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A thin insulated wire forms a plane spiral of N = 100 tight turns carrying a current I = 8 mA. The radii of inside and outside turns (Fig. 3.67) are equal to a = 50 mm and b = 100 mm. Find: 

(a) the magnetic induction at the centre of the spiral; 

(b) the magnetic moment of the spiral with a given current.

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(a) From Biot-Savart’s law, the magnetic induction due to a circular current carrying wire loop at its centre is given by,

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,

where  is the contribution of one turn of radius r and dN is the number of turns in the interval (r, r + dr)

i.e. 

Substituting in equation (1) and integrating the result over r between a and b, we obtain,

(b) The magnetic moment of a turn of radius 

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