An insulated wire is wound so that it forms a flat coil with N = 200 turns. The radius of the innermost turn is r₁ = 3 cm, and of the outermost turn r₂ = 6 cm. If 20 mA current flows in it then the magnetic moment will be 10⁻² A.m ^2 . The value of is _____.

Step-by-step solution

Let the number of turns per unit radial width be n = N r₂ - r₁ . Consider an elemental ring of radius r and width dr . The number of turns in this element is dN = n , dr . The magnetic moment of this elemental ring is dM = I , dN , A = I ( N r₂ - r₁ , dr ) ( r^2) . Integrating from r₁ to r₂ to find the total magnetic moment M : M = _ r₁ ^ r₂ I N r₂ - r₁ r^2 , dr = I N 3(r₂ - r₁) (r₂^3 - r₁^3) Substituting the given values: I = 20 10⁻³ A, N = 200 , r₁ = 0.03 m, and r₂ = 0.06 m: M = 20 10⁻³ 200 3(0.06 - 0.03) ( (0.06