(a) From Biot-Savart's law, the magnetic induction due to a circular current carrying wire loop at its centre is given by, B_{r} = \\dfrac {\\mu_{0}}{2r} i The plane spiral is made up of concentric circular loops, having different radii, varying from a to b . Therefore, the total magnetic induction at the centre, B_{0} = \\int \\dfrac {\\mu_{0}}{2r} dN .... (1) Where \\dfrac {\\mu_{0}}{2} i is the contribution of one turn of radius r and dN is the number of turns in the interval (r, r + dr) i.e. dN = \\dfrac {N}{b - a}dr Substituting in equation (1) and integrating the result over r between a and b , we obtain, B_{0} = \\int_{a}^{b} \\dfrac {\\mu_{0}i}{2r} \\dfrac {N}{(b - a)} dr = \\dfrac {\\mu_{0} iN}{2(b - a)} ln \\dfrac {b}{a} (b) The magnetic moment of a turn of radius r is p_{m} = i\\pi r^{2} and of all turns, p = \\int p_{m} dN = \\int_{a}^{b} i\\pi r^{2} \\dfrac {N}{b - a} dr = \\dfrac {\\pi iN (b^{3} - a^{2})}{3(b - a)} .