A wire of length l is used to form a coil. The magnetic field at its centre for a given current in it is minimum if the coil has

Options

A. `4 turn`
B. `2 turn`
C. `1 turn`
D. data is not sufficient

Explanation

To solve the problem of finding the number of turns in a coil that minimizes the magnetic field at its center for a given current, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a wire of length \( L \) that is formed into a coil, and we need to determine how many turns (n) will minimize the magnetic field at the center of the coil for a given current \( I \).

2. **Relate Length of Wire to Turns**: When the wire is formed into a coil with \( n \) turns, the total length of the wire is equal to the circumference of the coil multiplied by the number of turns. The circumference of a circular loop is given by \( 2\pi r \), where \( r \) is the radius of the coil. Therefore, we can write:
\[ L = n \cdot 2\pi r \]
From this, we can express the radius \( r \) in terms of \( L \) and \( n \):
\[ r = \frac{L}{2\pi n} \]

3. **Magnetic Field at the Center of the Coil**: The magnetic field \( B \) at the center of a circular coil with \( n \) turns carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 n I}{2r} \]
Substituting the expression for \( r \) from the previous step, we get:
\[ B = \frac{\mu_0 n I}{2 \left( \frac{L}{2\pi n} \right)} = \frac{\mu_0 n I \cdot 2\pi n}{2L} = \frac{\mu_0 \pi n^2 I}{L} \]

4. **Analyze the Magnetic Field**: From the equation \( B = \frac{\mu_0 \pi n^2 I}{L} \), we can see that the magnetic field \( B \) is directly proportional to \( n^2 \). This means that as the number of turns \( n \) increases, the magnetic field at the center also increases.

5. **Minimize the Magnetic Field**: To minimize the magnetic field \( B \), we need to minimize \( n \). The minimum value of \( n \) in the given options is \( 1 \) turn.

6. **Conclusion**: Therefore, the magnetic field at the center of the coil is minimum when the coil has **1 turn**.

### Final Answer:
The coil should have **1 turn** to minimize the magnetic field at its center for a given current.

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