# A toroidal ring, having 100 turns per cm of a thin wire is wound on a nonmagnetic metal rod of length 1 m and diameter

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A toroidal ring, having 100 turns per cm of a thin wire is wound on a nonmagnetic metal rod of length 1 m and diameter 1 cm. If the permeability of bar is equal to that of free space (µ0), calculate the magnetic field inside the bar (B) when the current (i) circulating through the turns is 1 A. Also determine the self-inductance (L) of the coil.

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Data : l = 1 m, d = 1 cm, n = 100 cm-1 = 104 m-1,
I = 100 A, µ0 = 4π × 10-7 H/m
The radius of cross section, r = $$\frac{d}{2}$$ = 0.5 cm
= 5 × 10-3 m
(a) Magnetic field inside the toroid,
B = µ0nI = (4π × 10-7)(104)(100)
= 0.4 × 3.142 = 1.257 T

(b) Self inductance of the toroid,
L = µ02πRn2A = µ0n2lA = µ0n2l(πr2)
= (4π × 10-7)(104)2(1) [π(5 × 10-3)2]
= π2 × 10-3 = 9.87 × 10-3 H = 9.87 mH