**Explanation:**

When a wire carrying a current forms a circular loop, it produces a magnetic field at the center of the loop. This magnetic field is known as magnetic induction or magnetic flux density. The value of the magnetic induction at the center of a single-turn circular loop is denoted by B.

When the same wire is looped into a coil of two turns and carries the same current, the magnetic induction at the center of the coil changes. Let's calculate the new value of magnetic induction at the center of the coil.

**Magnetic Field Due to a Single-turn Circular Loop:**

The magnetic field at the center of a single-turn circular loop can be calculated using Ampere's law. Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the product of the current enclosed by the loop and the permeability of free space (μ0).

In the case of a single-turn circular loop, the current enclosed by the loop is equal to the current through the loop. Therefore, we can write Ampere's law as:

∮B·dl = μ0 * I

where B is the magnetic field, dl is an infinitesimal length element along the loop, μ0 is the permeability of free space, and I is the current through the loop.

The magnetic field B is constant along the loop and is tangent to the loop. Therefore, the dot product B·dl is equal to B * dl, and the line integral can be simplified as:

B * ∮dl = μ0 * I

The line integral ∮dl is equal to the circumference of the loop, which is 2πr for a circular loop of radius r. Therefore, we can write:

B * 2πr = μ0 * I

Simplifying the equation, we get:

B = (μ0 * I) / (2πr)

**Magnetic Field Due to a Coil of Two Turns:**

Now, let's consider the case of a coil of two turns. The current through each turn of the coil is the same. Therefore, the total current through the coil is equal to the current through a single turn multiplied by the number of turns (I_total = I * 2).

Using Ampere's law for the coil of two turns, we can write:

B' * ∮dl = μ0 * I_total

where B' is the magnetic field at the center of the coil and ∮dl is the line integral along the loop formed by the coil.

The line integral ∮dl is equal to the circumference of the coil, which is 2πr for a circular coil of radius r. Therefore, we can write:

B' * 2πr = μ0 * I_total

Substituting the value of I_total, we get:

B' * 2πr = μ0 * I * 2

Simplifying the equation, we get:

B' = (μ0 * I) / (2πr)

Comparing the equations for B and B', we can see that the magnetic inductions at the centers of the single-turn circular loop and the coil of two turns are equal:

B = B'

Therefore, the new value of magnetic induction at the center of the coil of two turns is the same as the original value of magnetic induction at the center of the single-turn circular loop, which is B. Hence, the correct

I dont know the whole process of doing this type of questions..
just i know the short trick....
we have to SQUARE THE NUMBER OF TURNS ....
here number of turns=2... so 2^2=4...
new magnetic induction at the centre is 4B