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Question
A toroidal ring made from a bar, of length 1 m and diameter 1 cm, bent into a circle. It is wound tightly with 100 turns per centimeter. If the permeability of the bar is equal to that of free space, calculate
(a) the magnetic field inside the bar when the current through the turns is 100 A
(b) the self-inductance of the coil.
Solution
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 = 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 = `mu_02pi"Rn"^2"A" = mu_0"n"^2 l"A" = mu_0"n"^2l(pi"r"^2)`
`= (4pi xx 10^-7)(10^4)^2(1)[pi(5 xx 10^-3)^2]`
`= pi^2 xx 10^-3 = 9.87 xx 10^-3`H = 9.87 mH
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