Question

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

• A. proportional to $D$
• B. inversely proportional to $D$
• C. independent of $D$
• D. proportional to $L$

Hint:

Magnetic field inside a long solenoid depends only on the number of turns per unit length and the current.

Answer 1

The Correct Option is C

To determine the magnetic field at the center of a solenoid, we must consider the formula for the magnetic field inside a long solenoid:

\(B = \mu_0 \cdot n_{\text{total}}\cdot i\)

where:

• \(B\) is the magnetic field.
• \(\mu_0\) is the permeability of free space.
• \(n_{\text{total}}\) is the total number of turns per unit length of the solenoid.
• \(i\) is the current flowing through the solenoid.

For a solenoid with \(N\) turns and length \(L\), the number of turns per unit length is given by:

\(n_{\text{total}} = \frac{N \cdot n}{L}\)

Substituting this into the magnetic field formula:

\(B = \mu_0 \cdot \frac{N \cdot n}{L} \cdot i\)

From this expression, observe that:

• The magnetic field \(B\) is dependent on \(N\), \(n\), \(L\), and \(i\).
• Importantly, \(B\) is independent of the diameter \(D\) of the solenoid.

This leads to the conclusion that the magnetic field at the center of a long solenoid is:

Independent of \(D\).

Among the given options, the correct answer is independent of \(D\), confirming that the diameter of the solenoid does not influence the magnetic field at its center.