A long solenoid of length `L` has a mean diameter `D`. It has `n` layers of windings of `N` turns each. If it carries a current `‘i’` the magnetic field at its centre will be

Options

A. Proportional to D
B. Inversely proportional to D
C. Independent of D
D. Proportional to L

Answer

To find the magnetic field at the center of a long solenoid, we can use the formula for the magnetic field inside a long solenoid:

### Step-by-Step Solution:

1. **Understand the Formula**: The magnetic field \( B \) inside a long solenoid is given by the formula:
\[ B = \mu_0 n i \]
where:
- \( B \) is the magnetic field,
- \( \mu_0 \) is the permeability of free space (a constant),
- \( n \) is the number of turns per unit length of the solenoid,
- \( i \) is the current flowing through the solenoid.

2. **Determine the Number of Turns per Unit Length**: The total number of turns \( N \) in the solenoid can be expressed as:
\[ N = n \times L \]
where \( n \) is the number of turns per layer and \( L \) is the length of the solenoid.

3. **Calculate the Number of Turns per Unit Length**: The number of turns per unit length \( n \) can be calculated as:
\[ n = \frac{N}{L} \]

4. **Substitute into the Magnetic Field Formula**: Now, substituting \( n \) into the magnetic field formula:
\[ B = \mu_0 \left(\frac{N}{L}\right) i \]

5. **Analyze the Dependence on Parameters**: From the formula \( B = \mu_0 \frac{N}{L} i \), we can see that:
- The magnetic field \( B \) is directly proportional to the current \( i \).
- The magnetic field \( B \) is also directly proportional to the number of turns \( N \).
- The magnetic field \( B \) is inversely proportional to the length \( L \).
- Importantly, the magnetic field \( B \) does not depend on the diameter \( D \) of the solenoid.

6. **Conclusion**: Therefore, the magnetic field at the center of the solenoid is independent of the diameter \( D \).

### Final Answer:
The magnetic field at the center of the solenoid is independent of the diameter \( D \).

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