Question

A wire of length l is used to form a coil. The magnetic field at its centre for a given current in it is a minimum if the coil has
A. 4 turns
B. 2 turns
C. 1 turn
D. Data is not sufficient

Answer 1

Recall the expression for the magnetic field at the centre of the coil which has N number of turns. The magnetic field of the coil at the centre is proportional to the number of turns. Express the radius of the circular coil in terms of length of the wire and derive the expression for the magnetic field in terms of the length of the wire.

Complete step by step answer:
We know that moving current induces a magnetic field around the conductor. If we take a coil and current is supplied to the coil, the magnetic field induces at the centre of the coil and the magnitude of this magnetic field is given as,
$B = \dfrac{{{\mu _0}iN}}{{2R}}$ …… (1)
Here, ${\mu _0}$ is the permeability of the free space, I is the current, N is the number of turns of the coil and R is the radius of the coil.

Since the length l of the wire is used to make the n turns of the wire, we can write the length of the wire as circumference of the coil as,
$l = 2\pi RN$
$\Rightarrow R = \dfrac{l}{{2\pi N}}$
Substituting the above equation in equation (1), we get,
$\therefore B = \dfrac{{{\mu _0}\pi i{N^2}}}{l}$
Therefore, the magnetic field at the centre of the coil will be directly proportional to the square of the number of turns of the coil. We can see, the magnetic field at the centre of the coil will be the minimum if the term ${N^2} = 1$. Thus, the number of turns N must be equal to 1 so that the magnetic field will be the minimum.

So, the correct answer is option C.

Note: For every structure like solenoid, toroid and circular arc, the magnetic field at a certain point from the structure is always proportional to the number of turns of the coil. Therefore, students can answer this question using the fact that the magnetic field will be the minimum if the number of turns is equal to 1. If the expression for the magnetic field involves the term ‘number of the turns per unit length’, then the structure has solenoidal shape.