The electric current in a circular coil of four turns produces a magnetic induction `32T` at its centre.The coil is unwound and is rewound into a circular coil of single turn,the magnetic induction at the centre of the coil by the same current will be

Options

A. `8T`
B. `2T`
C. `4T`
D. `16T`

Answer

To solve the problem, we need to understand how the magnetic induction (magnetic field) at the center of a coil changes when the number of turns in the coil changes.

### Step-by-Step Solution:

1. **Understanding the Magnetic Induction Formula**:
The magnetic induction \( B \) at the center of a circular coil can be expressed using the formula:
\[ B = \frac{\mu_0 n I}{2R} \]
where:
- \( B \) is the magnetic induction,
- \( \mu_0 \) is the permeability of free space,
- \( n \) is the number of turns,
- \( I \) is the current,
- \( R \) is the radius of the coil.

2. **Given Values**:
- For the initial coil with 4 turns, the magnetic induction \( B_1 = 32 \, T \).
- Thus, we can write:
\[ B_1 = \frac{\mu_0 \cdot 4 \cdot I}{2R_1} \]

3. **Rewinding the Coil**:
- When the coil is unwound and rewound into a single turn, the number of turns \( n \) becomes 1.
- The radius of the new coil \( R_2 \) will be different, but we need to find the new magnetic induction \( B_2 \).

4. **Relating the Two Coils**:
- The total length of wire remains the same when the coil is unwound and rewound.
- The length of wire used in the first coil is \( L_1 = 2\pi R_1 \cdot 4 \) (for 4 turns).
- The length of wire used in the second coil is \( L_2 = 2\pi R_2 \cdot 1 \) (for 1 turn).
- Setting these equal gives:
\[ 2\pi R_1 \cdot 4 = 2\pi R_2 \cdot 1 \implies R_2 = 4R_1 \]

5. **Calculating the New Magnetic Induction**:
- Now substituting \( R_2 \) back into the formula for \( B_2 \):
\[ B_2 = \frac{\mu_0 \cdot 1 \cdot I}{2R_2} = \frac{\mu_0 \cdot I}{2(4R_1)} = \frac{\mu_0 \cdot I}{8R_1} \]

6. **Relating \( B_1 \) and \( B_2 \)**:
- We know from our first equation:
\[ B_1 = \frac{\mu_0 \cdot 4 \cdot I}{2R_1} \implies B_1 = \frac{2\mu_0 \cdot I}{R_1} \]
- Thus, we can express \( B_2 \) in terms of \( B_1 \):
\[ B_2 = \frac{B_1}{4} = \frac{32}{4} = 8 \, T \]

### Final Answer:
The magnetic induction at the center of the single-turn coil is \( 8 \, T \).