3}} = \frac{\mu_0 \cdot 3I \cdot 3}{2R} = \frac{9\mu_0 I}{2R} \]

6. **Relate New Magnetic Field to Initial Magnetic Field**:
We can express the new magnetic field in terms of the initial magnetic field \( B_0 \):
\[ B = 9 \left(\frac{\mu_0 I}{2R}\right) = 9B_0 \]

### Final Answer:
The new magnetic field at the center of the coil is:
\[ B = 9B_0 \]

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