Question

a square frame of side l carries a current i. The magnetic field at its centre is B. The same current is passed through a circular coil having the same perimeter as the square. The field at the center of the circular coil is B'. The ratio of B/B' is

Answer 1

Let's consider the square loop ABCD with side length = l.

We will use the formula
μ₀

where and are the angles subtended from the given point to the ends of the straight current carrying conductor.

Diagonals bisect each other at 90° for a square.
By symmetry, in the given scenario, and are each 45°.
For each side, the magnetic field at the centre = μ₀
= μ₀ = (say)

Now, each side of the loop carrying current i contributes to the magnetic field, in the same direction. There are four sides contributing the same.
So the resultant magnetic field


⇒ 4μ₀

Perimeter of the given loop = 4l
If this same loop is made into a circle, perimeter will be same.
Let the radius of the circle be r.
Hence, 2πr = 4l.
⇒r = 

For a circular loop carrying current i and of radius r,
The magnetic field at the centre (B) = μ₀

Therefore, B' = μ₀ = μ₀

Ratio asked for = B/B'.



And that's the answer.

[ANSWERED]

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