Question

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

Answer 1

Hint:Recall the expression for the magnetic field at the centre of the solenoid. The magnetic field is proportional to the number of turns per unit length of the solenoid. Check the dependence of the magnetic field on the diameter of the solenoid.

Complete answer:
We have given that the solenoid has length L and diameter D. As the current I start to flow through the solenoid, the induced magnetic field generates at the central axis of the solenoid. We have the expression for the magnetic field at the centre of the infinitely long solenoid due to current flowing through the solenoid,
\[B = {\mu _0}nI\]
Here, B is the magnetic field, n is the number of turns per unit length, \[{\mu _0}\] is the permeability of the free space and I is the current.

Since, n is the number of turns per unit length of the solenoid, we can express the above equation as,
\[B = \dfrac{{{\mu _0}NI}}{L}\]
From the above equation we can infer that the magnetic field at the centre of the solenoid is independent of the diameter or the radius of the solenoid. The magnetic field inside the solenoid remains the constant at each point. Also, from the above equation, we can see that the magnetic field is inversely proportional to the length of the solenoid and not directly proportional to the length of the solenoid.

So, the correct answer is option C.

Note: In the formula, \[B = {\mu _0}nI\], n is the number of turns per unit length and not the number of turns. Therefore, the magnetic field at the centre of the solenoid also depends on the length of the solenoid. Note that the solenoid must be very long so that the formula \[B = {\mu _0}nI\] remains valid.