Question

A long copper wire is wound in the form of a coil of radius r. A current of 2A is passed through this coil and the magnetic induction at the centre of this coil is noted. The same wire is now folded end to end and a coil of the same radius r is prepared and the same current is passed through it. The magnetic induction at the centre
A. Will be doubled
B. Will be halved
C. Will remain same
D. Will drop to zero

Answer 1

The copper wire has been wound to form a solenoid of N turns with the total length of the solenoid being supposed l initially. Then the wire's length is reduced to half, but the solenoid length remains the same as before.
Formula used:
The magnetic field strength at the centre of a solenoid of radius r is given as:
$B = \mu_0 n I$ ;
where I is the current passing through the coil and n is the number of turns per unit length.

Complete answer:
Consider the initial scenario: a wire is turned into a coil of radius r and a current of 2 A is passed through it.
The magnetic field produced at the center of the coil will be given as
$B = \mu_0 n I$.
Notice here that we have assumed the number of turns per unit length to be N and the total length of the solenoid to be L which gives us:
$n = \dfrac{N}{L}$.
Now, as the wire is followed end to end the total length of the wire has been reduced. The length of the solenoid formed however, is not reduced, it stays the same. Now as we try to wind this short wire onto the solenoid, we may find that two adjacent loops are more spread out than before or we can say that the total number of turns have been reduced to half due to the shortage of wire. This leads to a change in number of turns per unit length to be reduced to half, we can write:
$B' = \dfrac{\mu_0 n I}{2}$ .
Thus, the magnetic field gets reduced to half of its initial value,
$B' = \dfrac{B}{2}$.

So, the correct answer is “Option B”.

Note:
The term magnetic induction is also used in place of intensity of magnetic field, which is nothing but B. One can easily conclude the wrong answer in this question as the interpretation of the question is not clearly visible. From the question statement, one might assume that there is just a single loop created from the wire, but this interpretation is wrong as in the second scenario, the coil has been folded onto itself to give the loop of the same radius. Another way to get a wrong conclusion is to think that the wire, after being folded, will still have the same number of turns per unit length. Well, it is true that physically n is the same but think about the current and the direction in which it has to pass upon being folded. Therefore, upon folding the wire, it can be assumed to be consisting of just a single wire half of the original one.