A toroid is a coil of wire wrapped around a doughnut-shaped ring (a torus). For a tightly
wrapped toroid with N turns, the magnetic field lines of ⃗
B form concentric circles inside the
toroid, and the field is zero outside. Use Amp`ere’s law to find an expression for the magnetic
field strength B at a radial position r from the axis of the toroid.
1 Answer
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Any loop inside the toroidal coil that loops once around the "doughnut hole" of the torus incloses N turns of the wire. If the current in the wire is I, then the total current inclosed by the loop is NI.
By Ampère's law, the line integral of the magnetic field ℬ around the loop is the permeability, μ₀ = 4π·10⁻⁷ N/A, of free space times the total current NI inclosed by the loop:
∮ℬ·dℓ = μ₀NI.
In case the loop is a circle of radius r concentric with the torus, the magnetic field is parallel with the circle, and has constant magnitude B along the circle. So
∮ℬ·dℓ = 2πrB.
So
B = μ₀NI/(2πr).