A long wire is first bent into a circular coil of one turn and then into a circular coil of smaller radius, having n identical turns. If the same current passes in both the cases, find the ratio of the magnetric fields produced at the centre in the two cases.

Explanation

Let a wire of length L is first bent in a single circular loop of radius `R_1 = (L)/(2pi)` and a current I passes through it. Then the magnetic field at the centre of loop :
`B_1 = (mu_0 I)/(2 R_1) = (mu_0 I)/(2L) cdot 2pi " " ……..(i)`
When the same wire is bent into a circular coil of n turns then radius of coil `R_2 = L/(2 pi r)` . Now on passing same current I, the field at the centre point will be
`B_2 = (mu_0 nI)/(2 R_2) = (mu_0 n I)/(2L) cdot 2 pi r = (mu_0 n^2 I)/(2L) cdot 2 pi " ".............(ii)`
`implies (B_1)/(B_2) = 1/(n^2) " or " B_2 = n^2B_1`.