(a) An iron ring

a) According to ampere’s law, the line integral of magnetic field B around any closed circuit is equal to μ_o times the total current passing through this closed circuit. Mathematically,

∫B.dl = μ_oμ_rI_enclosed

Now ∫B.dl = B∫dl = BL , if B is constant and directed along the tangent to every point on the perimeter L of a closed curve.

For the field inside the ring, we can write

∫B.dl = B∫dl = B. 2πr

(r = radius of the ring)

Also, I_enclosed = (2πrn)I using equation (i)

Where n = number of turns per meter

∴ B. 2πr = μ_oμ_r (n. 2πr)I

∴ B = μ_oμ_rnI

b) The susceptibility of this material is positive and lies between 0 and 1. So it is a paramagnetic material. When a paramagnetic substance is placed in a uniform magnetic field the unpaired electron aligns itself with the external field. 
The modified magnetic field when a paramagnetic material is placed in a uniform magnetic field is shown below.