A long wire carries a steady current. It is bent into a circle of one turn and the magnetic field at the centre of the coil is B. It is then bent into a circular loop of N turns. What will be the magnetic field at the centre of the coil?

Let, length of wire = l, current = icase (i)The circumference of the circle is the length of the wire which is equal to i. e,l=2πR⇒R=12πThe magnetic field at the centre of this circle is,B=μ0i2R∴B=μ0iπl …(1)The same wire is bent into N circular coils. i.e.,N(2πr)=l⇒r=l2πNIn second case the magnetic field of the loop is,B1=Nμ022r⇒B1=Nμ0i×2πN2l=N2μ0iπl.....(2)From equation (1) and (2)B1=N2B