A long wire carries a steady current. When it is bent in a circular form, the magnetic field at its centre is B. Now if this wire is bent in a circular loop of n turns, what is the magnetic field at its centre ?

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(1) n²B. Given: one wire of fixed length L, first bent into a single turn of radius R, then into n turns of radius R'. Idea: the wire is the same piece both times, so winding it n times must shrink the radius. One turn: L=2π R and B=(μ₀ I)/(2R). n turns: L=n(2π R'), so R'=R/n. Field of the n-turn coil: B'=(μ₀ nI)/(2R')=(μ₀ nI)/(2R/n)=n²·(μ₀ I)/(2R). So B'=n²B.

A long wire carries a steady current. When it is bent in a circular form, the magnetic field at its centre is B. Now if this wire is bent in a circular loop of n turns, what is the magnetic field at its centre ?

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